Research in Computational Topology 2 (Association for Women in Mathematics Series, 30) 3030955184, 9783030955182

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Association for Women in Mathematics Series

Ellen Gasparovic Vanessa Robins Katharine Turner   Editors

Research in Computational Topology 2

Association for Women in Mathematics Series Volume 30

Series Editor Kristin Lauter Facebook Seattle, WA, USA

Focusing on the groundbreaking work of women in mathematics past, present, and future, Springer’s Association for Women in Mathematics Series presents the latest research and proceedings of conferences worldwide organized by the Association for Women in Mathematics (AWM). All works are peer-reviewed to meet the highest standards of scientific literature, while presenting topics at the cutting edge of pure and applied mathematics, as well as in the areas of mathematical education and history. Since its inception in 1971, The Association for Women in Mathematics has been a non-profit organization designed to help encourage women and girls to study and pursue active careers in mathematics and the mathematical sciences and to promote equal opportunity and equal treatment of women and girls in the mathematical sciences. Currently, the organization represents more than 3000 members and 200 institutions constituting a broad spectrum of the mathematical community in the United States and around the world. Titles from this series are indexed by Scopus.

More information about this series at https://link.springer.com/bookseries/13764

Ellen Gasparovic • Vanessa Robins Katharine Turner Editors

Research in Computational Topology 2

Editors Ellen Gasparovic Department of Mathematics Union College Schenectady, NY, USA

Vanessa Robins Research School of Physics Australian National University Canberra, ACT, Australia

Katharine Turner Mathematical Sciences Institute Australian National University Canberra, ACT, Australia

ISSN 2364-5733 ISSN 2364-5741 (electronic) Association for Women in Mathematics Series ISBN 978-3-030-95518-2 ISBN 978-3-030-95519-9 (eBook) https://doi.org/10.1007/978-3-030-95519-9 © The Author(s) and the Association for Women in Mathematics 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The second Women in Computational Topology (WinCompTop) Research Collaboration Workshop was held from July 1 to 5, 2019, at the Mathematical Sciences Institute (MSI) at the Australian National University (ANU). In addition to generous support from MSI as part of their special year on computational mathematics, the workshop was partially supported by the National Science Foundation and the Association for Women in Mathematics (AWM). Additional support for participant travel and accommodation was provided by the Australian Mathematical Sciences Institute. The 30 participants of the workshop included women from pure and applied mathematics, statistics, computer science, and physics. They represented women at all career stages from graduate student to professor, and arrived at the workshop in Canberra from Europe, the United States, and across Australia. The main goal of the workshop was to facilitate the formation of new and lasting research collaborations between junior and senior women working in the field of computational topology, growing and strengthening this cohort of women in the area. To that end, based on their research interests and backgrounds, each participant was assigned to one of four working groups headed by leading researchers in the field of computational and applied topology. The week commenced with a day of events that were open to anyone interested in learning more about computational topology and featured introductory talks from the group leaders and a poster session showcasing work from early-career participants. The remaining days included plenty of intense research time, with regular updates on progress from each group, and an everpopular excursion to look for kangaroos on the nearby Mt. Ainslie nature reserve. This single-blind peer-reviewed volume is the proceedings of the second WinCompTop workshop and features accepted submissions from each of the working groups as well as additional papers solicited from the wider community. The topics covered are broad in scope including theoretical developments in persistent homology, topological aspects of graphs and networks, and case studies in topological data analysis.

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The Multi-disciplinary Field of Computational Topology Computational topology is a modern, vibrant, diverse, and ever-evolving field of research that integrates computer science and algorithms with many areas of mathematics, such as algebraic and differential topology, algebra, geometry, category theory, statistics, probability, and functional analysis. Research directions range from the traditionally pure, such as questions pertaining to the decomposition, stability, and structure of homology groups, to the more applied, such as computational efficiency, parallelization questions, and intersections with statistics and machine learning methods. Though algorithms in topology are not new, the rise of computational power and an influx of data have led to increased demand for rigorous, provable, and fast algorithms in the area. In recent years, practitioners in topological data analysis have created tools to explore the topology of data. One such tool is persistent homology or persistence, a technique that may be described via a pipeline model from geometry through topology and algebra to discrete mathematics. More precisely, data are first modeled as objects in a metric space. The next step is to filter the data to obtain a family of nested topological spaces that capture the topological information at multiple scales. Homology with field coefficients is then applied to the filtration to obtain a graded module, called a persistence module, consisting of a family of vector spaces and linear maps between them. The final step is to encode the persistence module into a discrete object such as the persistence diagram or barcode. Persistent homology provides a multiscale stable summary of the topological features of the data. Another data analysis tool from computational topology is the mapper algorithm, which is an adaptation of the Reeb graph applicable to point cloud data. The field of computational topology has already been useful in understanding data from many applications. This is further demonstrated by the diversity of applications considered in this volume including the shape of proteins, motion tracking of particles in video, and electoral district analysis. Another common theme in this volume is exploring the added complexity within computational topology when direction information is included, such as the impact of direction on homology, homotopy, and nerve theorems.

Working Group Project Descriptions Each of the working groups from the second WinCompTop workshop initiated successful collaborations that led to four of the chapters featured in this proceedings volume. The working group led by Vanessa Robins studied questions motivated by the analysis of geometric structure in image data using persistent homology. There were two aspects to the project. One examined the effect on persistence diagrams of changing image resolution, and the other sought to resolve the mathematical

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relationship between filtrations built from a grayscale image using two different methods. Results from the second project are presented in this volume. The group proves a duality result for the two different cubical complex constructions used in topological image analysis, and uses this to establish a straightforward high-level algorithm to obtain the persistence diagram for one construction from the output of software implemented using the dual construction. Led by Claudia Landi, the second group focused on the two-fold goal of interpreting and visualizing the rank invariant for multi-parameter persistence modules. In their chapter, the authors use discrete Morse theory to compute the rank invariant and establish its fundamental relationship with the critical cells of a discrete gradient vector field. They introduce a procedure for fibering the multiparameter persistence rank invariant along lines of positive slope that a user may select. This then allows for the slicing up of the persistence space of a multiparameter persistence module into persistence diagrams. The team led by Kathryn Hess explored the use of topological descriptors of directed graphs. For the chapter in this volume, they compute Betti numbers and simplex counts of the directed flag complexes generated from many random graph models, and compare them statistically to well-established psuedometrics for directed graphs. The final working group led by Carina Curto studied a class of recurrent neural networks called combinatorial threshold-linear networks. They capture the combinatorial structure of a network by decomposing it into overlapping directional components and taking the nerve of this decomposition. In their chapter, they explicitly prove how the nerve constrains both the fixed points and the asymptotic dynamics.

Overview of the Contributed Chapters In addition to the four chapters from the WinCompTop workshop teams, this volume features eight additional chapters from members of the wider community. We briefly summarize them here. The first two contain theoretical developments in computational topology. The second two are related to algorithms for topological questions involving graphs or complexes, and the final four showcase various applications of topological data analysis. Given a zigzag persistence module, one can define a restricted version over a subinterval of the parameter space. Gasparovic et al. prove that distances between the restricted persistence modules are bounded by the distances between the original persistence modules and apply this to metric graphs and multi-parameter persistence. The chapter by Kolbe and Robins solves a hyperbolic tiling problem relevant to the self-assembly of polymers. They extend methods of tile gluing used in combinatorial tiling theory to the case of tiles that have infinite stabilizer groups.

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Their results and algorithms enumerate tilings of the Euclidean and hyperbolic planes consisting of a single type of unbounded, possibly branched, ribbon tile. There is considerable interest in simplifying the problem of determining when two directed paths are directed homotopic. To this end, Belton et al. develop a combinatorial approach to this problem by defining a link preserving directed collapse (LPDC), a combinatorial removal of certain cubes analogous to (and inspired by) elementary collapses of a free pair in the simple homotopy theory of J.H. Whitehead. The authors show that if, under certain conditions, the past link of all vertices of a cubical complex K is contractible, then the directed path space of K is also contractible. They also provide a computationally cost-efficient condition for determining whether a certain pair is an LPDC. Adams, Gibson, and Pfaffinger consider a pursuit-evasion process on a graph involving lions and contamination. In this process, when a lion arrives at a vertex, it clears the contamination at the vertex; when it leaves, contamination can spread from an adjacent vertex and re-infect it. The main question of the chapter is to decide how many lions are required to completely clear a graph of contamination, given any possible starting arrangement. To this end, the authors adapt existing methods on rectangular and square grids to triangulated rectangular and square grids, with diagonals added all in the same direction. They establish an upper bound for rectangles and a lower bound for squares on the number of lions required to clear contamination. They also use the Cheeger constant of a graph to establish lower bounds for two types of lion motion on a general graph. Motion tracks are one-dimensional trajectories extracted from wide field of view video images. The challenge of how to differentiate the trajectory of a target from that of a spurious object is addressed in the chapter by Emerson et al. The chapter presents a new approach using time-series embedding, persistent homology, and a k-nearest neighbor classifier on normalized segments of tracks. These initial results show great promise for further development and testing against larger collections of data. The geometry of protein molecules is a fundamental attribute that influences their function and is quantified in numerous ways. Hamilton et al. use persistent homology of the protein’s alpha-shape filtration to summarize the structure of approximately 3000 protein chains. The distribution of persistent homology summaries is compared with an existing method for quantifying topological structure. The comparison is made using the statistical method of angle-based joint and individual variation explained. Husain et al. consider the issue of electoral districting in terms of identifying, maintaining, and finding these communities. Focusing on Mecklenburg County, North Carolina, they applied the mapper algorithm to US Census data to cluster by demographics and mapped these geographically to identify communities of interest. Finally, the chapter by Zhou et al. presents an algorithm for “stitching” together two univariate mappers into a bivariate mapper. The authors examine informationtheoretic measures that aim to quantify topological notions of information gains. Their implementations for visualizing such topological gains for bivariate mapper graphs are publicly available on GitHub.

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Final Remarks and Acknowledgments The second workshop for Women in Computational Topology was an enormous success thanks to generous funding and support from the Australian Mathematical Sciences Institute, the National Science Foundation (NSF-CCF 1841455), the AWM ADVANCE grant (NSF-HRD 1500481), and the ANU’s Mathematics funding for the special year in Computational Mathematics. The WinCompTop research network steering committee, the organizers of the workshop, and the authors and editors of this volume are deeply grateful for all of the above support and for fostering our community of researchers. In particular, we appreciate the assistance provided by the AWM in running their valuable and insightful post-event survey of participants. The local organizers are deeply indebted to Brittany Joyce (ANU) for her expert assistance with all aspects of conference logistics, and to graduate students Yossi Bokor and Kelly Maggs for being local-area guides to our visitors. We are truly grateful to Springer and the AWM for the opportunity to create this volume as part of the ever-growing Association for Women in Mathematics Series. It would not exist without the many hours of painstaking work from all the authors, and the careful and thoughtful reports from our much-appreciated reviewers. Schenectady, NY, USA Canberra, ACT, Australia Canberra, ACT, Australia

Ellen Gasparovic Vanessa Robins Katharine Turner

Contents

The Persistent Homology of Dual Digital Image Constructions . . . . . . . . . . . . Bea Bleile, Adélie Garin, Teresa Heiss, Kelly Maggs, and Vanessa Robins

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Morse-Based Fibering of the Persistence Rank Invariant . . . . . . . . . . . . . . . . . . . Asilata Bapat, Robyn Brooks, Celia Hacker, Claudia Landi, and Barbara I. Mahler

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Local Versus Global Distances for Zigzag and Multi-Parameter Persistence Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ellen Gasparovic, Maria Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, and Lori Ziegelmeier Tile-Transitive Tilings of the Euclidean and Hyperbolic Planes by Ribbons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benedikt Kolbe and Vanessa Robins Graph Pseudometrics from a Topological Point of View . . . . . . . . . . . . . . . . . . . . Ana Lucia Garcia-Pulido, Kathryn Hess, Jane Tan, Katharine Turner, Bei Wang, and Naya Yerolemou

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Nerve Theorems for Fixed Points of Neural Networks. . . . . . . . . . . . . . . . . . . . . . . 129 Daniela Egas Santander, Stefania Ebli, Alice Patania, Nicole Sanderson, Felicia Burtscher, Katherine Morrison, and Carina Curto Combinatorial Conditions for Directed Collapsing . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Robin Belton, Robyn Brooks, Stefania Ebli, Lisbeth Fajstrup, Brittany Terese Fasy, Nicole Sanderson, and Elizabeth Vidaurre Lions and Contamination, Triangular Grids, and Cheeger Constants. . . . . 191 Henry Adams, Leah Gibson, and Jack Pfaffinger A Topological Approach for Motion Track Discrimination . . . . . . . . . . . . . . . . . 211 Tegan H. Emerson, Sarah Tymochko, George Stantchev, Jason A. Edelberg, Michael Wilson, and Colin C. Olson

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Persistent Topology of Protein Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 W. Hamilton, J. E. Borgert, T. Hamelryck, and J. S. Marron Mappering Mecklenburg County: Exploring Census Data for Potential Communities of Interest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Alisha Husain, Kristine Jones, Anthony Kolshorn, David Retchless, Kelemua Tesfaye, Courtney M. Thatcher, and Jim Thatcher Stitch Fix for Mapper and Topological Gains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Youjia Zhou, Nathaniel Saul, Ilkin Safarli, Bala Krishnamoorthy, and Bei Wang

The Persistent Homology of Dual Digital Image Constructions Bea Bleile, Adélie Garin, Teresa Heiss, Kelly Maggs, and Vanessa Robins

Abstract To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes, the two commonly used constructions (corresponding to direct and indirect digital adjacencies) can give different results for the same image. The two constructions are almost dual to each other, and we use this relationship to extend and modify the cubical complexes to become dual filtered cell complexes. We derive a general relationship between the persistent homology of two dual filtered cell complexes, and also establish how various modifications to a filtered complex change the persistence diagram. Applying these results to images, we derive a method to transform the persistence diagram computed using one type of cubical complex into a persistence diagram for the other construction. This means software for computing persistent homology from images can now be easily adapted to produce results for either of the two cubical complex constructions without additional low-level code implementation.

B. Bleile School of Science and Technology, University of New England, Armidale, NSW, Australia e-mail: [email protected] A. Garin · K. Maggs Laboratory for Topology and Neuroscience, École polytechnique fédérale de Lausanne (EPFL), Lausanne, Switzerland e-mail: [email protected]; [email protected] T. Heiss Institute of Science and Technology (IST) Austria, Klosterneuburg, Austria e-mail: [email protected] V. Robins () Research School of Physics, Australian National University, Canberra, ACT, Australia e-mail: [email protected] © The Author(s) and the Association for Women in Mathematics 2022 E. Gasparovic et al. (eds.), Research in Computational Topology 2, Association for Women in Mathematics Series 30, https://doi.org/10.1007/978-3-030-95519-9_1

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1 Introduction Persistent homology [10, 27] allows us to compute topological features of a space via a nested sequence of subspaces or filtration by returning a persistence diagram which reflects the connected components, tunnels, and voids appearing and disappearing as we sweep through the filtration. It has a wide range of applications in diverse contexts including digital images, used for example to study porous materials [22], hurricanes [24] or in medical applications [8]. While it is common to apply persistent homology to simplicial complexes arising from point sets, digital images are made up of pixels (in dimension d = 2) or voxels (for d ≥ 2) rendering cubical complexes the natural choice as they reflect the regular grid of numbers used to encode the image [15, 18]. Note that in some applications digital images are implicitly or explicitly given the structure of a simplicial complex [12, 23]. There are two ways to construct a cubical complex from an image I: The Vconstruction V (I) represents voxels by vertices and the T-construction T (I) represents voxels by top-dimensional cubes. These constructions are closely related to two different voxel connectivities of classical digital topology [17]. The Vconstruction corresponds to what is known in image analysis as direct adjacency, where voxels are connected if and only if their grid locations differ by 1, so that each voxel has 2d neighbours. For d = 2, pixels are 4-connected and the direct neighbours are to the left and right as well as above and below. The T-construction corresponds to indirect adjacency, where voxels are also connected diagonally, every voxel has 3d − 1 neighbours and pixels are 8-connected. It is well known that the choice of direct or indirect adjacency has an impact on the overall topological structure of a binary image and the critical points of a grayscale image function. The effects are particularly significant when the image has structure at a similar length-scale to the digital grid. This situation arises in images of objects with thin filaments (such as neurons or fractures), or after the application of morphological thinning algorithms. For example, a diagonal line that is one voxel thin, rendered as in Fig. 1, has just one connected component with respect to indirect adjacency but many components when viewed with direct adjacency. If the contrast between the line and the background is high (as shown), the difference between persistent homology found with the T- and V-constructions respectively will also be significant. Figure 3 shows another example of a small two-dimensional image where there is a dramatic difference between the persistent homology found using the V- and T- constructions. A further issue in classical digital topology is that a single choice of adjacency cannot be applied to both the foreground and background of a binary image (or the sub-level and super-level sets of a grayscale image) in a topologically consistent way. If the foreground is given indirect adjacency, the background must take direct adjacency (and vice versa) or intuitive topological results concerning sets and their complements in Rd fail to hold in the digital image [17]. Figure 2 illustrates the fact

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Fig. 1 Thin diagonal lines are often rendered like the digital grayscale image on the left. To preserve the topology of the line, indirect adjacency (illustrated with the T-construction in blue) needs to be chosen, because direct adjacency (illustrated with the V-construction in orange) breaks the line into disconnected pieces B

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Fig. 2 (a) A digital binary image with foreground pixels denoted by blue F’s and background pixels denoted by red B’s. (b) The foreground pixels are connected using indirect adjacency, and the background pixels using direct adjacency. If the background pixels are connected using indirect adjacency, the central red ‘B’ attaches to four pixels and the simple closed curve in blue no longer separates ‘inside’ from ‘outside’ violating the Jordan curve theorem. (c) the T-construction is applied to the foreground and the V-construction applied to the background to obtain cubical cell complexes with the same connectivity as the digital adjacency graph in (b)

that the foreground and background need different adjacency types for the Jordan curve theorem to hold, for example. The results from digital topology mentioned above suggest the existence of duality-like relationships between the V- and T-constructions applied respectively to the sub-level and super-level sets of a grayscale image. In fact, the resulting cubical complexes are almost dual. Vertices in the V-construction correspond to topdimensional cells in the T-construction and interior vertices of the T-construction correspond to top-cells in the V-construction. This paper establishes the precise nature of the duality-like relationship between the cubical complexes of the T- and V-constructions, filtrations of these induced by an image and its negative, and their persistence diagrams. We use these results to define simple algorithms that return the persistence diagram for V (I) from software that computes diagrams based on the T-construction and vice versa. Our results are based on a combinatorial notion of dual cell complexes and dual filtrations. We explain how the V- and

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T-constructions can be modified to obtain dual filtrations of the d-sphere. The relationship between the persistence diagrams of a grayscale digital image obtained via the V- and T-constructions then follows from an investigation of the effects of the required modifications on the persistence diagrams and the relationship between the persistent homology of dual filtrations on dual cell complexes. Our duality results make it possible to use the advantages of different software packages even when the cubical complex type is not the preferred one for the application at hand. For example, by using a streaming approach, the persistence software cubicle [26] is able to handle particularly large images that do not even need to fit into memory. However, it has only been implemented for the T-construction. Thanks to Sect. 6.2, cubicle can now be used to compute the Vconstruction persistence of an image that does not fit into memory and therefore cannot be processed by existing V-construction persistence software.

1.1 Related Work At its most basic level, the algebraic relationship between the persistent homology of two dual filtered cell complexes is similar to that between persistent homology and persistent relative cohomology. The latter corresponds to taking the anti-transpose of the boundary matrix [5] or equivalently, leaving the boundary matrix as is and applying the row reduction algorithm instead of the column-reduction algorithm [5, 11]. Lemma 1 shows that the same relationship applies to the boundary matrices of dual filtered cell complexes. Therefore, Theorem 2 can be viewed as a translation of the known bijection between the persistence pairs of persistent homology and relative persistent cohomology into the setting of dual filtrations. This theorem is the first step towards establishing the mapping between persistence diagrams of the T- and the V-construction of images in Sect. 6. Furthermore, it applies more generally to the persistent homology of dual filtered cell complexes without using the connection to persistent relative cohomology. The symmetry theorem of extended persistence diagrams [3] is also closely related to our Theorem 2 and Corollary 2. Given a manifold, X, without boundary and a function f : X → R, the extended persistent homology sequence starts with a filtration by sub-level sets f −1 (−∞, s] and continues with the relative homology of the pair (X, f −1 [r, ∞)), where s is an increasing threshold and r is a decreasing one. In [3] the symmetry theorem follows from a duality result for the extended persistence diagrams of a simplicial complex K whose underlying space is X with f defined on the vertices of K. As observed in [12], the cubical complex constructions used in digital image analysis do not satisfy the duality theorem of extended persistence because of inconsistencies such as those discussed in Fig. 2 for example. In short, the Partition Lemma of [3] fails for cubical complexes. The authors of [12] overcome this by constructing a simplicial complex from the digital image that

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consistently reflects the connectivity of both sub- and super-level sets, and use this to obtain the expected duality in the extended persistence diagram. In contrast, we work with the existing widely-implemented cubical complex constructions of digital images and establish results relating homology sequences of dual filtered complexes rather than the homology and relative homology sequences of sub- and super-level set filtrations of a single complex. This permits a simple high-level algorithm to transform between two regular (not extended) persistence diagrams computed from a digital image. This paper is a follow-up of an extended abstract [13] published in the Young Researcher Forum of SoCG 2020. A version of that with appendices included is available on arXiv. These results have already been cited as the basis for an algorithm implemented by the developers of the software cubical ripser [16].

1.2 Overview Our results are aimed at both pure and applied mathematicians who want to understand and use the relationship between the persistent homology of dual filtered cell complexes and particularly the two standard constructions of cubical complexes from digital images. In Sect. 2 we define dual cell complexes and dual filtered complexes as used throughout this paper, and provide a brief outline of the definitions and results of persistent homology. The reader who is not familiar with the theory of persistent homology should refer to [9, 27] for a more comprehensive introduction. Section 3 establishes the relationship between persistence diagrams of two dual filtered cell complexes. In Sect. 4, we describe and formalise the two standard cubical complexes used in topological computations on digital images. We explain how these two complexes (the T- and V-constructions described earlier) must be extended and modified to form dual filtered cell complexes with underlying space homeomorphic to the dsphere. The effects these modifications have on persistence diagrams are derived in Sect. 5. For the investigation of one of these effects we use the long exact sequence of a filtered pair of cell complexes arising from the category theoretic view of persistence modules. The last Sect. 6 states the results for persistence diagrams of digital images and explains how to compute the persistence diagram of the T-construction by simple manipulation of a persistence diagram computed using the V-construction, and vice versa. This gives a practical method for adapting the output from existing software packages that use one or the other construction to obtain the persistence diagram for the dual construction.

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2 Mathematical Background 2.1 Dual Cell Complexes and Filtrations CW-complexes [19] generalize simplicial complexes to allow cells that are not necessarily simplices but homeomorphic to open discs or balls, for example cubes instead of tetrahedra. A CW-complex is regular if the closure of each k-cell is homeomorphic to the closed k-dimensional ball D k . For the remainder of this paper a cell complex is a finite regular CW-complex. The dimension dim(X) of the cell complex X is the maximum dimension of cells in X and, writing X for the topological space as well as for the set of its cells, we obtain dim(X) = max{dim(σ ) | σ ∈ X}. Let X be a cell complex with cells τ and σ . If τ ⊆ σ then τ is a face of σ , and σ is a coface of τ , written as τ  σ . The codimension of a pair of cells τ  σ is the difference in dimension, dim(σ ) − dim(τ ). If σ has a face τ of codimension 1, we call τ a facet of σ , and write τ  σ . A function f : X → R on the cells of X is monotonic if it increases with the dimension, that is, f (σ ) ≤ f (τ ) whenever σ  τ. Definition 1 Two d-dimensional cell complexes X and X∗ are combinatorially dual if there is a bijection X → X∗ , σ → σ ∗ between the sets of cells such that 1. (Dimension Reversal) dim(σ ∗ ) = d − dim σ for all σ ∈ X. 2. (Face Reversal) σ  τ ⇐⇒ τ ∗  σ ∗ for all σ, τ ∈ X. Definition 2 A filtered (cell) complex (X, f ) is a cell complex X together with a monotonic function f : X → R. A linear ordering σ0 , σ1 , . . . , σn of the cells in X, such that σi  σj implies i ≤ j , is compatible with the function f when f (σ0 ) ≤ f (σ1 ) ≤ . . . ≤ f (σn ). Note that the monotonicity condition implies that, for r ∈ R, the sub-level set Xr := f −1 (−∞, r] is a subcomplex of X. The value f (σ ) determines when a cell enters the filtration given by this nested sequence of subcomplexes. The definition of a compatible ordering also implies that at each step in the sequence ∅ ⊂ { σ0 } ⊂ { σ0 , σ1 } ⊂ · · · ⊂ { σ0 , σ1 , . . . , σn } the union of cells is a subcomplex, and every sub-level set f −1 (−∞, r] appears somewhere in this sequence: f −1 (−∞, r] = f −1 (−∞, f (σi )] = { σ0 , σ1 , . . . , σi } for i = max{ i = 0, . . . , n | f (σi ) ≤ r }.

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Definition 3 Two filtered complexes (X, f ) and (X∗ , g) are dual filtered complexes if X and X∗ are combinatorially dual to one another and if there exists a linear ordering σ0 , σ1 , . . . , σn of the cells in X that is compatible with f and its ∗ , . . . , σ ∗ is compatible with g. dual ordering σn∗ , σn−1 0 Proposition 1 Suppose two functions f : X → R and f ∗ : X∗ → R satisfy f ∗ (σ ∗ ) = −f (σ ). Then (X, f ) and (X∗ , f ∗ ) are dual filtered complexes.

2.2 Persistent Homology When working with data, standard topological quantities can be highly sensitive to noise and small geometric fluctuations. Persistent homology addresses this problem by examining a collection of spaces, indexed by a real variable often representing an increasing length scale. These spaces are modelled by a cell complex X with a filter function f : X → R assigning to each cell the scale at which it appears.

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Given a filtered complex (X, f ), we obtain inclusions f −1 (−∞, r] → f −1 (−∞, s] of sub-level sets for r ≤ s. Applying degree-k homology with coefficients in Z/2Z to these inclusions yields linear maps between vector spaces Hk (f −1 (−∞, r]) → Hk (f −1 (−∞, s]). The resulting functor Hk (f ) : (R, ≤) → VecZ/2Z from the poset category (R, ≤) to the category of vector spaces over the field Z/2Z is called a persistence module, for details see [2]. As discussed in [2], Gabriel’s Theorem from representation theory implies that the persistence module Hk (f ) decomposes into a sum of persistence modules consisting of Z/2Z for r ∈ [b, d) connected by identity maps, and 0 elsewhere, called interval modules I[b,d) : Hk (f ) ∼ =



I[bl ,dl ) .

l∈L

Each interval summand I[bl ,dl ) represents a degree-k homological feature that is born at r = bl and dies at r = dl . If the final space X has non-trivial homology there are features that never die. These have dl = ∞ and the interval is called essential. The degree-k persistence diagram of f is the multiset Dgmk (f ) = { [bl , dl ) | l ∈ L}.

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We write [bl , dl )k ∈ Dgmk (f ) to denote the homological degree of an interval and define the persistence diagram of f as the disjoint union over all degrees: Dgm(f ) =

dim(X) 

Dgmk (f ).

k=0

Writing DgmF (f ) for the multiset of finite intervals with dl < ∞, and Dgm∞ (f ) for the remaining essential ones, we obtain Dgm(f ) = DgmF (f )  Dgm∞ (f ). 2.2.2

Computation

To compute the persistence diagram Dgm(f ) we choose an ordering σ0 , σ1 , . . . , σn of the cells in X that is compatible withf . Cells σi and σj appear at the same step in the nested sequence of sub-level sets f −1 (∞, r] r∈R if f (σi ) = f (σj ). For the following computations however, we must add exactly one cell at every step: ∅ ⊂ { σ0 } ⊂ { σ0 , σ1 } ⊂ · · · ⊂ { σ0 , σ1 , . . . , σn−1 } ⊂ { σ0 , σ1 , . . . , σn } = X. When adding the cells one step at a time, a cell of dimension k causes either the birth of a k-dimensional feature or the death of a (k − 1)-homology class [6], that is, each cell is either a birth or a death cell. A pair (σi , σj ) of cells where σj kills the homological feature created by σi is called a persistence pair. A persistence pair (σi , σj ) corresponds to the interval [f (σi ), f (σj )) ∈ DgmF (f ). Note that this interval can be empty, namely if f (σi ) = f (σj ). Empty intervals are usually neglected in the persistence diagram. A birth cell σi with no corresponding death cell is called essential, and corresponds to the interval [f (σi ), ∞) ∈ Dgm∞ (f ). Recall that presentations for the standard homology groups are found by studying the image and kernel of integer-entry matrices that represent the boundary maps taking oriented chains of dimension k to those of dimension (k − 1) [21]. In persistent homology, we work with the Z/2Z total boundary matrix D, which is defined by Di,j = 1 if σi  σj and 0 otherwise. Define j

j −1

rD (i, j ) = rank Di − rank Di

j

j −1

− rank Di+1 + rank Di+1

j

where Di = D[i : n, 0 : j ] is the lower-left sub-matrix of D attained by deleting the first rows up to i − 1 and the last columns starting from j + 1. Theorem 1 (Pairing Uniqueness Lemma [4]) Given a linear ordering of the cells in a filtered cell complex X, (σi , σj ) is a persistence pair if and only if rD (i, j ) = 1. The ranks are usually computed by applying the column reduction algorithm [9] j j to obtain the reduced matrix R and using the property that rank Di = rank Ri under the operations of the algorithm. The persistence pairs can then be read off easily

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since rR (i, j ) = 1 if and only if the ith entry of the j th column of the reduced matrix is the lowest 1 of this column. However, in this paper, we can work directly with rD . Corollary 1 If rD (i, j ) = 1 and rD (j, i) = 1 for all j then the cell σi is essential. Proof The fact that every cell is either a birth or a death cell implies that σi must be an unpaired birth or death cell. However, as every filtration begins as the empty set, there are no unpaired death cells.  

3

The Persistent Homology of Dual Filtered Complexes

Recall again that in standard homology and cohomology, the coboundary map is the adjoint of the boundary map. Hence, given a consistent choice of bases for the chain and cochain groups, their matrix representations are related simply by taking the transpose. In [5], another algebraic relationship is established between persistent homology and persistent relative cohomology, based on the observation that the filtration for relative cohomology reverses the ordering of cells in the total (co)boundary matrix. The same reversal of ordering holds for the dual filtered cell complexes defined here, so we obtain a similar relationship between the persistence diagrams. Our proof of the correspondence between persistence pairs in dual filtrations uses the matrix rank function and pairing uniqueness lemma in a similar way to the combinatorial Helmoltz-Hodge decomposition of [11]. Nonetheless, Theorem 2 interprets the underlying linear algebra in the setting of dual filtered complexes and only uses the concept of persistent homology without using the connection to persistent relative cohomology, which makes it more accessible. For this section suppose (X, f ) and (X∗ , g) are dual filtered cell complexes with n + 1 cells. Suppose that a linear ordering σ0 , σ1 , . . . , σn of the cells in X ∗ , . . . , σ ∗ is the dual linear is compatible with the filtration (X, f ), and that σn∗ , σn−1 0 ordering compatible with g. Let D be the total boundary matrix of X and D ∗ be the total boundary matrix of X∗ with their respective orderings. Remark 1 A useful indexing observation is that σi∗ is the (n − i)-th cell of the dual filtration. We denote by D ⊥ the anti-transpose of the matrix D, that is the reflection across ⊥ =D the minor diagonal: Di,j n−j,n−i . Anti-transposition is also the composition of standard matrix transposition with a reversal of the order of the columns and of the rows. Lemma 1 The matrix D ∗ is the anti-transpose D ⊥ of D, that is, ∗ ⊥ = Dn−j,n−i = Di,j . Di,j

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Proof The equivalences below follow from the definition of D, of dual cell complexes, and the remark above. ∗ ∗ ∗  σn−j ⇔ Di,j = 1. Dn−j,n−i = 1 ⇔ σn−j  σn−i ⇔ σn−i

Lemma 2 The sub-matrices defined in Sect. 2.2.2 satisfy (Di )⊥ = (D ⊥ )n−i n−j j

and thus rank Di = rank (D ⊥ )n−i n−j j

and rD (i, j ) = rD ⊥ (n − j, n − i). Proof The first statement follows from (Di )⊥ = (D[i : n, 0 : j ])⊥ = D ⊥ [(n − j ) : n, 0 : (n − i)] = (D ⊥ )n−i n−j . j

The second statement follows because anti-transposition is attained by composing the rank preserving operations of transposition and row and column permutations. The third statement follows from the second through: j

j −1

rD (i, j ) = rank Di − rank Di

j

j −1

− rank Di+1 + rank Di+1

⊥ n−i ⊥ n−i−1 = rank(D ⊥ )n−i + rank(D ⊥ )n−i−1 n−j − rank(D )n−j +1 − rank(D )n−j n−j +1

= rD ⊥ (n − j, n − i). Theorem 2 (Persistence of Dual Filtrations) Let (X, f ) and (X∗ , g) be dual filtered complexes with compatible ordering σ0 , σ1 , . . . , σn . Then 1. (σi , σj ) is a persistence pair in the filtered complex (X, f ) if and only if (σj∗ , σi∗ ) is a persistence pair in (X∗ , g). 2. σi is essential in (X, f ) if and only if σi∗ is essential in (X∗ , g). Proof Lemma 2 implies that rD (i, j ) = rD ∗ (n − j, n − i). Therefore, rD (i, j ) = 1 ⇔ rD ∗ (n − j, n − i) = 1. By the Pairing Uniqueness Lemma 1, the above implies that (σi , σj ) is a persistence pair whenever the (n − j )-th cell of the dual filtration (X∗ , g) is paired with the (n−i)-th, thus proving Part (1). For Part (2), Lemma 2 also tells us that the following two statements are equivalent:

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• Both rD (i, j ) = 1 and rD (j, i) = 1 for all j . • Both rD ∗ (n − j, n − i) = 1 and rD ∗ (n − i, n − j ) = 1 for all n − j . By Corollary 1, this means that σi is an essential cell in (X, f ) if and only if the (n − i)-th cell σi∗ is essential in the dual filtration (X∗ , g).   Corollary 2 Let (X, f ) and (X∗ , g) be dual filtered complexes. Then 1. [f (σi ), f (σj )) ∈ DgmkF (f ) ⇔ [g(σj∗ ), g(σi∗ )) ∈ Dgmd−k−1 (g). F 2. [f (σi ), ∞) ∈ Dgmk∞ (f ) ⇔ [g(σi∗ ), ∞) ∈ Dgmd−k ∞ (g). Proof Note that for a persistence pair (σi , σj ), found for an ordering compatible with the function f , the birth value is f (σi ) and the death value is f (σj ). The result then follows directly from Theorem 2.   Remark 2 It is worth noting that there is a dimension shift between essential and non-essential pairs coming from the fact that the birth cell defines the dimension of a homological feature. For finite persistence pairs, the birth cell changes from σi (of dimension k) to σj∗ (of dimension d − (k + 1)) in the dual, while for an essential cycle, the birth cell in the dual is σi∗ . This dimension shift also appears in our results on images later on.

4 Filtered Cell Complexes from Digital Images As described in the introduction, the motivating application for the duality results of this paper is grayscale digital image analysis. This section begins with the definition of grayscale digital images and describes the two standard ways to model such images by cubical complexes as well as the modifications required to make these dual filtered complexes. Definition 4 A d-dimensional grayscale digital image of size (n1 , n2 , . . . , nd ) is an R-valued array I ∈ Mn1 ×n2 ×...×nd (R). Equivalently, it is a real-valued function on a d-dimensional rectangular grid I : I = 1, n1  × 1, n2  × . . . × 1, nd  → R where 1, ni  is the set { k ∈ N | 1 ≤ k ≤ ni }. The index set, I , of I is also called the image domain. Recall that elements p ∈ I are called pixels when d=2, voxels otherwise and the value I(p) ∈ R is the grayscale value of p.

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One would like to use persistent homology to analyse such images via their sublevel sets. However, the canonical topology on I ⊆ Zd ⊂ Rd makes it a totally disconnected discrete space. To induce a meaningful topology on the image that better represents the perceived connectivity of the voxels, grayscale digital images are modelled by regular cubical complexes [18]. Definition 5 An elementary k-cube σ ⊂ Rd is the product of d elementary intervals, σ = e1 × e2 × . . . × ed such that k of the intervals have the form ei = [li , li + 1] and d − k are degenerate, ei = [li , li ]. A cubical complex X ⊂ Rd is a cell complex consisting of a set of elementary k-cubes, such that all faces of σ ∈ X are also in X, and such that all vertices of X are related by integer offsets.

4.1 Top-cell and Vertex Constructions There are two common ways to build a filtered cubical complex from an image I : I −→ R. One method is to represent the voxels as vertices of the cubical complex as in [22]. We call this cubical complex the vertex construction, or Vconstruction for short. The second method takes voxels as top-dimensional cells: we call it the top-cell construction, or T-construction. It is shown in [18], that the vertex construction corresponds to the graph-theoretical direct adjacency used in traditional digital image processing and the top-cell construction to the indirect adjacency model. These adjacency models are also respectively referred to as the open and closed digital topologies. An example of how each construction is built from an image is given in Fig. 3. The explicit definitions of such constructions are given below. Definition 6 Given a d-dimensional grayscale digital image I : I → R, of size (n1 , n2 , . . . , nd ), the V-construction is a filtered cell complex (V (I ), V (I)) defined as follows. 1. V (I ) is a cubical complex built from an array of (n1 − 1) × . . . × (nd − 1) elementary d-cubes and all their faces. 2. The vertices υ (0) ∈ V (I ) are indexed exactly by the elements p ∈ I , and we define the function V (I) firstly on these vertices as, V (I)(υ (0) ) = I(p). Then for an elementary k-cube σ , the function takes the maximal value of its vertices

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13

Fig. 3 Top: The V- and T-constructions generated by an image I : I → R with the values of I indicated on the vertices and the top-dimensional cells, respectively. Middle: the filtration V (I) : V (I ) → R and the corresponding persistence pairs. Bottom: the filtration T (I) : T (I ) → R and the corresponding persistence pairs

V (I)(σ ) = max V (I)(υ (0) ). υ (0) σ

This ensures that V (I) is monotonic with respect to the face relation on V (I ). Definition 7 Given a d-dimensional grayscale digital image I : I → R, of size (n1 , n2 , . . . , nd ), the T-construction is a filtered cell complex (T (I ), T (I)) defined as follows. 1. T (I ) is a cubical complex built from the array of n1 ×. . .×nd elementary d-cubes and all their faces. 2. The d-cells τ (d) ∈ T (I ) are indexed exactly by the elements p ∈ I , and we define the function T (I) firstly on these top-dimensional cells as, T (I)(τ (d) ) = I(p).

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Then for an elementary k-cube σ , the function takes the smallest value of all adjacent d-cubes, T (I)(σ ) = min T (I)(τ (d) ). σ τ (d)

This ensures that T (I) is monotonic with respect to the face relation on T (I ). The next section describes how to modify the original image and take quotients to obtain dual complexes and filtrations.

4.2 Modifications for Duality The cubical complexes defined using the top-cell and vertex constructions are not strictly dual to each other in the standard context of a rectangular digital image domain due to the presence of a boundary. If the rectangular image domain happens to be the unit cell for a periodic structure, then we can identify opposite faces and form the d-torus. On this d-torus, V (I ) and T (I ) are dual cubical complexes with vertices and d-cubes indexed by I . Taking V (I) as the function on V (I ) and T (−I) as the function on T (I ), we obtain dual filtered complexes and can immediately apply Corollary 2 to deduce the persistence pairs of one filtration from the other. Otherwise, the more commonly encountered situation is that the image is a simple convex domain in Rd . A standard approach to handling the boundary is to form a quotient identifying the boundary to a point so that the convex domain becomes a subset of the d-sphere. To obtain dual cell complexes on the d-sphere, we increase the size of the image domain before taking the quotient modulo the boundary. The image function is assigned a large arbitrary value on these extra voxels and dual filtered complexes are obtained by considering I in one construction and −I in the other as detailed in the following definitions and results. Let I : I → R be a grayscale digital image with index set I = 1, n1  × . . . × 1, nd , and set N > max I(p). p∈I

Definition 8 The padded image IP : I P → R has image domain I P = 0, n1 + 1 × . . . × 0, nd + 1 and image function I (p) = P

 I(p), for p ∈ I N,

for p ∈ I P \ I.

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Padded Image

Fig. 4 The transformation of the V- and T-constructions into dual cell complexes (right) using the padded image (left) and a mapping from V (I P ) to T (I ) (center)

As shown in Fig. 4, the padded image is simply obtained by adding a shell of N -valued voxels to I. We apply the V- and T-constructions to the padded image to obtain the functions V (IP ) : V (I P ) → R and T (IP ) : T (I P ) → R, respectively. Let T (I ) ∂T (I ) κ (d) denote the cell complex obtained from T (I ) by attaching a d-cell κ (d) along the boundary ∂T (I ) (see [14, p.5] for details). Furthermore, let V (I P )/∂V (I P ) be the quotient cell complex obtained by identifying all points of the boundary (see [14, p.8] for details). Note that both these modifications create cells that are not elementary cubes. Lemma 3 Given a rectangular digital image domain, I , the quotient of the padded V-construction modulo its boundary, V (I P )/∂V (I P ), is the combinatorial dual of T (I ) ∂T (I ) κ (d) . Proof Each elementary k-cube, σ ∈ V (I P ) takes the form σ = e1 × . . . × ed ,

ei = [li , li + 1] or ei = [pi , pi ]

where k of the elementary intervals are non-degenerate with li ∈ {0, . . . , ni } and (d − k) are degenerate with pi ∈ {0, . . . , ni + 1}. Note that σ ∈ ∂V (I P ) if at least one degenerate interval has pi = 0 or (ni + 1). Now consider the following cell constructed from σ : σ ∗ = e1∗ × . . . × ed∗ ,

ei∗ = [li + 12 , li + 12 ] or [pi − 12 , pi + 12 ]

with li and pi as defined above. This cell has k degenerate intervals and (d −k) nondegenerate ones so σ ∗ is an elementary (d − k)-cube. If we insist that σ ∈ ∂V (I P ), then we see that pi ∈ {1, . . . , ni }, and the degenerate coordinate values (li + 12 ) ∈ { 12 , 32 . . . , (ni + 12 )}. Thus we obtain a bijection between k-cells in V (I P ) \ ∂V (I P ) and (d − k)-cells in T (I ). Mapping the 0-cell [∂V (I P )] ∈ V (I P )/∂V (I P ) to the d-cell attached to ∂T (I ) yields a dimension reversing bijection between all cells of V (I P )/∂V (I P ) and those of T (I ) ∂T (I ) κ (d) .

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The next step is to confirm that the face relations between cells in V (I P )/∂V (I P ) are mapped to coface relations in T (I ) ∂T (I ) κ (d) . By the construction above, all interior face relations for V (I P ) map to coface relations for T (I ). Given that only cells in the boundary belong to [∂V (I P )], this correspondence is inherited by the quotient. Hence, the last detail we need to check is that the vertex [∂V (I P )] in V (I P )/∂V (I P ) has dual face relations to the d-cell κ (d) attached to the boundary of T (I ) in T (I ) ∂T (I ) κ (d) . This is equivalent to the statement that [∂V (I P )]  σ in V (I P )/∂V (I P ) ⇔ σ ∗  κ (d) in T ∂T κ (d) . Now, σ ∗ is a face of κ (d) if and only if σ ∗ ∈ ∂T (I ), which means at least one of the degenerate elementary intervals of σ ∗ has li + 12 = 12 or (ni + 12 ). This makes li = 0 or ni , so the corresponding elementary interval in the dual cell σ is ei = [li , li + 1] = [0, 1] or [ni , ni + 1]. This forces σ ∩ ∂V (I P ) = ∅, so that [∂V (I P )]  σ . The converse implication follows in the same manner, and we are done.   Lemma 4 Given a rectangular digital image domain, I , the quotient of the padded T-construction modulo its boundary, T (I P )/∂T (I P ), is the combinatorial dual of V (I P ) ∂V (I P ) κ (d) . Proof This follows from the same arguments as the previous lemma with the roles of T and V reversed. Note that we pad the V-construction before attaching the cell κ (d) to account for the fact that T (I ) naturally has more cells than V (I ).   We have described how the two cubical complex models can be augmented to form dual cell complexes of the d-sphere. We now show how to obtain dual filtered cell complexes by comparing the image function on one construction with its negative on the other. The details are made precise in the lemmata below. First note that the function V (−IP ) is constant on ∂V (I P ) so it induces a func(−IP ) : V (I P )/∂V (I P ) → R with V (−IP )([∂V (I P )]) tion on the quotient space, V P = −N and agreeing with V (−I ) on all other cells. Similarly, the function T (I) extends to a function T(I) on T (I ) ∂T (I ) κ (d) with T(I)(κ (d) ) = N . Lemma 5 For each σ ∈ T (I ) ∂T (I ) κ (d) and dual cell σ ∗ ∈ V (I P )/∂V (I P ) we have (−IP )(σ ∗ ). −T(I)(σ ) = V Proof Firstly, suppose dim σ = d and σ = κ (d) , the d-cell attached to the boundary. Suppose p ∈ I is the corresponding element of the image domain, so that I(p) = T(I)(σ ). The dual cell σ ∗ ∈ V (I P )/∂V (I P ) corresponds to the same voxel but is given the negative value (−IP )(σ ∗ ) = −I(p) = −T(I)(σ ). V

The Persistent Homology of Dual Digital Image Constructions

17

For the remaining d-cell κ (d) , with dual [∂V (I P )]∗ , the function values satisfy (−IP )([∂V (I P )]). −T(I)(κ (d) ) = −N = V Lastly, suppose σ ∈ T (I ) ∂T (I ) κ (d) and dim σ < d. By construction, it follows that −T(I)(σ ) = − min T(I)(τ (d) ) = max −T(I)(τ (d) ) τ (d) σ

τ (d) σ

(−IP )(σ ∗ ) (−IP )(υ (0) ) = V = max V υ (0) σ ∗

 

as required.

(IP ) on Now define the functions T(−IP ) on T (I P )/∂T (I P ) and V P (d) V (I ) ∂V (I P ) κ similarly to those above. Lemma 6 For each σ ∈ V (I P ) ∂V (I P ) κ (d) and dual cell σ ∗ ∈ T (I P )/∂T (I P ) we have (IP )(σ ) = T(−IP )(σ ∗ ). −V Proof Similar to Lemma 5 with the roles of V and T interchanged.

 

Corollary 3 For a grayscale digital image I : I → R (−IP )) 1. The filtered complexes (T (I ) ∂T (I ) κ (d) , T(I)) and (V (I P )/∂V (I P ), V are dual. (IP )) and (T (I P )/∂T (I P ), T(−IP )) 2. The filtered complexes (V (I P )∂V (I P ) κ (d) , V are dual. Proof This follows directly from applying Lemma 5 for part (1) and Lemma 6 for part (2), then Proposition 1.  

5 Persistence Diagrams of the Modified Filtrations In the previous section, we showed that the T- and V-constructions built from an image can be modified via padding, cell attachment and quotient operations to become dual cell complexes of the d-sphere. Our next step is to examine the effect of such operations on the persistence module and diagram. As in Sect. 4, suppose I : I → R is a grayscale digital image and N > max I. The specific operations we study are 1. Padding an image I : I → R with a outer shell of N-valued pixels, then forming the V- and T-constructions. 2. Attaching a d-cell to the boundary of V (I P ) or to T (I ) with value N .

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3. Taking the quotient modulo the boundary in the negative padded filtration, i.e. (−IP ) and from T (−IP ) to T(−IP ). changing from V (−IP ) to V Of these, the first two have relatively transparent effects on the persistent homology of the filtered spaces. Padding the image as in (1) does not change the persistence diagrams; attaching a d-cell as in (2) creates an essential d-cycle with birth at N. We summarise these formally as follows. Proposition 2 For a grayscale digital image I : I → R 1. Dgm(V (IP )) = Dgm(V (I)) and Dgm(T (IP )) = Dgm(T (I)) 2. (I)) = Dgm(V (I)) ∪ { [N, ∞)d } Dgm(V and Dgm(T(I)) = Dgm(T (I)) ∪ { [N, ∞)d } The remaining operation to investigate is the third, namely the effect of taking the quotient modulo the boundary. For this we need some machinery we will now introduce.

5.1 Long Exact Sequence of a Filtered Pair To examine the effect of taking quotients on persistence diagrams, we use the description of persistence modules as functors together with a long exact sequence (LES) for these. Given a pair of cell complexes (X, A) with A ⊆ X, the LES is i

p

δ

. . . → Hk (A) − → Hk (X) − → Hk (X, A) − → Hk−1 (A) → . . . where Hk (X, A) denotes the relative homology of the pair. Similarly, suppose we have a filtered cell complex (X, f ) and a sub-complex A ⊆ X. Then the restriction f |A : A → R induces a filtered sub-complex (A, f |A ) and, at each index r ∈ R, we obtain a pair (Xr , Ar ), where Xr = f −1 (−∞, r] and Ar = f |−1 A (−∞, r]. Theorem 4.5 in [25] states that the R-indexed collection of short exact sequences (SES) of cellular chain complexes 0 → C∗ (Ar ) → C∗ (Xr ) → C∗ (Xr , Ar ) → 0 yields a LES of persistence modules: Theorem 3 (Long Exact Sequence for Relative Persistence Modules) Given a monotonic function f : X → R and a sub-complex A ⊆ X, there is a long exact

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sequence of persistence modules . . . → Hk (f |A ) → Hk (f ) → Hk (f, f |A ) → Hk−1 (f |A ) → Hk−1 (f ) → . . . where Hk (f, f |A ) denotes the persistence module given by r → Hk (Xr , Ar ). Here the persistence modules are functors from the poset category (R, ≤) to VecZ/2Z and the morphisms are natural transformations obtained from the standard connecting homomorphisms. The linear transformations are induced by inclusions and projections at each filtration index. The kernels and cokernels of the morphisms are themselves persistence modules defined by taking the kernel or cokernel at each filtration index, with maps between corresponding vector spaces at different filtration indices induced by inclusions. This result is implicit in the recent work in [1, 20], where it follows as corollary of the fact that persistence modules form an abelian category whereby the snake lemma holds.

5.2 Persistence of the Image with Boundary Identified The remaining operation to investigate is that of taking the quotient of a padded image modulo the boundary filtered with the negative of the image function. We state the result in terms of a space X homeomorphic to the d-dimensional closed disc D d , filtered by a function f : X → R taking a constant minimal value, min f = −N, on the boundary of X so that Lemma 7 applies to both T- and V-constructions. Using the long exact sequence of a pair, we show that the (d − 1)cycle with interval [−N, max f )d−1 representing the boundary is removed while a d-cycle with interval [max f, ∞)d is added. Lemma 7 Take a monotonic function f : X ∼ = D d → R with σ ∈ ∂X ⇒ f (σ ) = −N = min f and induced quotient map f : X/∂X → R. Then   Dgm(f) = Dgm(f ) \ { [−N, max f )d−1 } ∪ { [max f, ∞)d }. Proof For pairs of cell complexes the relative homology groups are naturally isomorphic to the reduced homology groups H˜ k (f˜) of the quotient [14, p.124]. Naturality implies that the result extends to persistence modules and that the reduced persistence modules differ only by the essential interval I[−N,∞) in degree 0. To compute the reduced persistence modules H˜ k (f˜) of the quotient, we therefore consider the LES of the filtered pair (f, f |∂X ) αk

αk−1

. . . → Hk (f |∂X ) −→ Hk (f ) → Hk (f, f |∂X ) → Hk−1 (f |∂X ) −−→ Hk−1 (f ) → . . .

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where αk is the map induced by the inclusion ∂X ⊆ X. Taking the cokernel of αk and the kernel of αk−1 the LES yields the SES 0 → Coker(αk ) → Hk (f, f |∂X ) → Ker(αk−1 ) → 0. First assume d > 1 and note that, in this case,  I[−N,∞) ∼ Hk (f |∂X ) = 0

for k = d − 1, 0 otherwise

∼ Ker(αk ) = 0 for k = d − 1, 0. For αd−1 , the image of the Thus Im(αk ) = essential (d − 1)-cycle of the boundary dies once all cells in (X, f ) have been filtered at function value max f . Hence Im(αd−1 ) ∼ = I[−N,max f )

and

Ker(αd−1 ) ∼ = I[max f,∞)

As −N = min f we conclude α0 (I[−N,∞) ) = I[−N,∞) , so that Im(α0 ) ∼ = I[−N,∞)

and

Ker(α0 ) = 0

Since X is homeomorphic to a d-dimensional disc, Hd (f ) = 0. Hence Coker(αd ) = 0 and, for k = d, the SES implies H˜ d (f˜) ∼ = Hd (f, f |∂X ) ∼ = Ker(αd−1 ) ∼ = I[max f,∞) . For 0 ≤ k < d the persistence module on the right of the SES is trivial. Thus

H˜ k (f˜) ∼ = Hk (f, f |∂X ) ∼ = Coker(αk ) ∼ =

⎧ ⎪ ⎪ ⎨Hd−1 (f )/I[−N,max f ) ⎪ ⎪ ⎩

for k = d − 1

Hk (f )

for 0 < k < d − 1

H0 (f )/I[−N,∞)

for k = 0

and the result follows for d > 1. For d = 1 the only non-trivial persistence module of the boundary is H0 (f |∂X ) ∼ = I[−N,∞) ⊕ I[−N,∞) and we obtain Im(α0 ) ∼ = I[−N,∞) ⊕ I[−N,max f )

and

Ker(α0 ) ∼ = I[max f,∞)

For k = 1 we proceed as above and, for k = 0, the SES yields   H˜ 0 (f˜) ∼ = H0 (f, f |∂X ) ∼ = Coker(α0 ) ∼ = H0 (f )/ I[−N,∞) ⊕ I[−N,max f ) As above we conclude H0 (f˜) ∼ = H0 (f )/I[−N,max f ) .

 

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Corollary 4 For a d-dimensional image I : I → R (−IP )) = Dgm(V (−IP )) \ { [−N, − min I)d−1 } ∪ { [− min I, ∞)d } Dgm(V and Dgm(T(−IP )) = Dgm(T (−IP )) \ { [−N, − min I)d−1 } ∪ { [− min I, ∞)d } Proof This follows from Lemma 7 applied to f = V (−IP ) and f = T (−IP ) respectively, using max V (−IP ) = − min I and max T (−IP ) = − min I.  

6 Duality Results for Images In this section, we explicitly describe the relationship between the diagrams of both the T- and V-constructions. Software to compute persistent homology of an image I : I → R typically builds one of the two constructions implicitly so the results in this section provide a solution to the problem of how to use software based on the V-construction to compute a persistence diagram with respect to the T-construction, and vice versa. For the algorithms that we define in this section we assume the following subroutines given a grayscale digital image I. 1. PAD(I, N): returns the image padded with an outer shell of N -valued voxels. 2. NEG(I): multiplies each voxel value by −1. 3. max(I), min(I): returns the maximum and minimum voxel values of I respectively. 4. VCON(I), TCON(I): returns the persistence diagrams Dgm(V (I)) and Dgm(T (I)) of the V- and T-constructions of the image respectively.

6.1 From the V-Construction to the T-Construction Suppose we have software that computes the persistent homology of a ddimensional grayscale digital image I using the V-construction. The following theorem states that the persistence diagram of the T-construction for I can be calculated directly from the pairs in that of the V-construction of the negative padded image. Theorem 4 (T from V) For a grayscale digital image I : I → R the diagrams of the V- and T-constructions satisfy DgmF (T (I)) = { [−q, −p)d−k−1 | [p, q)k ∈ DgmF (V (−IP )) } \ { [min I, N)0 }

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and Dgm∞ (T (I)) = { [min I, ∞)0 }. Proof That Dgm∞ (T (I)) = { [min I, ∞)0 } follows from the fact that T (I ) ∼ = D (d) and the first cell in the filtration occurs at value min T (I) = min I. For the finite case: DgmF (T (I)) = DgmF (T(I)) (−IP )) } = { [−q, −p)d−k−1 | [p, q)k ∈ DgmF (V = { [−q, −p)d−k−1 | [p, q)k ∈ DgmF (V (−IP )) \ { [−N, − min I)d−1 } } = { [−q, −p)d−k−1 | [p, q)k ∈ DgmF (V (−IP )) } \ { [min I, N )0 }

where the equalities follow from Proposition 2, Corollary 2 and Corollary 4 respectively.   The structure of the algorithm follows immediately from the theorem and is summarised below: Algorithm 1 Computing the T-construction persistence diagram with Vconstruction software Require: An image I and the subroutine VCON. 1: Dgm(T (I)) ← { [min(I), ∞)0 } 2: N ← max(I) + C  choose C to ensure N  max(I) 3: −IP ← NEG(PAD (I, N)) 4: Dgm(V (−IP )) ← VCON(−IP )  Apply V-construction software. 5: for [p, q)k in Dgm(V (−IP )) with p = −N do 6: Dgm(T (I)) ← Dgm(T (I)) ∪ { [−q, −p)d−k−1 } 7: return Dgm(T (I))  Output T-construction persistence diagram.

6.2 From the T-Construction to the V-Construction In the other direction, suppose we have software that computes the persistent homology of a d-dimensional grayscale digital image I : I → R using the T-construction. The following theorem states that a persistence diagram for the Vconstruction of I can be calculated directly from the pairs computed for the negative padded image using the T-construction. Theorem 5 (V from T) For a grayscale digital image I : I → R the diagrams of the V- and T-constructions satisfy

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DgmF (V (I)) = { [−q, −p)d−k−1 | [p, q)k ∈ DgmF (T (−IP )) } \ { [min I, N)0 } and Dgm∞ (V (I)) = { [min I, ∞)0 }. ∼ Proof That Dgm∞ (V (I)) = { [min I, ∞)0 } follows from the fact that V (I ) = D (d) and the first cell in the filtration occurs at time min I. For the finite case, we have that DgmF (V (I)) = DgmF (V (IP )) (IP )) = DgmF (V = { [−q, −p)d−k−1 | [p, q)k ∈ DgmF (T(−IP )) } = { [−q, −p)d−k−1 | [p, q)k ∈ DgmF (T (−IP )) \ { [−N, − min I)d−1 } } = { [−q, −p)d−k−1 | [p, q)k ∈ DgmF (T (−IP )) } \ { [min I, N)0 }

where the equalities follow from Proposition 2, Corollary 2 and Corollary 4 respectively.   The structure of the algorithm follows immediately from the theorem and is summarised below: Algorithm 2 Computing the V-construction persistence diagram with Tconstruction software Require: An image I and the subroutine TCON. 1: Dgm(V (I)) ← { [min(I), ∞)0 } 2: N ← max(I) + C  choose C to ensure N  max(I) 3: −IP ← NEG(PAD (I, N)) 4: Dgm(T (−IP )) ← TCON (−IP )  Apply T-construction software. 5: for [p, q)k in Dgm(T (−IP )) with p = −N do 6: Dgm(V (I)) ← Dgm(V (I)) ∪ { [−q, −p)d−k−1 } 7: return Dgm(V (I))  Output V-construction persistence diagram.

Example 1 Suppose we are working with the two dimensional digital grayscale image given in Fig. 3 and have only the software to compute the T-construction. We depict the filtration of T (−IP ) in Fig. 5, and the corresponding intervals in the persistence module. Similarly, we show the filtered V-construction V (I) in Fig. 6. The reader may confirm that the correspondence between the intervals is accurately described by Theorem 5.

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Fig. 5 The filtration T (−IP ) and intervals of the persistence diagram Dgm(T (−IP )) for the image I : I → R of Fig. 3

Fig. 6 The filtration V (I) and intervals of the persistence diagram Dgm(V (I)) for the image I : I → R of Fig. 3

7 Discussion Our results clarify the relationship between the two cubical complex constructions commonly used in digital image analysis software and provide a simple method to use software that implements one construction to compute a persistence diagram for the other. This permits a user’s choice of adjacency type for their images to depend on that appropriate to the application rather than on the type of construction used in available efficient persistence software. In addition to facilitating this application, the results of Sects. 3 and 5 may be of independent interest for the following reasons. Theorem 2 is a new interpretation of a duality relationship that manifests in many contexts such as the correspondence between persistent homology and persistent relative cohomology [5], the duality theorem of extended persistence [3], and a discrete Helmoltz-Hodge decomposition [11]. In [13], we show that the filtered discrete Morse chain complexes also exhibit this duality. Further investigations in this area may reveal other interpretations of this relationship. The results of Sect. 5 are formulated specifically for the case of an image with its domain homeomorphic to a closed ball, but could be extended to spaces with more interesting topology. We anticipate that the long exact sequence of a pair can be used to derive a relationship between filtered cell complexes that satisfy conditions for duality if their boundaries can be capped or quotiented as in Sect. 4 to obtain a manifold.

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Further applications of the relationship between the T- and V-constructions can be found. In particular, a discrete gradient vector field built on V (I ) is easily transformed into a dual one on T (I ) [13]. If the gradient vector field is built to be consistent with the grayscale image I according to the algorithm of [22], then the dual gradient vector field on T (I ) will be consistent with −I. This means the skeletonisation and partitioning algorithms of [7] can now be adapted to work on images where the T-construction is preferred. As discussed in Sect. 1.1, the duality and symmetry results of extended persistence do not apply in the cubical setting because the Partition Lemma fails. However, using the T-construction for homology and the V-construction for relative homology (or vice versa), we can derive an extension of extended persistence to cubical filtrations of images satisfying duality and symmetry similar to the results in [3]. Analyzing properties of this definition and comparing this approach to that in [12] would be interesting future work. There are also interesting questions about algorithm performance to explore. The results of Sect. 6 suggest that persistence diagram computation from grayscale images should have the same average run time independent of the choice of T- and V-construction. If the T-construction executes faster on a particular image, then the V-construction should execute faster on the negative of the image. To answer this question fully requires a careful analysis of the effects of taking the anti-transpose of the boundary matrix on the run time of the matrix reduction algorithm and the extra cells added when padding the image. Acknowledgments This project started during the Women in Computational Topology workshop held in Canberra in July of 2019. All authors are very grateful for its organisation and the financial support for the workshop from the Mathematical Sciences Institute at ANU, the US National Science Foundation through the award CCF-1841455, the Australian Mathematical Sciences Institute and the Association for Women in Mathematics. AG is supported by the Swiss National Science Foundation grant CRSI I 5_177237. TH is supported by the European Research Council (ERC) Horizon 2020 project “Alpha Shape Theory Extended” No. 788183. KM is supported by the ERC Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 859860. VR was supported by Australian Research Council Future Fellowship FT140100604 during the early stages of this project.

References 1. Bubenik, P., Mili´cevi´c, N.: Homological algebra for persistence modules. Found. Comput. Math. (2021). https://doi.org/10.1007/s10208-020-09482-9 2. Chazal, F., De Silva, V., Glisse, M., Oudot, S.: The Structure and Stability of Persistence Modules. Springer, Berlin (2016). https://doi.org/10.1007/978-3-319-42545-0 3. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Extending persistence using Poincaré and Lefschetz duality. Found. Comput. Math. 9, 79–103 (2009) 4. Cohen-Steiner, D., Edelsbrunner, H., Morozov, D.: Vines and vineyards by updating persistence in linear time. In: Proceedings of the Twenty-Second Annual Symposium on Computational Geometry, SCG ’06, pp. 119–126. (2006). https://doi.org/10.1145/1137856. 1137877

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5. De Silva, V., Morozov, D., Vejdemo-Johansson, M.: Dualities in persistent (co)homology. Inverse Problems 27(12), 124003 (2011). https://doi.org/10.1088/0266-5611/27/12/124003 6. Delfinado, C.J.A., Edelsbrunner, H.: An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere. Comput. Aided Geom. Design 12(7), 771–784 (1995) 7. Delgado-Friedrichs, O., Robins, V., Sheppard, A.: Skeletonization and partitioning of digital images using discrete Morse theory. IEEE Trans. Pattern Anal. Mach. Intell. 37(3), 654–666 (2015). https://doi.org/10.1109/TPAMI.2014.2346172 8. Dunaeva, O., Edelsbrunner, H., Lukyanov, A., Machin, M., Malkova, D., Kuvaev, R., Kashin, S.: The classification of endoscopy images with persistent homology. Pattern Recogn. Lett. 83, 13–22 (2016). https://doi.org/10.1016/j.patrec.2015.12.012 9. Edelsbrunner, H., Harer, J.: Persistent homology - a survey. In: Surveys on Discrete and Computational Geometry: Twenty Years Later. American Mathematical Society, 257–282 (2008). 10. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28(4), 511–533 (2002) 11. Edelsbrunner, H., Ölsböck, K.: Tri-partitions and bases of an ordered complex. Discrete Comput. Geom. 64, 759–775 (2020). https://doi.org/10.1007/s00454-020-00188-x 12. Edelsbrunner, H., Symonova, O.: The adaptive topology of a digital image. In: Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012, pp. 41–48. IEEE, Piscataway (2012) 13. Garin, A., Heiss, T., Maggs, K., Bleile, B., Robins, V.: Duality in persistent homology of images. Extended abstract SoCG YRF (2020) 14. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002) 15. Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology. Springer, New York (2004) 16. Kaji, S., Sudo, T., Ahara, K.: Cubical ripser: Software for computing persistent homology of image and volume data. ArXiv 2005.12692 (2020) 17. Kong, T., Rosenfeld, A.: Digital topology: Introduction and survey. Comput. Vis. Graph. Image Process. 48(3), 357–393 (1989). https://doi.org/10.1016/0734-189X(89)90147-3 18. Kovalevsky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process. 46(2), 141–161 (1989). https://doi.org/10.1016/0734-189X(89)90165-5 19. Lundell, A.T., Weingram, S.: The Topology of CW Complexes. Springer, Berlin (1969) https:// doi.org/10.1007/978-1-4684-6254-8 20. Miller, E.: Homological algebra of modules over posets. ArXiv 2008.00063 (2020) 21. Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley Publishing Company, Menlo Park (1984) 22. Robins, V., Wood, P., P Sheppard, A.: Theory and algorithms for constructing discrete Morse complexes from grayscale digital images. IEEE Trans. Pattern Anal. Mach. Intell. 33, 1646– 1658 (2011). https://doi.org/10.1109/TPAMI.2011.95 23. Tierny, J., Favelier, G., Levine, J.A., Gueunet, C., Michaux, M.: The Topology ToolKit. IEEE Transactions on Visualization and Computer Graphics (Proc. of IEEE VIS) (2017). https:// topology-tool-kit.github.io/ 24. Tymochko, S., Munch, E., Dunion, J., Corbosiero, K., Torn, R.: Using persistent homology to quantify a diurnal cycle in hurricanes. Pattern Recogn. Lett. 133, 137–143 (2020). https://doi. org/10.1016/j.patrec.2020.02.022 25. Varli, H., Yilmaz, Y., Pamuk, M.: Homological properties of persistent homology. ArXiv 1805.01274 (2018) 26. Wagner, H.: Cubicle: Streaming computations for persistent homology of large greyscale images (2018). https://bitbucket.org/hubwag/cubicle/src/master/ 27. Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)

Morse-Based Fibering of the Persistence Rank Invariant Asilata Bapat, Robyn Brooks, Celia Hacker, Claudia Landi, and Barbara I. Mahler

Abstract Although there is no doubt that multi-parameter persistent homology is a useful tool to analyze multi-variate data, efficient ways to compute these modules are still lacking in the available topological data analysis toolboxes. Other issues, such as interpretation and visualization of the output, remain difficult to solve. Software visualizing multi-parameter persistence diagrams is currently only available for 2-dimensional persistence modules. One of the simplest invariants for a multi-parameter persistence module is its rank invariant, defined as the function that counts the number of linearly independent homology classes that live in the filtration through a given pair of values of the multi-parameter. We propose a step towards interpretation and visualization of the rank invariant for persistence modules for any given number of parameters. We show how discrete Morse theory may be used to compute the rank invariant, proving that it is completely determined by its values at points whose coordinates are critical with respect to a discrete Morse gradient vector field. These critical points partition the set of all lines of positive slope in the parameter space into equivalence classes such that the rank invariant along lines in the same class are also equivalent. We show that we can deduce the A. Bapat Australian National University, Canberra, ACT, Australia e-mail: [email protected] R. Brooks Boston College, Chestnut Hill, MA, USA e-mail: [email protected] C. Hacker EPFL, Lausanne, Switzerland e-mail: [email protected] C. Landi () Università di Modena e Reggio Emilia, Modena, Italy e-mail: [email protected] B. I. Mahler University of Oxford, Oxford, UK e-mail: [email protected] © The Author(s) and the Association for Women in Mathematics 2022 E. Gasparovic et al. (eds.), Research in Computational Topology 2, Association for Women in Mathematics Series 30, https://doi.org/10.1007/978-3-030-95519-9_2

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persistence diagrams for all lines in a given class from the persistence diagram of a representative line in that class.

1 Introduction Digital data are being produced at a constantly increasing pace, and their availability is changing the approach to science and technology. The fundamental hypothesis of Topological Data Analysis is that data come as samples taken from an underlying shape, and unveiling such shape is important for understanding the studied phenomenon. Topological shape analysis amounts to determining nontrivial topological holes in any dimension. Computational Topology provides tools to derive specific signatures—topological invariants—which depend only on topological features of the shape of data and are robust to local noise [12]. Among them, persistent homology [6, 26] stands out as most useful, having already found numerous applications in a diverse range of fields [3, 4, 15, 19, 24]. The first step in the persistence pipeline is to build a family, called a filtration, of nested simplicial complexes that model the data at various scales by varying one or more parameters. The second step focuses on the maps induced in homology by the simplicial inclusions to extract invariants such as the persistence module. The third step is to use persistence invariants as a source of feature vectors in machine learning contexts, the final goal being to use the acquired topological information to improve the understanding of the underlying data. An important feature of this pipeline is its robustness to noise in the input data [12]. Some systems warrant analysis across multiple parameters, so it is important to focus on multi-parameter persistence [8] where the filtration may depend on any number of real-valued parameters, and n-tuples of parameter values have an inherited partial order. Unlike single-parameter persistent homology, which is completely described by a persistence diagram, multi-parameter persistence modules contain more information than it is possible to handle, understand, and visualize easily. Therefore, it is convenient to summarize them by simpler invariants. Among the various invariants considered for a multi-parameter persistence module, such as the blockcodes of [13] and the multi-graded Betti numbers of [21], one of the simplest invariants is the rank invariant [8], defined as the function that counts the number of linearly independent homology classes that live in the filtration through any given pair of values of the multi-parameter. Theoretically, computation of the rank invariant can be carried out by fibering it along lines with positive slope in an n-dimensional space, where n is the number of parameters [9]. Indeed, the rank invariant of the restriction of a persistence module to an increasing line is completely described by a persistence diagram. The union of all such persistence diagrams forms a compact object called the persistence space [11]. Practically though, one needs to restrict the number of relevant lines to a reasonable number. For example, the state-of-the-art tool for rank invariant visualization, RIVET [20],

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achieves reduction to some template lines using Betti tables, takes O(m3 ) run-time (with m the number of simplices), and is limited to two parameters. Given the dimensional, computational, and interpretability limits of the currently available methods, multi-parameter persistence is not yet a viable option for data analysis. It is thus worth exploring different approaches that could enable faster computation and better understanding of the rank invariant in a dimension-agnostic way. To this end, we propose to exploit the information contained in the critical cells of a discrete gradient vector field consistent with the given multi-filtration as a means to enhancing geometrical understanding and computational efficiency of multi-parameter persistence. Evidence for the usefulness of Morse theory in multi-parameter persistence is given in [1, 2, 23]. In these papers, discrete Morse theory is used to reduce the multi-parameter persistence input data size by substituting the original simplicial complex with a Morse complex containing only critical cells but having the same persistence as the initial complex. This approach takes advantage of the fact that the number of critical cells is very small in comparison with m, the total number of simplices. Computations are sped up by reducing the number of cells, while the run-time complexity of obtaining a discrete gradient vector field compatible with the filtration is m · s 2 , where s denotes the maximum number of simplices in a vertex star. Although in the worst case s may be as large as m, in many applications it is negligible in comparison to m, so that the complexity may be considered linear in m. Moreover, as shown by tests in [23], this global reduction also avoids repeating the retrieval of the same null persistence pairs when fibering the rank invariant along different lines, improving time performances proportionally to the number of lines: the larger the number of lines, the more convenient the reduction to the Morse complex. Supported by such empirical evidence, in this paper we investigate the theoretical connection between critical cells of discrete Morse theory and rank invariant computation along lines, with the goal of improving the available methods for fibering the rank invariant of multi-parameter persistence in a way that is computationally efficient, geometrically interpretable, and readily visualizable. To this end, we exploit the correspondence between critical cells of a discrete gradient vector field and topological changes in filtrations compatible with it to: 1. show that the rank invariant for n-parameter persistence modules can be computed by selecting a small number of values in the parameter space Rn determined by the critical cells of the discrete gradient vector field, and using these values to reconstruct the rank invariant for all other possible values in the parameter space (Theorem 1); 2. use such critical values to define an equivalence relation among lines so that representatives from each equivalence class form a convenient selection of lines for fibering the rank invariant (Definition 14); 3. show that fibering the rank invariant along lines in the same equivalence class yields persistence diagrams that can be obtained one from the other by simple bijections between critical values (Theorem 2);

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4. present a method based on the previous results for computing any fiber of the rank invariant using only finitely many template fibers (Sect. 5). We emphasize that all of our results hold for any number of parameters, thus improving the state-of-the-art methods, which allow for only two. Moreover, our method’s input requires only the critical cells of a discrete gradient vector field, so that the required pre-processing can be achieved in just linear time according to various available algorithms [1, 2, 23]. Finally, the connection to Morse theory allows for a more immediate geometric interpretation and visualization of the topological features detected by persistence as pairs of critical cells in the given simplicial complex. The outline of this paper is as follows: In Sect. 2, we outline background information which will be necessary for the reader to understand the rest of the text. Section 3 focuses on computing the rank invariant, supported by lemmas and diagram-chasing. Section 4 explains the computation of the persistence space along with delving into fibering the rank invariant along equivalent lines. We conclude with Sect. 5 by laying out a method for fibering the rank invariant of persistence by lines and suggesting possible applications. Finally, in the Appendix we discuss a result that allows us to develop an algorithm which chooses a representative line for each equivalence class in the case of 2-parameter persistence modules. The generalization of this algorithm to higher-dimensional persistence modules remains future work.

2 Notation and Definitions These definitions are partially based on the definitions in [17, 20] and [14]. Let K be a field. For computational purposes, K is often taken to be finite. Define the following partial order on Rn : for u = (ui ), v = (vi ) ∈ Rn , we say that u  v (resp. u ≺ v) if and only if ui ≤ vi (resp. ui < vi ) for all i. The poset (Rn , ) will be our parameter space. Definition 1 (Persistence Module) A persistence module V over the parameter space Rn is an assignment of a K–vector space Vu to each u ∈ Rn , and transition maps i u,v : Vu → Vv to pairs of points u  v ∈ Rn , satisfying the following properties: • i u,u is the identity map for all u ∈ Rn . • i v,w ◦ i u,v = i u,w for all u  v  w ∈ Rn . A persistence module over Rn is also known as an n-parameter persistence module or an nD persistence module. Definition 2 (Rank Invariant) For n ≥ 1, let Hn ⊂ Rn × Rn be the subset of pairs (u, v) such that u  v. Let V be an nD persistence module. Then the rank invariant of V is a function ρV : Hn → Z, defined as

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ρV (u, v) = rank(i u,v ). Persistence modules most often arise from filtered simplicial complexes. Definition 3 (Finite Simplicial Complex) A finite simplicial complex K is a collection of subsets of a finite set V0 , such that: • all singletons of V0 are in K, and • if β ∈ K and α ⊂ β, then α ∈ K. The elements of V0 are called vertices, and the elements of K are called simplices. If α ∈ K contains p + 1 vertices, then we say α has dimension p, sometimes denoted α (p) . If β ∈ K and α ⊂ β, then we say α is a face of β and β is a coface of α, and we denote this by α < β. If α is a codimension one face of β, we say that α is a facet of β and β is a cofacet of α. If α is a face of a dimension p simplex β, and it is not a face of any other p dimensional simplex, then we say that α is a free face of β. Definition 4 (Filtration) Let K be a finite simplicial complex. An n-parameter filtration of K is a collection of subcomplexes K = {K u }u∈Rn of K, such that K u ⊆ K v whenever u  v, and moreover that

K u. K= n u∈R A complex with such a filtration is called a multi-filtered simplicial complex or an n-filtered simplicial complex. Remark 1 Since K is finite, there is some u ∈ Rn such that K = K u . Definition 5 (Entrance Value) We say that a simplex σ in K has entrance value u ∈ Rn if

Kw σ ∈ Ku − wu,u=w

This value tells us where σ has entered the filtration. Note that if the entrance value of σ is u, then σ ∈ K v for all u  v, and σ ∈ / K w for all w  u such that u = w. Therefore, if u is an entrance value of σ , then u is a (possibly non-unique) minimal value of the set {v ∈ Rn | σ ∈ K v }. According to the definition of filtration, if a simplex σ enters the filtration at u, all of its faces have to be in K u as well. The faces can enter jointly with σ or at earlier values of the filtration. However, entrance values are not guaranteed to exist. To avoid pathological situations in which a simplex does not have any entrance values, such as in the 1-parameter filtration of K defined by K u = ∅ for each u ≤ 0 and K u = K for each u > 0, we will only consider filtrations as follows.

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Definition 6 (Tameness) A filtration of a simplicial complex K is said to be tame if each simplex of K has at least one entrance value. Remark 2 For a tame filtration, it is guaranteed that there exists some u ∈ Rn such that K u = ∅. Example 1 Consider a function f : K → Rn that is monotonic with respect to the face relation. That is, α < β ∈ K implies that f (α)  f (β). Every such function gives rise to a tame filtration, by defining the filtered pieces of K to be sublevel sets as follows: K v = f −1 ({u ∈ Rn : u  v}). On the other hand, not all filtrations of K as in Definition 4 define a function f : K → Rn , even under the tameness assumptions, because the entrance value of a simplex may not be unique. This motivates the following definition that can be found in [7]. Definition 7 (One-criticality) An n-parameter filtration K of K is said to be onecritical if every simplex of K has a unique entrance value in K. Proposition 1 Each tame one-critical filtration of a simplicial complex K is the sublevel set filtration of a monotonic function f : K → Rn . Conversely, if f : K → Rn is monotonic, then its sublevel sets form a tame and one-critical filtration of K. Proof Let K be a simplicial complex and K a tame and one-critical filtration of K. We construct a monotonic function fK : K → R whose sublevel set filtration is precisely K. For each simplex σ ∈ K, set fK (σ ) to be the unique entrance value of σ in K with respect to K, which exists and is unique because the filtration is tame and one-critical. Let α, β ∈ K such that α < β. Recall that if β ∈ K u for some f (β) u ∈ Rn , then α ∈ K u as well. In particular, α ∈ K K . Since fK (α) is the unique minimum value u such that α ∈ K u , it must be the case that fK (α)  fK (β), and therefore fK is monotonic. Conversely, let f : K → Rn be a monotonic function. As before, set K to be the set of all sublevel sets K v = f −1 ({u ∈ Rn : u  v}). Let σ be any simplex of K. It is clear that σ ∈ K f (σ ) . We prove that f (σ ) is the unique entrance value of σ . Note that it is indeed an entrance value: if u  f (σ ) and u = f (σ ), then f (σ ) ∈ / K u . Suppose that v is any entrance value of σ , which means v that σ ∈ K and σ ∈ / K u for any u = v such that u  v. Since σ ∈ K v , we must have f (σ )  v. Since σ ∈ / K u for any u = v such that u  v, it must be the case that f (σ ) = u for any such u. Therefore v = f (σ ). Since every simplex σ ∈ K has the unique entrance value f (σ ), we obtain a tame and one-critical filtration of K from the sublevel sets of f .  

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To avoid pathologies, we always assume the filtrations considered in this paper are tame and one-critical. Now think of the poset Rn = (Rn , ) as a category where the objects are elements of Rn , and the morphisms are given by the order relation. Let SC be the category of finite simplicial complexes with inclusions as the morphisms. A filtered simplicial complex can be thought of as a functor K : Rn −→ SC. Further, for each i ∈ Z, we can take the i-th homology of a simplicial complex to obtain a vector space. The two functors above can be composed to obtain a functor from Rn to VectK , the category of K-vector spaces. By Definition 1, a persistence module can be viewed as a functor from Rn to VectK , so the construction above is a special case of a persistence module. This motivates the following definition. Definition 8 (Persistent Homology) For an integer i, the i-th multi-parameter persistent homology module is the persistence module Hi K : Rn −→ VectK defined as the composition of the filtration functor K with the i-th homology functor for simplicial complexes:

We now introduce the basic constructions of discrete Morse theory, which we will heavily use in the remainder of the paper. More details may be found in [14]. Definition 9 (Discrete Vector Field) A discrete vector field V on K is a collection of pairs of simplices (α, β) of K where α is a facet of β, such that each simplex is in at most one pair of V. Given a discrete vector field V on a simplicial complex K, a V-path is a sequence of simplices of dimensions p and p + 1, α0 , β0 , α1 , β1 , α2 , . . . , βr , αr+1 such that for each i = 0, . . . , r, we have (αi , βi ) ∈ V and βi > αi+1 = αi . A path is called a non-trivial closed path if r > 0 and α0 = αr+1 . Definition 10 (Gradient Vector Field) A discrete vector field V on a simplicial complex K is called a gradient vector field if it contains no non-trivial closed Vpaths. A simplex σ ∈ K is critical if it is not paired in V. An example of such a discrete gradient vector field can be found in Fig. 1. Definition 11 (Elementary Collapse) Let K1 , K2 be simplicial complexes such that K2 ⊂ K1 , and K1 \ K2 = {α, β} where α is a free facet of β. Then the combinatorial deformation retract of K1 to K2 given by removing α and β is called an elementary (simplicial) collapse. The pair (α, β) is called a collapsing pair.

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Fig. 1 An example of discrete gradient vector field. The pairs in the vector field are denoted by arrows and the critical cells are marked by a circle Fig. 2 The complex on the right is obtained by the one on the left by first collapsing the pair (α, β), then the pair (γ , δ)

˜

˜

Remark 3 The definition of elementary collapse is also valid for CW-complexes. Example 2 In Fig. 2 we can see how to collapse the complex in Fig. 1 using the given discrete gradient vector field. The first collapsing pair is formed by the edge α and the triangle β. The second collapse uses the pair (γ , δ), given by the edge δ and the vertex γ . An elementary simplicial collapse is a homotopy equivalence between two simplicial complexes, which in turn induces an isomorphism on the level of homology. Therefore, if K1 and K2 are related through a series of elementary collapses, then Hi (K2 ) ∼ = Hi (K1 ) for all i. In the case when a simplicial complex has no free faces available, it is still possible to simulate an elementary collapse, which in this case is called internal, by first removing a critical cell in order to obtain a free face, and then reinserting it after updating the incidence relations. The cell complex obtained in this case may no longer be simplicial, but internal collapses still induce isomorphisms in homology. To turn simplicial collapses from just homology preserving into persistent homology preserving transformations, it is convenient to confine ourselves to considering gradient vector fields compatible with filtrations. Definition 12 (Consistency) A discrete gradient vector field V on a simplicial complex K is said to be consistent (or compatible) with a filtration K = {K u }u∈Rn of K if the following condition is satisfied: ∀(σ, τ ) ∈ V, σ ∈ K u ⇐⇒ τ ∈ K u . In this case, we say that a value u ∈ Rn is a critical value of K if it is the entrance value of a critical cell of V. Example 3 In Fig. 3 we see a bifiltration, i.e. a filtration with 2 parameters, of a finite simplicial complex. This filtration is one-critical and the gradient vector field is consistent with the given filtration.

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3

2

1

0

0

1

2

3

Fig. 3 Each box corresponds to a multi-parameter in R2 , showing the simplices that are present at the corresponding step of the filtration. As before, the pairs of the discrete gradient vector field are indicated by arrows and the critical cells by circles. As we will see later on, we are interested in the entrance values of the critical cells. When the critical cells enter the filtration, they are denoted in a darker red, whereas they are denoted in a lighter orange for larger values, so as to not lose track of them throughout the filtration

3 Computing the Rank Invariant Let K be a finite simplicial complex, V be a discrete gradient vector field on K, and K = {K u }u∈Rn be a one-critical n-parameter filtration on K which is consistent with V. Then, for each integer i, there is an n-parameter persistence module Vi where the K-vector space associated to u ∈ Rn is Hi (K u ), the i-th homology group of K u . Furthermore, for u  v ∈ Rn , i u,v is the induced map, on the level of homology, of the inclusion of K u into K v . The goal of this section is to identify a finite subset of values in Rn from which the rank invariant of Vi can be computed. In other words, we want some finite U ⊂ Rn such that, for all u  v ∈ Rn , there exists u  v ∈ U such that ρVi (u, v) = ρVi (u, v). Define C to be the set of critical values of V: C = {u ∈ Rn |σ is critical in V, the entrance value of σ is u}.

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Fig. 4 Taking the set of critical values C to consist of {c1 , c2 , c3 }, C = C ∪ {c4 }. The grey area contains all values x  u, and u¯ = c2 2

=¯

4

1

3

Let C be the closure of C under least upper bound. Theorem 1 states that our candidate set U as described above is exactly given by C. In order to identify each u ∈ Rn with an element of C, define u = max{u ∈ C|u  u}. Example 4 Figure 4 shows the set of critical values and its closure in the parameter space for the bifiltration in Example 3. We can see there that for the value u the corresponding u¯ is the critical value c2 . Lemma 1 Let U be a non-empty finite set closed under least upper bound. For all u in Rn , the set U = {u ∈ U |u  u}, if non-empty, admits a (unique) maximum. Proof From the fact that U is non-empty and finite, it admits maximal elements. Let a and b be maximal elements in U . Because U is closed under least upper bound, there is c in U such that a, b  c  u. Hence, c ∈ U . Thus, c = a = b by maximality of a and b in U .   The idea behind the definition of C is as follows. Since the filtration K has a consistent discrete gradient vector field V, the tools of discrete Morse theory may be used to identify which elements of K are guaranteed to be homotopy equivalent and therefore have isomorphic homology groups. Lemma 2 can be used to show the existence of a simplicial collapse, induced by V, between certain elements of K; this simplicial collapse is a homotopy equivalence. Lemma 2 Let K = {K u }u∈Rn be a one-critical filtration and V a discrete gradient vector field on a finite simplicial complex K consistent with K. For u ∈ Rn , let u = max{u ∈ C|u  u}, with C the set of critical values of V and C its closure under least upper bound. If K u − K u is non-empty, then it contains two simplices σ, τ such that (σ, τ ) ∈ V and σ is a free facet of τ . Proof Because K u − K u is non-empty and finite, we can take τ ∈ K u − K u of maximal dimension. Denote by l.u.b.(u, v) the least upper bound of two elements ˆ u)  u. u, v ∈ Rn . If τ is critical, there is a critical value uˆ such that u = l.u.b.(u, Since l.u.b.(u, ˆ u) ∈ C, this contradicts the definition of u.

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Thus, it must be that τ is non-critical, and belongs to a vector (σ, τ ) ∈ V. Because V is consistent with K, σ must belong to K u − K u as well, and since τ is of maximal dimension, it must be that σ is a face of τ . Hence, each of the simplices of maximal dimension in K u − K u belongs to a vector (σ, τ ) ∈ V with σ also in K u − K u , and there are finitely many of them because K is finite. We claim that one of the pairs of simplices (σ, τ ) ∈ V, with τ of maximal dimension in K u − K u , must be a collapsing pair, meaning σ is a free facet of τ . Supposing that this is not the case, we create a non-trivial cyclic path in V, contradicting the assumption that V is a discrete gradient vector field. Choose a pair (σ1 , τ1 ) ∈ V of simplices in K u − K u , with τ1 of maximal dimension. If σ1 is not a free facet of τ1 , there exists another cofacet τ2 ∈ K u such that σ1 < τ2 = τ1 . As V is consistent with K, σ1 cannot be added to the filtration after τ2 , so it must be that τ2 ∈ K u − K u as well. Moreover, since the dimension of τ1 is maximal and σ1 is a facet of τ2 , τ1 and τ2 must have the same dimension. Thus also τ2 is of maximal dimension. Thus, τ2 is in some non-critical pair (σ2 , τ2 ) ∈ V of simplices in K u − K u . Finally, since V is a discrete gradient vector field, σ1 can only exist in one pair of V, so that σ2 = σ1 . We may iterate the above argument. After an appropriate finite number of iterations, we obtain the following V -path: σ1 < τ1 > σ2 < τ2 > · · · < τn−1 > σn < τn > σn+1 . Since there are only finitely many possible choices for σi , it must be that σi = σj for some i = j ∈ {1, . . . , n + 1}. The portion of the V -path between σi and σj is non-trivial and cyclic. This is a contradiction, as V is a discrete gradient vector field. Thus, for some pair (σ, τ ) ∈ V of simplices in K u − K u with τ of maximal dimension, σ must be a free facet of τ , proving our claim.   We may now prove the main result of the section, which gives a formula for computing the rank invariant for any pair (u, v), using elements of C. Theorem 1 Let K be a finite simplicial complex, and let K = {K u }u∈Rn be a filtration on K. Suppose V is a discrete gradient vector field on K consistent with K. Let Vi = Hi (K). Then, for all u  v, ρVi (u, v) = ρVi (u, v) with u = max{u ∈ C|u  u}, v = max{v ∈ C|v  v} if {u ∈ C|u  u} is non-empty, and ρVi (u, v) = 0 otherwise.

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Proof Let u, v ∈ Rn . Because C is non-empty and C contains all least upper bounds, if {u ∈ C|u  u} = ∅, then K u = ∅, yielding ρVi (u, v) = 0. Otherwise, if {u ∈ C|u  u} is non-empty, then also {v ∈ C|v  v} is so, and u, v exist and are unique by Lemma 1. Set K0 := K u . If K0 = K u , then obviously Hi (K u ) = Hi (K u ) for all i. Otherwise, K0 − K u is non-empty and by Lemma 2 there is a pair (σ1 , τ1 ) ∈ V of simplices of K0 − K u such that σ1 is a free facet of τ1 . Let K1 be the simplicial complex obtained by performing the elementary collapse of σ1 onto τ1 ; note that this means that the map induced on homology by the inclusion of K1 into K0 is an isomorphism: Hi (K1 ) ∼ = Hi (K0 ). Now, either K1 = K u , in which case obviously u Hi (K1 ) = Hi (K ) for all i, or, we may restrict V to K 1 and consider the filtration of K1 induced from that of K0 . By applying Lemma 2 to K1 , there is a vector in V whose elementary collapse gives K2 such that the map induced by the inclusion of K2 into K1 is an isomorphism: Hi (K2 ) ∼ = Hi (K1 ). By induction, for any r ≥ 1, either Kr = K u , in which case obviously Hi (Kr ) = Hi (K u ) for all i, or, we may restrict V to Kr and consider the filtration of Kr induced from that of Kr−1 . By applying Lemma 2 to Kr , there is a vector in V whose elementary collapse gives Kr+1 such that the map induced by the inclusion of Kr+1 into K1 is an isomorphism: Hi (Kr+1 ) ∼ = Hi (Kr ). By the finiteness of K, there must be a value of r such that Kr = K u , yielding that the map induced by the inclusion of K u into K u is an isomorphism for all i: Hi (K u ) ∼ = Hi (K u ). Analogous argument works for v: Hi (K v ) ∼ = Hi (K v ). Because the above isomorphisms are induced by inclusions, the following diagram commutes and gives equality of the ith rank invariants of the pairs (u, v) and (u, v):

 

4 Computing the Persistence Space For the sake of visualization, the rank invariant of an n-parameter persistence module V can be completely encoded as a multiset of points known as a persistence diagram when n = 1 [12] and as a persistence space when n ≥ 1 [11]. By completeness of the encoding we mean that the rank invariant can be exactly

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reconstructed from the persistence space (cf. the k-Triangle Lemma in [12] and the Representation Theorem in [11]). While the persistence space is easier to visualize than the rank invariant (as it is a set of points rather than a function), there is the added complication that the persistence space for an n-parameter persistence module lives in a 2n-dimensional space. Therefore, it is convenient to visualize the persistence space along fibers [9]. For example, PHOG [5] and RIVET [20] visualize the persistence space of a 2parameter persistence module by fibering it through lines. The goal of this section is to propose a computational procedure to recover such a fibration along lines for persistence modules with any number of parameters, by using critical values of gradient vector fields. We start by reviewing the necessary definitions and properties. A point (u, v) ∈ Hn belongs to the persistence space spc(V) if and only if its multiplicity μV (u, v) :=

min

e"0 u+e≺v−e

ρV (u + e, v − e) − ρV (u − e, v − e) +

(1)

−ρV (u + e, v + e) + ρV (u − e, v + e)

(2)

is positive. This corresponds to the number of independent cycles that, along a positive direction in the parameter space, appear at u and become boundaries at v. Similarly, a point (u, ∞) belongs to the persistence space of V if and only if its multiplicity μV (u, ∞) := min ρV (u + e, v) − ρV (u − e, v) e"0 vu

(3)

is positive. This corresponds to the number of independent cycles that, along a positive direction in the parameter space, appear exactly at u and persist for every larger value of the parameter. In both cases, the multiplicity can be computed by fixing a direction for e and only varying its length (with alternate sums of the ranks decreasing as the length decreases). Two convenient directions for e are the diagonal direction and the v − u direction. Moreover, for points at infinity, the multiplicity is reached for increasing values of v. In particular, for n = 1, the persistence space is the persistence diagram of a 1-parameter persistence module. In terms of intervals in a persistence module bar decomposition, points in Hn of positive multiplicity correspond to finite intervals, and points at infinity of positive multiplicity correspond to infinite intervals.

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4.1 Restriction of a Persistence Module to Lines Given a line L contained in the parameter space Rn , each point u ∈ L can be written as u = mt + u0 , with u0 a fixed starting point on L, m ∈ Rn a fixed velocity vector, and t a real parameter. If 0 ≺ m, we say that L has positive slope. For an n-parameter persistence module V and a line L ⊆ Rn with positive slope, the restriction of V to L is the persistence module VL that assigns Vu to each u ∈ L, and whose transition maps iLu,v : Vu → Vv for u  v ∈ L are the same as in V. Once a parametrization u = mt + u0 of L is fixed, the persistence module VL is isomorphic to the 1-parameter persistence module, by abuse of notation denoted by VL that assigns to each t ∈ R the vector space (VL )t = Vu and to s < t ∈ R the transition map i s,t = iLu,v = i u,v . By construction, for u = ms + u0 and v = mt + u0 , it holds that ρV (u, v) = ρVL (s, t). Hence, the multiplicity of a point (u, v) ∈ Hn in spc(V) coincides with that of (s, t) ∈ H1 in dgm(VL ): μV (u, v) = μVL (s, t). In conclusion, the persistence space spc(V) can be viewed as the fibered union of infinitely many persistence diagrams dgm(VL ), each associated with a line L with positive slope.

4.2 Critical Values Determine the Persistence Space Our next goal is to demonstrate that, for a persistence module V obtained from a tame and one-critical filtration K of a simplicial complex, points of the persistence space spc(V) are completely determined by the critical values of a discrete gradient vector field V compatible with K. This claim is proven in Proposition 3. The underlying idea to prove it is as follows. As spc(V) can be viewed as the fibered union of infinitely many persistence diagrams dgm(VL ), with each VL obtained by restricting V to a line L with positive slope, the filtration K may also be restricted to L. This way we obtain a 1-parameter filtration KL , and VL turns out to be the persistence module of KL . Moreover, if K has a compatible discrete gradient vector field V, then this discrete gradient vector field is inherited by KL . Each critical cell of V has an entrance value in KL (as L has positive slope). As is the case with K, the entrance values in KL of the critical cells in C identify elements in KL where the filtration may undergo a change in homotopy type and therefore a change in homology. Therefore, to determine spc(V), it is enough to identify the entrance values of critical cells of K in the restricted filtrations KL . To this end, we introduce the following notation.

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For every point u in Rn , let S+ (u) be the positive cone with vertex u: S+ (u) = {v ∈ Rn : u  v}. The boundary of the positive cone, ∂S+ (u), decomposes into disjoint faces. In particular, ∂S+ (u) can be partitioned by non-empty subsets A of [n] = {1, 2, . . . , n} in the following way. For ∅ = A ⊆ [n], define / A}. SA (u) = {(x1 , . . . , xn ) ∈ Rn |xi = ui for i ∈ A, xj > uj for j ∈

Then, for A = B ⊆ [n], SA ∩ SB = ∅, and ∂S+ (u) =

SA .

∅=A⊆[n]

Example 5 If n = 2 and u = (u1 , u2 ), the faces of ∂S+ (u) consist of the vertex u and the two half-lines exiting from u rightwards and upwards, respectively, as shown in Fig. 5. ∂S+ (u) = S{1,2} (u) ∪ S{1} (u) ∪ S{2} (u) = {(u1 , u2 )} ∪ {(x1 , x2 ) ∈ R2 |x1 = u1 , x2 > u2 } ∪ {(x1 , x2 ) ∈ R2 |x1 > u1 , x2 = u2 }. It will be useful to consider the projection of points in the parameter space onto lines with positive slope (cf. [18]). Definition 13 (Push) Given a line L ⊆ Rn with positive slope, for every u ∈ Rn define pushL (u) := L ∩ ∂S+ (u). Proposition 2 (Properties of Push) Some properties of pushL (u) are (see also Fig. 6): Fig. 5 The positive cone S+ (u) of u ∈ R2 and the decomposition of its boundary into S{1} , S{2} and S{1,2} , which correspond respectively to the vertical boundary, horizontal boundary, and u

{1}

=

+(

{1,2}

)

{2}

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Fig. 6 The push of u along the line L

push ( )

1. pushL (u) consists of exactly one point because L has positive slope. 2. There is a unique non-empty subset AL u of [n] such that pushL (u) = L ∩ SALu (u). For ease of notation, we concisely write SL (u) meaning SALu (u). 3. u  pushL (u) with equality only when u ∈ L. 4. pushL (u) is the smallest point on L which is greater than or equal to u; smaller points on L are either incomparable or less than u. 5. If u  v, then pushL (u)  pushL (v). 6. Let u  v. Let ∅ = A, B ⊆ [n] such that pushL (u) ∈ SA (u) and pushL (v) ∈ SB (v). We have: (a) SA (u) ∩ SB (v) = ∅ implies that A ⊆ B. (b) pushL (u) = pushL (v) if and only if SA (u) ∩ SB (v) = ∅. Proof Properties 1, 2, 3 and 4 are immediate. Proof of Property 5 Suppose not; then u  v and pushL (v) ≺ pushL (u). Note that, since both pushL (u) and pushL (v) are points on L with positive slope, pushL (v) ≺ pushL (u) if and only if each coordinate is strictly less than, i.e., (pushL (v))i < (pushL (u))i for all i ∈ [n]. Also, since L has positive slope, there must exist at least one j ∈ [n] such that uj = (pushL (u))j . Combining these, we have vj ≤ (pushL (v))j < (pushL (u))j = uj , which contradicts u  v. Thus, the claim holds. Proof of Property 6(a) Suppose there exists y ∈ SA (u) ∩ SB (v). By definition, yi > vi ≥ ui for all i ∈ / B; yi > ui implies that i ∈ / A. Thus, i ∈ / B implies i ∈ / A, and the contrapositive must also be true, j ∈ A implies j ∈ B. Note that the converse is not necessarily true; one could have A = B = {1} but u1 < v1 , so that SA (u) ∩ SB (v) = ∅.

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Fig. 7 The double bar of u with respect to the line L is the value c4 ∈ C. In this case we also have c¯¯3L = u¯¯ L = c4 2

4

= =

1

3

Proof of Property 6(b) Indeed, pushL (u) = pushL (v) implies that SA (u) ∩ SB (v) = ∅. Now, suppose SA (u) ∩ SB (v) = ∅. By Property 6(a), this implies that A ⊆ B, and ui = vi for i ∈ A. Now, since pushL (u) and pushL (v) both belong to L, a line with positive slope, either pushL (u) = pushL (v) or pushL (u)j < pushL (v)j for all j ∈ [n]. And, since pushL (u) ∈ SA (u) and pushL (v) ∈ SA (v), pushL (u)j = uj = vj = pushL (v)j for all j ∈ A. Therefore, pushL (u) = pushL (v).   Recall the notations of Theorem 1 where a single bar on some value u ∈ Rn for which {u ∈ C|u  u} is non-empty denotes the greatest value in C less than or equal to that value: u¯ := max{u ∈ C|u  u}. We also introduce a double-bar notation that depends on a given line L with positive slope (see also Fig. 7): u¯¯ L := max{u ∈ C|u  u and SL (u) ∩ SL (u ) = ∅}. Lemma 3 For every u ∈ C, it holds that pushL (u) = u¯¯ L . Proof First we note that for all u ∈ C, using notation from Theorem 1, pushL (u) = max{u ∈ C|u  pushL (u)} = max{u ∈ C|u  u  pushL (u)}, as u ∈ C, and u  pushL (u). Note that u  u implies pushL (u)  pushL (u ), and u  pushL (u) implies that pushL (u )  pushL (pushL (u)) = pushL (u). So, we may write

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pushL (u) = max{u ∈ C|u  u and pushL (u) = pushL (u )}. L (u) By definition, AL u and Au are the unique subsets such that pushL (u) ∈ SAL u and pushL (u ) ∈ SAL (u ). Hence, u  u and pushL (u) = pushL (u ) if and only if u

SALu (u) ∩ SAL (u ) = ∅. So finally we obtain u

pushL (u) = max{u ∈ C|u  u and SALu (u) ∩ SAL (u ) = ∅} u

= u¯¯ L . This proves the claim, recalling that the notation SL (u) is a shorthand for SALu (u).

 

Lemma 4 For all lines L with positive slope, and for all u ≺ v ∈ C, we have ρV (pushL (u), pushL (v)) = ρV (u¯¯ L , v¯¯ L ). Proof Using Theorem 1 and Lemma 3, we obtain ρV (pushL (u), pushL (v)) = ρV (pushL (u), pushL (v)) = ρV (u¯¯ L , v¯¯ L ).   Lemma 5 For all lines L with positive slope, and all u ∈ L, we have L

u¯ = (u) ¯ . L

L

Proof By definition of the double-bar notation, u¯  (u) ¯ and SL (u) ¯ ∩ SL ((u) ¯ ) = ∅. Hence, by Proposition 2.6(b),   L . ¯ = pushL (u) ¯ pushL (u) L

Additionally, Proposition 2.3 implies that (u) ¯

  pushL

 , and, by Propo(u) ¯ L

sition 2.5, we have pushL (u) ¯  pushL (u) because u¯  u. Moreover, as u ∈ L, pushL (u) = u. So finally we have L



L

u¯  (u) ¯  pushL (u) ¯ L

 = pushL (u) ¯  pushL (u) = u. L

As (u) ¯ ∈ C, the above ineqalities imply that u¯ = (u) ¯ by definition of u. ¯

 

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Our next goal is to prove that critical values of the discrete vector field on K determine points of the persistence diagram of the restriction along a line through the parameter space. Given such a line L, we may define pushL (C) = {pushL (c) | c ∈ C}. Note that, since C is finite, so is pushL (C). We can order the elements of pushL (C) as c1 , c2 , . . . , cm , with ci ≺ ci+1 . Proposition 3 Let L be a line with positive slope. Let pushL (C) = {c1 , c2 , . . . , cm } be increasingly ordered. For all points u ≺ v on L, it holds that: (i) If u = ci and v = cj , then μV (u, v) = ρV (ci , cj −1 ) − ρV (ci−1 , cj −1 ) − ρV (ci , cj ) + ρV (ci−1 , cj ), and μV (u, v) = 0 if u or v not in C. (ii) If u = ci , then μV (u, ∞) = ρV (ci , cm ) − ρV (ci−1 , cm ), and μV (u, ∞) = 0 if u not in C. Proof From C = ∅ we get pushL (C) = ∅. Note that we may partition the line L by points of pushL (C). For each ci ∈ pushL (C), we have ci = pushL (ci ). i Indeed, by the bar notation, d  ci  ci for all d ∈ push−1 L (c ). Thus,

ci = pushL (d)  pushL (ci )  pushL (ci ) = ci . We first consider the case when u ∈ L and u ≺ c1 . In this case, μV (u, v) = 0 for all v  u, and μV (u, ∞) = 0. Indeed, we can take e " 0 small enough so that, for all 0 ≺ e  e, we have u − e ≺ u ≺ u + e ≺ c1 . Hence, Theorem 2 ρV (u − e , v) = ρV (u + e , v) = 0 for all v  u. We now consider the case of a point u ∈ L such that c1  u. Let ci be the maximal element in pushL (C) such that ci  u. In this case we claim that u = ci . Indeed, suppose not. Since ci  u, we must have ci  u. Therefore, ci = pushL (ci )  pushL (u)  pushL (u) = u.

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Since pushL (u) is in pushL (C) and ci is maximal in pushL (C) such that ci  u (by assumption), we must have that ci = pushL (u). But then u  ci ,with u ∈ C, implying u  ci by the bar notation. So, it must be that u = ci . Now, we are ready to prove the first statement in the case c1  u. First, suppose that u is not in pushL (C). Then there is a maximal element in pushL (C) such that ci ≺ u. If i = m, there exists e " 0 such that ci ≺ u − e ≺ u ≺ u + e ≺ ci+1 ; if i = m, then there exists e " 0 such that cm ≺ u − e ≺ u ≺ u + e, and the above inequalities hold for all 0 ≺ e  e. Moreover, by the above claim, we have that for all such e , u − e = u + e = ci . Therefore, μV (u, v) =

min

e"0 u+e≺v−e

ρV (u + e, v − e) − ρV (u − e, v − e)+ − ρV (u + e, v + e) + ρV (u − e, v + e)

=

min

e"0 u+e≺v−e

ρV (u + e, v − e) − ρV (u − e, v − e)+ − ρV (u + e, v + e) + ρV (u − e, v + e)

=

min

e"0 u+e≺v−e

ρV (ci , v − e) − ρV (ci , v − e)+ − ρV (ci , v + e) + ρV (ci , v + e)

= 0. Similarly, if v is not in pushL (C), we obtain μV (u, v) = 0. Now, if cj = v and ci = u for ci , cj ∈ pushL (C), then we can find e " 0 small enough such that both ci−1 ≺ ci − e ≺ ci ≺ ci + e ≺ ci+1 and cj −1 ≺ cj − e ≺ cj ≺ cj + e ≺ cj +1 (note that if j = m, then the second set of inequalities does not have the final “≺ cj +1 ” term).

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Additionally, for all 0 ≺ e ≺ e: • • • •

cj − e = cj −1 , cj + e = cj , ci − e = ci−1 , and ci + e = ci . Using these, we find that: μV (u, v) =

min

e"0 ci + e ≺ cj − e

ρV (ci + e, cj − e) − ρV (ci − e, cj − e)+ − ρV (ci + e, cj + e) + ρV (ci − e, cj + e)

=

min

e"0 ci + e ≺ cj − e

ρV (ci + e, cj − e) − ρV (ci − e, cj − e)+

− ρV (ci + e, cj + e) + ρV (ci − e, cj + e) =

min

e"0 ci + e ≺ cj − e

ρV (ci , cj −1 ) − ρV (ci−1 , cj −1 ) − ρV (ci , cj ) + ρV (ci−1 , cj )

= ρV (ci , cj −1 ) − ρV (ci−1 , cj −1 ) − ρV (ci , cj ) + ρV (ci−1 , cj ) = ρV (ci , cj −1 ) − ρV (ci−1 , cj −1 ) − ρV (ci , cj ) + ρV (ci−1 , cj ).

To prove the second statement, we first suppose that u is not in pushL (C), and that ci is the maximal element in pushL (C) such that ci ≺ u. Then, as in the proof of the first statement, we can find 0 ≺ e such that, for all 0 ≺ e  e, u − e = u + e = ci . Thus, μV (u, ∞) = min ρV (u + e, v) − ρV (u − e, v) e"0 vu

= min ρV (u + e, v) − ρV (u − e, v) e"0 vu

= min ρV (ci , v) − ρV (ci , v) e"0 vu

= 0. If u = ci for some ci ∈ pushL (C), then we can find 0 ≺ e such that, for all 0 ≺ e  e, u − e = ci−1 and u + e = ci . We also note that for all v ∈ L such that cm ≺ v, v = cm . Thus,

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μV (u, ∞) = min ρV (u + e, v) − ρV (u − e, v) e"0 vu

= min ρV (u + e, v) − ρV (u − e, v) e"0 vu

= min ρV (ci−1 , v) − ρV (ci , v) vu

= min ρV (ci−1 , v) − ρV (ci , v) vu

= min ρV (ci−1 , v) − ρV (ci , v). vu

= min ρV (ci−1 , cm ) − ρV (ci , cm ) vu

= ρV (ci−1 , cm ) − ρV (ci , cm ) = ρV (ci−1 , cm ) − ρV (ci , cm ).  

4.3 Grouping Fibers of Persistence Spaces by Equivalence We now use critical values to partition the set of all lines of Rn into equivalence classes, as illustrated in Fig. 8, such that the persistence diagrams associated to lines in the same class are easily obtainable from each other by a bijective correspondence. Definition 14 (Reciprocal Position) Two lines L, L ⊆ Rn with positive slope are said to have the same reciprocal position with respect to u if and only if pushL (u) and pushL (u) belong to the same face of ∂S+ (u). Given a non-empty subset U of Rn , we write L ∼U L , if L and L have the same reciprocal position with respect to u for all u ∈ U . Example 6 Figure 8 shows two examples of the equivalence classes of lines yielded by the set C of Example 3. Lines with the same reciprocal position with respect to C are characterized by the property of hitting the same face of the positive cone of u for each u ∈ C: L Lemma 6 L ∼C L if and only if AL u = Au for all values u ∈ C.

Proof Recall that SL (u) = SALu (u) and AL u is the unique non-empty subset of [n] such that pushL (u) = L ∩ SALu (u). Therefore, pushL (u) ∈ SALu (u) and pushL (u) ∈ SAL (u). u

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Fig. 8 The dashed lines represent the boundaries of the positive cones of the values in C. Here L and L have the same reciprocal position with respect to C but L does not

⬙

2

4

1

3

By definition, L ∼C L if and only if, for all u ∈ C, pushL (u) and pushL (u) belong to the same face of ∂S+ (u), i.e. SALu (u) = SAL (u) u

for all u ∈ C, which, by the disjointness of faces of ∂S+ (u), means exactly that L AL   u = Au for all u ∈ C. Proposition 4 Given a non-empty subset U in R2 , L ∼U L defines an equivalence relation on the set of lines with positive slope. Proof We can define AL (U ) = {AL u }u∈U . By Lemma 6, L ∼U L if and only if AL (U ) = AL (U ). Using this equivalent definition of L ∼U L , it is clear that ∼U is reflexive, transitive, and symmetric, and therefore an equivalence relation.   The rank invariant on equivalent lines satisfies the following condition. Proposition 5 If u ≺ v ∈ L, with {u ∈ C|u  u} non-empty, and L ∼C L , then it holds that ¯ pushL (v)). ¯ ρV (u, v) = ρV (pushL (u), Proof Since u  v, it follows from Theorem 1 that ρV (u, v) = ρV (u, v). L

L

¯ and v¯ = (v) ¯ by Lemma 5, implying As u, v ∈ L, u¯ = (u) 

L

L

¯ , (v) ¯ ρV (u, v) = ρV (u)

 .

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As u, v ∈ C, and u  v implies u  v, from Lemma 4 we get L

L

¯ , (v) ¯ ) = ρV (pushL (u), pushL (v)) . ρV ((u)   Lemma 7 If L ∼C L , then pushL (u) = pushL (u) for all values u ∈ C. Proof By Lemma 3, pushL (u) = max{u ∈ C|u  u and SL (u) ∩ SL (u ) = ∅}, pushL (u) = max{u ∈ C|u  u and SL (u) ∩ SL (u ) = ∅}. So the claim follows because SL (u) = SL (u) and SL (u ) = SL (u ) by Lemma 6.

 

Lemma 8 If L ∼C L , then pushL (u) ¯ = u¯ for all values u ∈ L. Proof It follows by successively applying Lemmas 7, 3, and 5.

 

Lemma 9 If L ∼C L , then the correspondence σ : pushL (C) → pushL (C) defined by σ (d) = pushL (push−1 L (d)) for all d ∈ pushL (C) is an order preserving ¯ with d¯ = max{u ∈ C : u  d} bijective function. In particular, σ (d) = pushL (d) as usual for all d ∈ pushL (C). Proof Let d ∈ pushL (C). Then there exists at least one c ∈ C such that d = pushL (c). We first show that, for all c ∈ push−1 L (d), ¯ . pushL (c) = pushL (d)

(4)

By definition of double bar, we have that SL (c¯¯L ) ∩ SL (c) = ∅. So, as L ∼C L implies that SL (c¯¯L ) = SL (c¯¯L ) and SL (c) = SL (c) by Lemma 6. This means that SL (c¯¯L ) ∩ SL (c) = ∅ . Therefore, by Proposition 2.6(b), we have that pushL (c¯¯L ) = pushL (c) . Now note that d¯ = pushL (c) = c¯¯L by Lemma 3, and so ¯ = pushL (c¯¯L ) = pushL (c) . pushL (d) Equality (4) implies that σ is a well defined function because d¯ is unique by ¯ is also unique by Property 1 of Proposition 2. Lemma 1 and pushL (d)

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Now for any d ∈ pushL (C), there exists (at least one) c ∈ C such that d = pushL (c ) and σ (pushL (c )) = pushL (c ), showing that σ : pushL (C) → pushL (C) is surjective. We can analogously define a function τ : pushL (C) → pushL (C) by setting τ (d ) = pushL (push−1 L (d )) for all d ∈ pushL (C). Now we prove that σ and τ are bijective by showing that σ is the inverse of τ : For all d ∈ pushL (C), −1 σ (τ (d )) = σ (pushL (push−1 L (d ))) = pushL (pushL (d )) = d ,

and, similarly, for all d ∈ pushL (C), −1 τ (σ (d)) = τ (pushL (push−1 L (d))) = pushL (pushL (d)) = d .

Finally, we show that σ is order-preserving: Assume that d, e ∈ pushL (C) with d  e. Then d¯ ∈ C and d¯  d  e by definition of bar, and therefore d¯  e¯ = max{u ∈ C|u  e} . Hence, by Proposition 2.5, we have that ¯  pushL (e) ¯ = σ (e) , σ (d) = pushL (d) as required.   Lemma 10 Let L ∼C L be equivalent lines with positive slope. For any ∈ 1 2 m i i pushL (C) = {c , c , . . . , c } increasingly ordered, let d = σ (c ) with σ as in Lemma 9. Then, ci

μV (d i , d j ) = μV (ci , cj ) for ci ≺ cj , and μV (d i , ∞) = μV (ci , ∞). Proof Assuming ci ≺ cj ∈ pushL (C), by Proposition 3(i), μV (ci , cj ) = ρV (ci , cj −1 ) − ρV (ci−1 , cj −1 ) − ρV (ci , cj ) + ρV (ci−1 , cj ). On the other hand, setting d h = σ (ch ) for h ∈ {i, i − 1} and d k = σ (ck ) for k ∈ {j, j − 1} yields

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ρV (ch , ck ) = ρV (ch , ck ) (by Theorem 1 ) = ρV (pushL (ch ), pushL (ck ))

(by Lemma 8)

= ρV (pushL (ch ), pushL (ck ))

(by Theorem 1)

= ρV (d h , d k )

(by definition).

Therefore, μV (ci , cj ) = ρV (d i , d j −1 )−ρV (d i−1 , d j −1 )−ρV (d i , d j )+ρV (d i−1 , d j ) = μV (d i , d j )

with the second equality holding by Proposition 3(i) applied to L . Analogously, by Proposition 3(ii), we can see that μV (ci , ∞) = ρV (ci , cm ) − ρV (ci−1 , cm ) = ρV (d i , d m ) − ρV (d i−1 , d m ) = μV (d i , ∞).

  Theorem 2 Let L ∼C L be equivalent lines with positive slope, parametrized by L : u = ms + u0 and L : u = m s + u0 , respectively. Let dgm(VL ) and dgm(VL ) be the persistence diagrams of the restrictions of V to L and L , respectively. Then, there exists a multi-bijection (that is, a bijection between sets of points with multiplicities), γ : dgm(VL ) → dgm(VL ) such that, for all (s, t) ∈ dgm(VL ), γ (s, t) = (s , t ) ∈ dgm(VL ) with • s ∈ R such that m s + u0 = pushL (ms + u0 ), and • t ∈ R such that m t + u0 = pushL (mt + u0 ) if t ∈ R, while t = ∞ if t = ∞. Proof For (s, t) ∈ dgm(VL ) with s < t < ∞, we have μVL (s, t) = μV (u, v) with u = ms + u0 and v = ms + u0 in L. By Proposition 3, we can see that (u, v) will be of the form (ci , cj ) where ci ≺ cj ∈ pushL (C). By Lemma 10, the bijection σ : pushL (C) → pushL (C) such that σ (ci ) = d i with d i = pushL (ci ) for 1 ≤ i ≤ m satisfies μV (ci , cj ) = μV (d i , d j ) and μV (ci , ∞) = μV (d i , ∞). The parametrization of L uniquely determines values s < t ∈ R such that d i = m s + u0 and d j = m t + u0 . Since μV (s , t ) = μV (d i , d j ) = μV (ci , cj ) = μVL (s, t) L and μV (s , ∞) = μV (d i , ∞) = μV (ci , ∞) = μVL (s, ∞), L

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we see that σ induces the desired multi-bijection γ : dgm(VL ) → dgm(VL ) (s, t) → (s , t ).

 

5 Conclusions and Discussion Based on the results of this paper, we can derive a method to fiber the rank invariant of multi-parameter persistence along lines of positive slope chosen by a user. This way, the persistence space of an n-parameter persistence module is sliced into persistence diagrams. The method consists of an offline preprocessing step, where we compute the representative lines and their persistence diagrams and an interactive step where we can compute the rank invariant for any chosen line in real time. Starting from a discrete gradient vector field V consistent with the multi-filtration at hand as input data, the offline step requires: • computing the set C of entrance values of the critical cells of V, • taking the closure C of C with respect to their least upper bound, • partitioning the set of lines with positive slope by the equivalence relation ∼C and picking a representative line from each equivalence class (for example, following the procedure shown in Appendix), and • storing the persistence diagrams of the restriction of filtration to each representative line. Having pre-computed these data, the interactive part: • takes as input from the user a line L with positive slope, • detects the representative line L0 of its equivalence class with respect to ∼C , and • computes the persistence diagram relative to L by pushing onto L the bars of the persistence diagram relative to L0 . The correctness of the method is guaranteed by Theorem 2. The method requires additional routines from computational geometry in order to efficiently detect representatives of equivalence classes of lines, computing the bars of points, and pushing points onto lines. In Appendix, we propose a method applied to 2-parameter persistence modules to find representative lines for the equivalence classes defined in Sect. 4. The method is based on a bijection between segments linking points of C and lines cutting C in two non-empty subsets. Moreover, the method requires routines for the persistence diagram computation such as those implemented in [16, 22], or [25]. It is worth noticing that, based on the tests using [22] presented in [23], it is more efficient to compute persistence diagrams of the persistence module restricted to lines starting with the Morse complex obtained from V rather than directly from the original cell complex. In

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other words, the gradient vector field V is used twice: its critical cells allow us both to determine representative lines and to reduce persistence computation In conclusion, the presented method allows for: computational efficiency, by requiring only linear asymptotic time complexity to obtain the input gradient field, e.g. with the algorithm of [23]; theoretical improvements, by permitting any number of parameters; data analysis and understanding advantages, by making explicit the correspondence between persistence features and critical cells. For future work we plan to extend the algorithm to a larger number of parameters. The rank invariant fibering along lines is central also in the definition of the matching distance, a metric on rank invariants of multi-parameter persistence modules [9, 10] ensuring their stability. In [17], the exact computation of the matching distance is achieved for at most two parameters in polynomial runtime in the number of simplices requiring O(m11 ) runtime and O(m4 ) memory, with m denoting the number of simplices. Motivated by the practical need of decreasing the number of operations and increasing the number of allowed parameters, our next project will be to extend equivalence classes of lines to pairs of persistence modules and to apply the method presented here to the matching distance exact computation in any number of parameters.

Appendix: Enumerating Equivalence Classes of Lines In this appendix we describe an algorithm to enumerate the equivalence classes of lines with positive slope with respect to a set C in R2 . The results of this section will be initially presented in a slightly more general form. Consider a set P of n points in the real plane, not necessarily in general position. We say that any division of P into two non-empty subsets via a line not passing through any point of P is a cut in the plane. Note that a cut determines an equivalence class of lines if and only if the cut can be realized by a line of positive slope.

Cuts Are Determined by Primitive Pairs A key observation is that we can associate a certain pair of points in P to any given cut. We say that a pair of distinct points in P is primitive if no other point of P lies in the interior of the segment joining the two points. Proposition 6 There is a bijection between the primitive pairs of points in P and the cuts of P . Proof We give an explicit bijection, as follows: given a primitive pair of points in P , we can rotate the line joining the two points clockwise by a small amount, around the midpoint of the segment. It is possible to choose a small enough angle so that this rotated line does not pass through any point of P . Moreover, the two original points lie on opposite sides of this line, and so this line defines a cut.

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Fig. 9 Choosing the initial center of rotation a

v

Let us now prove that to every cut we can associate a pair of points as above. Let {A, B} be a cut, where A and B are non-empty disjoint subsets of P with P = A∪B. We can represent this cut by a dividing line L. Now fix a direction v orthogonal to L. Translate L along v until it hits at least one point of P for the first time. Let H be the set of points that L hits. If H has only one element, set a to be the unique element of H . If H has more than one element, then set a to be the one with the following property: for any other point a1 ∈ H , → is positive. In other words, if we the dot product of the vector v with the vector − aa 1 consider v to be the positive direction along the x-axis, then a is the point with the minimum y-coordinate (see Fig. 9). Note that until the line hits a, it still defines the same cut. Without loss of generality, assume that a ∈ A. Taking a to be the centre of rotation, rotate the line L counter-clockwise until it hits another point of P . Note that the line defines a cut of all points of P other than those on this line, and it is the same cut except for these points. If the line simultaneously hits multiple points, then exactly one of the following is true of these points: they all points lie in A, they all lie in B, or they include both points in A and in B. In the third case, a lies between the rest of the points in A and those in B on the line. Now we have an algorithm to generate the primitive pair a, b for the cut L, as follows. 1. If at least one point hit lies in B, set b to be the point of B hit by the line that is closest to a, as shown in Fig. 10 (Left). In this case, rotating the line ab clockwise around the midpoint of the segment ab gets us back the original cut. This is because the line yields the original cut excluding the points of P that lie on it, and therefore rotating it slightly clockwise restores these points to the correct subsets (either A or B). We are done. 2. If all points hit lie in A, reset the center of rotation to be the point of A hit by the line that is farthest away from a as in Fig. 10 (Right). Rename this point as a. Continue to rotate the line counter-clockwise around the new centre of rotation until it hits another point of P , and repeat the above steps until the algorithm terminates.

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new center of rotation

Fig. 10 Left: Rotating L around a will hit at least one point in B. Right: Rotating L around a hits only points in A

The algorithm terminates when the rotating line hits a point of B. This always happens, for the following reason. As the line makes a full rotation around the convex hull of A, it sweeps through the entire plane except for the convex hull of A, while B is a non-empty set outside the convex hull of A.   Remark 4 There is a similar, counter-clockwise, bijection between cuts and primitive pairs: rotate a line segment joining a primitive pair by a small amount counter-clockwise. Thus each cut corresponds to a “clockwise primitive pair”, and another “counter-clockwise primitive pair”. These pairs are distinct unless all points of P lie on a single line. Correspondingly, there is a counter-clockwise version of the algorithm explained in the previous proposition (see Fig. 11). We use this fact in the next section.

Achieving Positive Slope We have determined that every primitive pair of points in P determines a cut by rotating the line segment through this pair slightly clockwise about the midpoint of the segment. This is the clockwise primitive pair associated to this cut. Similarly, every primitive pair of points determines another cut by rotating the line segment through this pair slightly counter-clockwise around the midpoint of the segment. This is the counter-clockwise primitive pair associated to this cut. These are shown in Fig. 11. The clockwise and counter-clockwise primitive pairs associated to any cut can be found by the algorithm in the previous section, and its variant explained in Remark 4 respectively. We now tackle the problem of determining whether such a cut can be defined by a line of positive slope. The answer is given by Algorithm 1. We will need some preparatory lemmas to prove the correctness of the algorithm. We use the same setup for all of these lemmas, as follows. Let (a1 , b1 ) and (a2 , b2 ) be the clockwise and anti-clockwise primitive pairs respectively for a cut {A, B}. It may be the case that either a1 = a2 or b1 = b2 , but we suppose that not both are true. This supposition is true unless all points in P lie on a line.

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2

1

1

1

2

1

=

2

2

Fig. 11 The clockwise and counter-clockwise primitive pairs associated to the cut {A, B}. The pair (a1 , b1 ) is the clockwise primitive pair and (a2 , b2 ) is the counter-clockwise primitive pair

Algorithm 1 An algorithm to determine whether a cut can be obtained by a line of positive slope 1: Let (a1 , b1 ) and (a2 , b2 ) be the clockwise and anti-clockwise primitive pairs respectively for a cut {A, B}. 2: Let m1 and m2 be the slopes of the lines a1 b1 and a2 b2 respectively. 3: if a1 = b1 and a2 = b2 then 4: return true 5: end if 6: if 0 < m1 ≤ ∞ then 7: return true 8: else if 0 ≤ m2 < ∞ then 9: return true 10: else 11: We have −∞ < m1 ≤ 0 and −∞ ≤ m2 < 0. 12: if m1 < m2 then 13: return true 14: else 15: return false 16: end if 17: end if

Let p be the intersection point of the segments a1 b1 and a2 b2 . Note that p must exist, for the following reason. If all four points are distinct, then a2 and b2 lie on opposite sides of the line a1 b1 , and so the segments intersect somewhere in their interiors. Otherwise, if a1 = a2 (resp. b1 = b2 ), then p = a1 = a2 (resp. p = b1 = b2 ).

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Lemma 11 Suppose that the four points a1 , a2 , b1 , b2 are distinct. If we start at the line a1 b1 and rotate clockwise around p until we hit a2 b2 , every intermediate line defines the cut {A, B}. Proof Let L1 be the line a1 b1 and L2 be the line a2 b2 . Note that each of the two lines has a well-defined “A” side and a well-defined “B” side: the side of the line on which the remaining points of A lie is the “A” side, and the side on which the remaining points of B lie is the “B” side. The lines L1 and L2 cut up the plane into four open cones. We can label these cones as CAA , CAB , CBB , CBA , where for example CBA is the intersection of the “B” side of L1 with the “A” side of L2 . The cones CAB and CBA contain no points of P . This is precisely because these cones lie on the “A” side of one of the lines and on the “B” side of the other. Moreover, their closures only intersect at the point p. It is clear that any line rotated clockwise around p starting from L1 until we hit L2 , excluding L1 and L2 itself, lies completely in the set CAB ∪ CBA ∪ {p} and has a1 on its “A” side and b1 on its “B” side. So any such line continues to define the same cut {A, B}.   Lemma 12 Suppose that a1 = a2 . If we start at the line a1 b1 and rotate clockwise around a1 = a2 until we hit a1 b2 keeping track of the trace of b1 under this rotation, then every intermediate line, except for a1 b2 itself, defines the cut {A, B} after a sufficiently small clockwise rotation about the midpoint between a1 and the trace of b1 . Proof As in the proof of Lemma 11, let L1 be the line a1 b1 and L2 be the line a1 b2 , and notice that L1 and L2 cut up the plane into four open cones CAA , CAB , CBB , CBA . As in the proof of Lemma 11, the cones CAB and CBA contain no points of P and their closures only intersect in the point of intersection of L1 and L2 , which is a1 in this case. It is clear that any line rotated clockwise around a1 starting from L1 until we hit L2 , excluding L1 and L2 itself, lies completely in the set CAB ∪ CBA ∪ {a1 } and has b1 on its “B” side, so defines the cut {A, B} excluding a1 . A sufficiently small clockwise rotation of such a line about the midpoint between a1 and the trace of b1 moves a to its “A” side without crossing any other points in P , so the resulting line indeed defines the cut {A, B}.   Lemma 13 Let S be the set of possible slopes of lines obtained by starting at the line a1 b1 and rotating clockwise through p until we hit the line a2 b2 , excluding the slopes of the lines a1 b1 and a2 b2 themselves. Then the slope of any line L that defines the same cut {A, B} lies in S. Proof For the proof of this lemma, the four points a1 , a2 , b1 , b2 need not all be distinct. First note that if L is any line defining the cut {A, B}, then it intersects the interiors of the segments a1 b2 and a2 b1 . This is precisely because {a1 , a2 } and {b1 , b2 } lie on opposite sides of L.

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Now suppose L is any line that intersects the interiors of the segments a1 b2 and a2 b1 . These are two opposite sides of the (possibly degenerate) quadrilateral a1 a2 b1 b2 . Therefore L must also intersect the interiors of the diagonals of this (possibly degenerate) quadrilateral, namely the segments a1 b1 and a2 b2 . In particular, because L intersects both a1 b1 and a2 b2 , it cannot have slope equal to either m1 or m2 . The set of possible slopes of lines in the plane can be identified with the real projective line, by noting that slopes can lie between [−∞, ∞] with ∞ = −∞. We now have a continuous map a1 b2 × a2 b1 → RP1 ,

(5)

defined by mapping an ordered pair of points to the slope of the line joining the two points. By the previous argument, the image of this map lies in RP1 \ {m1 , m2 }, which has two connected components. Since the domain is connected, the image of the map must lie in exactly one of the connected components. The set S is precisely one of the two connected components: we start at m1 , rotate clockwise until we hit m2 . The other connected component is obtained by rotating counter-clockwise starting at m1 until we hit m2 . We already know by either Lemma 11 or Lemma 12 (depending on whether or not the four points a1 , a2 , b1 , b2 are distinct) that there are points in the image of the map in Equation 5 that lie in S. By connectedness, all lines that intersect the interiors of a1 b2 and a2 b1 have slopes that lie in S. In particular, all lines that define the same cut {A, B} have slopes that lie in S.   Now we can prove the correctness of the algorithm. Proposition 7 Algorithm 1 correctly determines whether a cut can be obtained by a line of positive slope. Proof Recall that (a1 , b1 ) and (a2 , b2 ) are the clockwise and anti-clockwise primitive pairs respectively for a cut {A, B}. Recall that m1 and m2 are the slopes of the lines a1 b1 and a2 b2 respectively. We treat each step of the algorithm in order. First, a1 = a2 and b1 = b2 if and only if all points of P lie on a single line. In this case it is clearly always possible to achieve any cut by a line of positive slope. Now assume that not all points of P lie on a single line, which implies that either a1 = a2 or b1 = b2 . This is the setting of the previous lemmas. If 0 < m1 ≤ ∞, then a small clockwise rotation of the line a1 b1 has positive slope. Since the rotated line determines the desired cut {A, B}, we are done. Similarly, if 0 ≤ m2 < ∞, then a small counter-clockwise rotation of the line a2 b2 has positive slope. Since the rotated line determines the desired cut {A, B}, we are done. Now suppose that −∞ < m1 ≤ 0 and −∞ ≤ m2 < 0. Let p be the intersection point of a1 b1 and a2 b2 . Let S be the set of possible slopes of lines obtained by starting at the line a1 b1 and rotating clockwise through p until we hit the line a2 b2 .

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Suppose first that −∞ < m1 < m2 < 0. As we sweep clockwise from a1 b1 , we begin at m1 , decrease slope until we hit a vertical line with slope −∞ = ∞, and then decrease again from ∞ until we cross 0 down to m2 . In particular, at least one of the intermediate lines has positive slope. If the points a1 , a2 , b1 , b2 are all distinct, then by Lemma 11 we have a line of positive slope that gives the cut {A, B}. If two of the four points are equal, then Lemma 12 states that a sufficiently small clockwise rotation of one of the intermediate lines (which will also have positive slope) gives the cut {A, B}. Now suppose that m1 ≥ m2 . In this case, the set S consists only of negative numbers: these are the slopes starting from m1 and decreasing down to m2 . By Lemma 13, we see that there is no line of positive slope that defines this cut.  

Cuts Through a Fixed Point In order to address equivalence classes of lines that pass through a given point c of C, we now say that a division of a non-empty set of points P of the plane, with c ∈ / P , into two disjoint subsets A and B, of which at most one can be empty, via a line passing through the given point c and disjoint from P , is a c-cut of P . Reciprocally, we say that a line through c and a point of P is a c-primitive line of P. Proposition 8 There is a bijection between the c-primitive lines of P and the c-cuts of P . Proof We construct an explicit bijection, called the clockwise bijection, as follows. Given a c-primitive line of P , we can rotate this line clockwise by a small amount, around c, so that this rotated line does not pass through any point of P . This line defines a c-cut. Vice versa, with every c-cut of P we can associate a c-primitive line by rotating the line realizing the c-cut counter-clockwise until it hits some point of P , which exists because P is non-empty. Note that by a completely symmetric argument we also have a counter-clockwise bijection.   We have determined that any c-cut L of P can be determined by rotating both a c-primitive line L1 clockwise and a c-primitive line L2 counter-clockwise. Let m1 and m2 be the slopes of L1 and L2 , respectively. In the case m1 = m2 , because L1 and L2 both pass though c, we have L1 = L2 . In this case all points of P belong to L1 and positive slope can be achieved by rotation around c for every value of m1 . In the case when m1 = m2 , Algorithm 1 applied to c-primitive lines instead of primitive pairs of points achieves the goal. Indeed, again, L1 and L2 have a well defined A-side and B-side, and L − {c} is contained in CAB ∪ CBA . Then the argument follows as in Proposition 7.

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Retrieving Representatives Lines With reference to the equivalence relation on lines defined by their reciprocal position with respect to the set C of critical values and their least upper bounds as given in Definition 14, our goal is to retrieve a representative line with positive slope for each possible equivalence class. Recall from Lemma 6 that two lines belong to the same equivalence class with respect to C if and only if they hit the positive cone of each point of C at the same facet. There are three possible situations for lines in the same class: (i) There is only one line in the equivalence class passing through two points c and c of C; (ii) the lines in the considered equivalence class contain exactly one point of C, say c; (iii) the lines in the considered equivalence class do not contain any point of C. Case (i) can be easily solved by taking lines through all possible pairs of distinct points c and c in C, provided that c  c , paying attention to not taking the same line multiple times if there are more than two points on the same line. Case (ii). In this case, each such line partitions C − {c} into two subsets A and B. For each equivalence class of lines for which A and B are both empty, c is the only point of c, so there is only one such equivalence class and any line through c with positive slope is a representative of it. In the case when at least one between A and B are non-empty, we can obtain a representative line by applying Algorithm 1 as explained in the previous subsection. Note that since C is closed under least upper bound, the case where −∞ < m1 < m2 < 0 (line 12 in Algorithm 1) cannot occur. Case (iii). In this case, each such line partitions C into two subsets A and B. For each equivalence class of lines for which A and B are both non-empty, we can obtain a representative by applying the algorithm presented in section “Achieving Positive Slope”. For the case when either A or B is empty, the other one is necessarily equal to C. There are exactly two such equivalence classes of lines depending on whether the lines hit all the positive cones of points of C at their horizontal or vertical facets. As a representative of the first class, we can take a line parallel to the diagonal of R2 passing to a point with abscissa greater than the maximum abscissa of points of C, and ordinate smaller than the minimum ordinate of points of C. Symmetrically, as a representative of the second class, we can take a line parallel to the diagonal of R2 passing to a point with abscissa smaller than the minimum abscissa of points of C, and ordinate greater than the maximum ordinate of points of C. Acknowledgments This research began at the 2019 Women in Computational Topology (WinCompTop) workshop in Canberra. We thank Ashleigh Thomas and Elizabeth Stephenson for joining us in the initial discussions during that week. The results of this paper have been presented within the Summer 2020 AATRN Seminars. We thank Anand Deopurkar, Anthony Licata, and Nicholas Proudfoot for helpful conversations related to Appendix.

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References 1. Allili, M., Kaczynski, T., Landi, C.: Reducing complexes in multidimensional persistent homology theory. J. Symb. Comput. 78, 61–75 (2017) 2. Allili, M., Kaczynski, T., Landi, C., Masoni, F.: Acyclic partial matchings for multidimensional persistence: algorithm and combinatorial interpretation. J. Math. Imaging Vis. 61(2), 174–192 (2019) 3. Bendich, P., Marron, J.S., Miller, E., Pieloch, A., Skwerer, S.: Persistent homology analysis of brain artery trees. Ann. Appl. Stat. 10(1), 198–218 (2016) 4. Bhattacharya, S., Ghrist, R., Kumar, V.: Persistent homology for path planning in uncertain environments. IEEE Trans. Robot. 31(3), 578–590 (2015) 5. Biasotti, S., Cerri, A., Giorgi, D., Spagnuolo, M.: PHOG: Photometric and geometric functions for textured shape retrieval. Comput. Graph. Forum 32(5), 13–22 (2013) 6. Carlsson, G.: Topology and data. Bull. Amer. Math. Soc. (N.S.) 46(2), 255–308 (2009) 7. Carlsson, G., Singh, G., Zomorodian, A.: Computing multidimensional persistence. In: Algorithms and Computation. Lect. Notes Comput. Sci., vol. 5878, pp. 730–739. Springer, Berlin (2009) 8. Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. Discrete Comput. Geom. 42(1), 71–93 (2009) 9. Cerri, A., Di Fabio, B., Ferri, M., Frosini, P., Landi, C.: Betti numbers in multidimensional persistent homology are stable functions. Math. Methods Appl. Sci. 36(12), 1543–1557 (2013) 10. Cerri, A., Ethier, M., Frosini, P.: The coherent matching distance in 2D persistent homology. In: Computational Topology in Image Context. Lect. Notes Comput. Sci., vol. 9667, pp. 216–227. Springer, Berlin (2016) 11. Cerri, A., Landi, C.: Hausdorff stability of persistence spaces. Found. Comput. Math. 16(2), 343–367 (2016) 12. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007) 13. Dey, T.K., Xin, C.: Generalized persistence algorithm for decomposing multiparameter persistence modules. J. Appl. Comput. Topology (2022). https://doi.org/10.1007/s41468-02200087-5 14. Forman, R.: A user’s guide to discrete Morse theory. Sém. Lothar. Combin. 48, 1–35 (2002) 15. Green, S.B., Mintz, A., Xu, X., Cisewski-Kehe, J.: Topology of our cosmology with persistent homology. Chance 32(3), 6–13 (2019) 16. GUDHI library, geometry understanding in higher dimensions. http://gudhi.gforge.inria.fr/ 17. Kerber, M., Lesnick, M., Oudot, S.: Exact computation of the matching distance on 2parameter persistence modules. In: SoCG 2019. LIPIcs, vol. 129, pp. 46:1–46:15 (2019) 18. Landi, C.: The Rank Invariant Stability via Interleavings. Association for Women in Mathematics Series, vol. 13, pp. 1–10. Springer, Berlin (2018) 19. Lee, Y., Barthel, S.D., Dłotko, P., Moosavi, S.M., Hess, K., Smit, B.: Quantifying similarity of pore-geometry in nanoporous materials. Nat. Commun. 8(1) (2017) 20. Lesnick, M., Wright, M.: Interactive Visualization of 2-D Persistence Modules. arXiv:1512.00180v1 (2015) 21. Lesnick, M., Wright, M.: Computing minimal presentations and bigraded Betti numbers of 2-parameter persistent homology. arXiv:1902.05708 (2019) 22. Phat (persistent homology algorithm toolbox). https://github.com/blazs/phat 23. Scaramuccia, S., Iuricich, F., De Floriani, L., Landi, C.: Computing multiparameter persistent homology through a discrete Morse-based approach. Comput. Geom. 89, 101623 (2020) 24. Sinhuber, M., Ouellette, N.T.: Phase coexistence in insect swarms. Phys. Rev. Lett. 119, 178003 (2017) 25. TTK the topological toolkit. https://topology-tool-kit.github.io/ 26. Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2004)

Local Versus Global Distances for Zigzag and Multi-Parameter Persistence Modules Ellen Gasparovic, Maria Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, and Lori Ziegelmeier

Abstract In this paper, we establish explicit and broadly applicable relationships between persistence-based distances computed locally and globally. In particular, we show that the bottleneck distance and the Wasserstein distance between two zigzag persistence modules restricted to an interval is always bounded above by the distance between the unrestricted versions. While this result is not surprising, it could have potential practical implications. We give two related applications for metric graph distances, as well as an extension for the matching distance between multi-parameter persistence modules. We also prove a similar restriction inequality for the interleaving distance between two multi-parameter persistence modules.

E. Gasparovic () Department of Mathematics, Union College, Schenectady, NY, USA e-mail: [email protected] M. Gommel North Central College, Naperville, IL, USA e-mail: [email protected] E. Purvine Pacific Northwest National Laboratory, Seattle, WA, USA e-mail: [email protected] R. Sazdanovic North Carolina State University, Raleigh, NC, USA e-mail: [email protected] B. Wang School of Computing, University of Utah, Salt Lake City, UT, USA e-mail: [email protected] Y. Wang University of California, San Diego, CA, USA e-mail: [email protected] L. Ziegelmeier Macalester College, Saint Paul, MN, USA e-mail: [email protected] © The Author(s) and the Association for Women in Mathematics 2022 E. Gasparovic et al. (eds.), Research in Computational Topology 2, Association for Women in Mathematics Series 30, https://doi.org/10.1007/978-3-030-95519-9_3

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1 Introduction Assessing similarity or dissimilarity of complex objects is an important problem that is ubiquitous in science, engineering, and mathematics. The complexity of the input objects can make this problem quite difficult; for instance, comparing graphs or even trees via the edit-distance can be NP-hard [6]. With the recent development of applied and computational topology, a new paradigm for comparing complex objects is to first map them to easier-to-compare yet still meaningful topological summaries, and then compare the resulting topological summaries as a proxy. Indeed, one such topological summary is the persistence diagram obtained via persistent homology. By choosing appropriate filtrations, input objects are represented by so-called persistence modules. In this paper, we consider the more general zigzag persistence modules [7], which admit a unique decomposition up to isomorphism into certain elementary pieces (called indecomposable modules) that are intervals. The information encoded by these intervals can be combinatorially represented by its persistence diagram. Such a persistence diagram serves as a topological summary representation of the input object, enabling one to compare multiple input objects by comparing their associated persistence diagrams using metrics such as the bottleneck distance [12] or Wasserstein distance [14].

1.1 Motivation In practice, there are many applications where one might be interested in only computing a local or restricted summary. For a first example, consider the question of determining or approximating graph motif counts. A graph motif is a subgraph on a small number of vertices contained within a larger, more complex graph. Graph motifs have proven useful for characterizing networks in domains such as biology [23] and cyber security [19]. The standard problem of counting the number of small motifs or patterns within a graph is equivalent to the subgraph isomorphism problem, which is NP-complete. Since restricted persistence modules reveal information about the local structure of a space, we posit that the restricted modules for a metric graph (see Sect. 4) can be utilized in a similar manner to the way in which graph motifs are currently used, e.g., as inputs to classification algorithms or anomaly detection algorithms in time-varying data [19, 21]. For a second example, consider persistent local homology, which studies a multiscale notion of homology within a local neighborhood of the data relative to its boundary. It has applications in road network analysis [2], data visualization [24], graph reconstruction [1, 11], and clustering and stratification learning [3, 5]. Furthermore, persistent local homology extracts local geometric and topological information in data, which may then be used as input to machine learning algorithms [4]. In such applications, there is often a choice as to the size of local neighborhood that one should use, and a natural question to ask is how the local

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information relates to the global topological summaries. This paper aims to provide some answers to this question.

1.2 Our Contributions After a brief review of the relevant background information in Sect. 2, we show in Sect. 3 that the bottleneck distance and the Wasserstein distance between two zigzag persistence modules restricted over an interval of parameter values is always bounded from above by the distance between the unrestricted versions (Theorem 1) and state a corollary in the case of level set zigzag persistence (Corollary 2). As an example of how such a bound could be useful: imagine that one wishes to compare the persistence profiles of two very large data sets but finds that it is prohibitively computationally expensive. Instead, one could compute a restricted version of the bottleneck distance as an approximation to the global distance. As the interval size increases, the bottleneck (or Wasserstein) distance between the restricted versions approaches the distance for the global versions. In an opposite direction, our result means that the global distance between two modules does not “wash out” local information encoded in the same module. In Sect. 4, we discuss some further implications of our results, by establishing two results involving distance inequalities in the special case of metric graphs (Corollary 3 and Corollary 4), as well as by extending our result to the matching distance of multi-parameter persistence modules [20]. Finally, we close by providing a restriction inequality result (Theorem 2) on the interleaving distance between multi-parameter persistence modules. In this case, we are not operating on the level of persistence diagrams but rather on the modules themselves.

2 Background and Definitions Our treatment of zigzag persistence is brief; for more details, see [7] and [8]. A zigzag diagram of topological spaces X1 , X2 , . . . , Xn is a sequence X1 ↔ X2 ↔ · · · ↔ Xn where each bidirectional arrow between two topological spaces represents a continuous function mapping either forwards or backwards. Applying the p-th homology functor with coefficients in a field K yields a zigzag diagram of vector spaces Hp (X1 ; K) ↔ Hp (X2 ; K) ↔ · · · ↔ Hp (Xn ; K), known as a zigzag module, denoted as X, from which zigzag persistence may be computed. Since the maps in a zigzag diagram are allowed to go in either direction, the resulting zigzag module is the most general form of 1-dimensional persistence.

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A zigzag module decomposes into intervals X ∼ =



I[bj , dj ], where each I[bj , dj ]

j ∈J

is defined as 0 ←→ · · · ←→ 0 ←→ K ←→ · · · ←→ K ←→ 0 · · · ←→ 0 with nonzero values in the range [bj , dj ]. We will use DgX to denote the resulting persistence diagram of a fixed homology dimension p. Typically, a persistence diagram is considered to be a set of points {(b, d)} for which b < d. Proposition 2.12 of [7] implies that restricting the module X to the range [r1 , r2 ] (denoted X[r1 , r2 ]) yields a decomposition as the direct sum of the intervals in X restricted to [r1 , r2 ]; that is,  I([bj , dj ] ∩ [r1 , r2 ]). (1) X[r1 , r2 ] ∼ = j ∈J

In particular, any summand I([bj , dj ]) of X with [bj , dj ] ∩ [r1 , r2 ] = ∅ becomes a zero module in the direct sum for X[r1 , r2 ]. Moreover, the length of each summand, i.e., the number of spaces in the sequence, does not change in the restriction, meaning that the spaces of summands are 0 outside of the interval [r1 , r2 ]. The bottleneck distance between two persistence diagrams is equal to δ if there exists a matching between the points of the two diagrams (where points are allowed to be matched to diagonal elements) such that any pair of matched points are at distance at most δ. Formally, for a fixed homology dimension, the bottleneck distance is given by dB (DgX, DgY) =

inf

sup ||x − μ(x)||∞ ,

μ:DgX→DgY x∈DgX

where μ ranges over all bijections between the two diagrams [17]. In order to compute the bottleneck distance, one adds countably many copies of the diagonal {(x, x) : x ∈ R}, which may intuitively correspond to topological features that are born and simultaneously die (and thus, never really exist at all). This allows for a point in one persistence diagram to be matched to the diagonal if it is far away from any point in the other diagram, and also accounts for the fact that two persistence diagrams may have different numbers of off-diagonal points. The q-th Wasserstein distance is defined as ⎡ dq (DgX, DgY) = ⎣

inf

μ:DgX→DgY



⎤1/q ||x − μ(x)||∞ ⎦ q

.

x∈DgX

We conclude this section by defining a projection map that keeps track of the points in the global persistence diagram that disappear in the restricted version. We next define a restriction map for a persistence diagram and then show that this is the persistence diagram of a restricted zigzag module.

Local Versus Global Distances for Zigzag and Multi-Parameter Persistence Modules Fig. 1 An illustration of the six cases for the projection map . Green points are the images of the blue points under

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E B

D r2 A C

r1

F r1

r2

Definition 1 Given I = [r1 , r2 ] ⊂ R, we let DgXI denote the restriction of the persistence diagram DgX to the interval I defined as the output of the following projection map (see Fig. 1): : DgX → DgXI ⎧ ⎪ ⎪ ⎨(max(b, r1 ), min(d, r2 )) (Types A, B, C, D) (b, d) → (b, b) if r2 ≤ b (Type E) ⎪ ⎪ ⎩(d, d) if d ≤ r (Type F) 1

Notice that points like E and F in Fig. 1 correspond to features that are born and die outside of the interval I (either completely before or completely after). The restriction result cited above from [7] would not include points (E) or (F ) in its diagram. However, since both (E) and (F ) are on the diagonal, including them in DgXI does not change the bottleneck distance between two restricted diagrams. We formalize this in the following Lemma. Note that we could choose to map to any point on the diagonal, such as a point in the support of [r1 , r2 ], but the choice of (b, b) and (d, d) simplifies the proof of Theorem 1 below. Lemma 1 Let I = [r1 , r2 ] ⊂ R. The restriction of the persistence diagram DgX to the interval I is equal to the persistence diagram of the restricted module X[r1 , r2 ], i.e., DgXI = DgX[r1 , r2 ].

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I[bj , dj ]. Then by (1) we have that DgX[r1 , r2 ] contains the

j ∈J

following set of points: DgX[r1 , r2 ]   (max(bj , r1 ), min(dj , r2 )) if r1 ≤ bj ≤ r2 or r1 ≤ dj ≤ r2 = : (bj , dj ) ∈ DgX . (r1 , r2 ) if bj < r1 ≤ r2 < dj

The first case aligns with types A, B, and C in Fig. 1, and the second case is type D. Note that this does not take into account those intervals for which [bj , dj ] ∩ [r1 , r2 ] = ∅. To create DgXI we map them to (bj , bj ) or (dj , dj ) depending on whether they fall before the interval I or after. But these points are not represented directly in DgX[r1 , r2 ]. Rather, to complete the proof we rely on the fact that persistence diagrams by design contain infinitely many copies of the diagonal {(x, x) : x ∈ R}. So, both diagrams contain the same mapped points of type A, B, C, and D, and infinitely many copies of the diagonal.  

3 Bottleneck and Wasserstein Distances in the Local vs. Global Settings In this section, we prove our main result relating the bottleneck distance (resp. Wasserstein distance) between persistence diagrams with the bottleneck (resp. Wasserstein) distance between their interval-restricted versions. Theorem 1 Let X and Y be two zigzag modules and let DgX and DgY be their corresponding zigzag persistence diagrams. Consider the interval I = [r1 , r2 ] ⊂ R and let DgXI and DgY I be the restrictions of these diagrams to I . Then dB (DgXI , DgY I ) ≤ dB (DgX, DgY ) and dq (DgXI , DgY I ) ≤ dq (DgX, DgY ). Proof Let μ ⊆ DgX × DgY be a partial matching, where any unpaired point in one of the persistence diagrams is matched to the nearest point (in the L∞ norm) on the diagonal = {(x, x) : x ∈ R}. Then μ can be modeled as a bipartite graph between finite sets of points from the two diagrams. We define μI ⊆ DgXI × DgYI such that, for each (p, q) ∈ μ, we have ( (p), (q)) ∈ μI . Since we are simply relabeling the coordinates of points in the bipartite graph of μ to define μI , μI remains a partial matching. What is left to

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show is that the maximal distance between the matched points in μI is less than that for μ. That is, we want to show that for the partial matching μI , sup ( (p), (q))∈μI

|| (p) − (q)||∞ ≤ sup ||p − q||∞ .

(2)

(p,q)∈μ

We show that $ (p) − (q)$∞ ≤ $p − q$∞ ,

(3)

for all matched p and q. This is then sufficient to establish the claimed inequality. Consider two points p ∈ DgX and q ∈ DgY such that (p, q) ∈ μ. Let p = (bp , dp ) and q = (bq , dq ). By definition of L∞ distance, we have $p − q$∞ = max(|bp − bq |, |dp − dq |). First, if both p and q are points of type A, B, C, or D as in Definition 1 and Fig. 1, then (bp , dp ) = (max(bp , r1 ), min(dp , r2 )) and (bq , dq ) = (max(bq , r1 ), min(dq , r2 )). Thus, we have $ (p)− (q)$∞ = max(| max(bp , r1 )−max(bq , r1 )|, | min(dp , r2 )−min(dq , r2 )|). To establish inequality (3), we need only show that | max(bp , r1 ) − max(bq , r1 )| ≤ |bp − bq |, and likewise that | min(dp , r2 ) − min(dq , r2 )| ≤ |dp − dq |. Suppose without loss of generality that bp ≤ bq , so that one of the following three cases holds: (i) bp ≤ bq ≤ r1 , (ii) bp ≤ r1 ≤ bq , or (iii) r1 ≤ bp ≤ bq . In all three cases, it is immediate that | max(bp , r1 ) − max(bq , r1 )| ≤ |bp − bq |. Similar reasoning shows that | min(dp , r2 ) − min(dq , r2 )| ≤ |dp − dq |. Second, if either p or q is a point of type E or F, the non-trivial cases arise when such a point is paired with a point of type B or C. For instance, suppose q = (bq , dq ) is of type B, so that (q) = (bq , r2 ). If p = (bp , dp ) is of type E, we have (p) = (bp , bp ) and || (p) − (q)||∞ = max{|bp − bq |, bp − r2 }. Since bq ≤ r2 ≤ bp , this implies that the horizontal distance between the projections must be larger than the vertical distance. Therefore, || (p) − (q)||∞ = bp − bq ≤ max{bp − bq , |dp − dq |} = ||p − q||∞ . If p = (bp , dp ) is of type F, then (p) = (dp , dp ) and || (p) − (q)||∞ = max{bq − dp , r2 − dp }. Since bp ≤ dp ≤ r1 ≤ bq ≤ r2 ≤ dq , the horizontal distances satisfy bq − dp ≤ bq − bp and the vertical distances satisfy r2 − dp ≤ dq − dp , yielding the desired inequality. The case analysis for the remaining pairings proceeds in a similar manner. Therefore, if one takes μ to be the infimum over all matchings with respect to the bottleneck distance between DgX and DgY, and the cost of μI is smaller, then the bottleneck distance between DgXI and DgYI will only be smaller still. This concludes the proof of the bottleneck distance portion of the theorem. The Wasserstein result follows similarly.   Corollary 1 Let I = [I , rI ], and J = [J , rJ ] be two intervals, with dist (I, J ) = max{|I − J |, |rI − rJ |}. Given zigzag persistence modules X and Y , dB (DgXI , DgY J ) ≤ dB (DgX, DgY ) + dist (I, J ).

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Proof The result follows from combining Theorem 1 with the inequality dB (DgXI , DgXJ ) ≤ dist (I, J ) and the triangle inequality.   Given an R-valued function, there is a natural construction of a level set zigzag persistence module [8] that sweeps its level sets from bottom to top [18] in a sense that we will describe in the next paragraph. Before going into those details, our motivation for introducing level set zigzag persistence is as follows. Level set zigzag persistence can be used to compute the ordinary persistent homology of an R-valued function with good space efficiency. In particular, the level set zigzag module is related to the ordinary (extended) persistence module via the MayerVietoris pyramid [8, Figure 3], where the zigzag sequence and the ordinary sequence are shown to contain the same information in their persistent homology. Therefore, one could use the algorithm for zigzag persistent homology to compute extended persistence while using space that depends only on the size of the largest level set instead of the entire domain [8, 22]. With this motivation in mind, we now discuss level set zigzag persistence a bit further. Given a topological space X and a continuous function f : X → R, (X, f ) is said to be of Morse type if, for the finite set of critical values a1 < a2 < . . . < an of f , the open intervals (−∞, a1 ), (a1 , a2 ), . . . , (an−1 , an ), (an , ∞) are such that for each interval I , f −1 (I ) is homeomorphic to Y × I for some compact and locally connected space Y with f serving as the projection onto I [8]. The homeomorphisms should extend to continuous functions on Y × I¯, where I¯ is the closure of I in R, and each Xt = f −1 (t) should also have finitely-generated homology, where Xt denotes the level set of f for any t ∈ R. Let XI = f −1 (I ) denote the slice of X which f maps to the interval I ⊂ R. If I = [a, b], we may denote this as Xba . Then, given (X, f ) of Morse type with critical values ai as above, we choose arbitrary si satisfying −∞ < s0 < a1 < s1 < a2 < · · · < sn−1 < an < sn < ∞. The level set zigzag persistence of (X, f ) is defined to be the zigzag persistence for the sequence Xss00 → Xss10 ← Xss11 → Xss21 ← · · · → Xssnn−1 ← Xssnn . We denote the persistence diagram by Dgf . We now state a straightforward corollary to Theorem 1 in the level set zigzag persistence setting, which we will refer to again in Sect. 4. Corollary 2 Let f : X → R and g : Y → R be Morse type functions defined on topological spaces X and Y, and for an interval I = [r1 , r2 ], let Dgf I and Dgg I be the restrictions of the level set zigzag persistence diagrams Dgf and Dgg to the interval I . Then dB (Dgf I , Dgg I ) ≤ dB (Dgf, Dgg).

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4 Applications to Metric Graphs and d-Parameter Persistence 4.1 Metric Graphs For uses of Corollary 2, we turn to the metric graph setting. Metric graphs commonly arise when studying road networks as well as biological or chemical structure graphs. Given a graph G with a set of vertices and edges, a length function on the edges, and a geometric realization |G| of the graph, one may specify a metric on G by taking the minimum length of any path between any pair of points (not necessarily vertices) in the geometric realization. Given a base point v ∈ |G|, the geodesic distance function fv : |G| → R is given by fv (x) = dG (v, x). Then Dgfv denotes the 0-dimensional level set zigzag persistence diagram induced by fv . Equivalently, Dgfv is the union of the 0- and 1-dimensional extended persistence diagrams for fv (see [13] for the details of extended persistence). Corollary 2 can be used to compare local neighborhoods of two different metric graphs, G1 and G2 , with base points v ∈ G1 and u ∈ G2 . In particular, given fv : |G1 | → R and gu : |G2 | → R, we have dB (Dgfv I , Dggu I ) ≤ dB (Dgfv , Dggu ) for any real interval I . Typically, for comparing local neighborhoods, I = [0, r]. The following corollary gives a stability-type result for comparing two local neighborhoods within a single metric graph. Corollary 3 Let G be a metric graph with geometric realization |G|. For a fixed interval I and points u, v ∈ |G|, we have dB (DgfuI , DgfvI ) ≤ dG (u, v). Proof By Corollary 2, dB (Dgfu I , Dgfv I ) ≤ dB (Dgfu , Dgfv ). Since fu , fv : |G| → R are two Morse type functions, dB (Dgfu , Dgfv ) ≤ ||fu − fv ||∞ by the level set zigzag stability theorem of [8]. Furthermore, by the triangle inequality, for any x ∈ |G|, |dG (x, u) − dG (x, v)| ≤ dG (u, v), meaning that ||fu − fv ||∞ ≤ dG (u, v). Putting everything together proves the claim.   Another application of Corollary 2 is as follows. Define  : |G| → SpDg, (v) = Dgfv , where SpDg denotes the space of persistence diagrams. Given metric graphs (G1 , dG1 ) and (G2 , dG2 ), their persistence distortion distance [15] is dP D (G1 , G2 ) := dH ((|G1 |), (|G2 |)), where dH denotes the Hausdorff distance. In other words, dP D (G1 , G2 )  = max

 sup

inf

D1 ∈(|G1 |) D2 ∈(|G2 |)

dB (D1 , D2 ),

sup

inf

D2 ∈(|G2 |) D1 ∈(|G1 |)

dB (D1 , D2 ) .

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Note that the diagram Dgfv contains both 0- and 1-dimensional persistence points, but only points of the same dimension are matched under the bottleneck distance. A local version of the persistence distortion distance, which we will denote by dPr D , may be defined as follows: for each base point v, only consider the distance function to points within a fixed intrinsic radius r. Specifically, let r : |G| → SpDg, r (v) = Dgfv[0,r] . Then dPr D (G1 , G2 ) := dH (r (|G1 |), r (|G2 |). Corollary 4 If r ≤ r , then dPr D (G1 , G2 ) ≤ dPr D (G1 , G2 ). Proof Let D1r be the persistence diagram for some base point v ∈ |G1 |, where the geodesic distance function is computed in the interval [0, r]. Let D1r be the persistence diagram for the same base point, but where the distance function is computed in the interval [0, r ]. Define D2r and D2r similarly for some base point in |G2 |. By viewing Dir as a restriction of Dir for i = 1, 2, we can apply Theorem 1 to show that dB (D1r , D2r ) ≤ dB (D1r , D2r ). Since our choice of base points was arbitrary, this inequality holds for persistence diagrams across all choices of base points in |G1 | and |G2 |. Therefore, using the definition of the local version of the persistence distortion distance, we can conclude that dPr D (G1 , G2 ) ≤   dPr D (G1 , G2 ).

4.2 Multi-Parameter Persistence Theorem 1 can be applied to d-parameter persistence modules on any topological space (not restricted to the level set or metric graph settings). A d-parameter persistence module is indexed by a d-dimensional family of vector spaces, {Xu }u∈Rd , together with a family of linear maps {ρX (u, v) : Xu → Xv }uv such that for u  v  w ∈ Rd , we have ρX (u, u) = idXu and ρX (v, w) ◦ ρX (u, v) = ρX (u, w) [9]. Here, u  v if and only if ui ≤ vi for i = 1, . . . , d, where ui and vi are the coordinates of u and v. Any line L in the set of all lines of Rd with direction m = (m1 , . . . , md ) such that min mi is strictly positive gives a one-parameter slice i

of the d-parameter persistence module. Given two d-parameter persistence modules X and Y, we define their matching distance [20] to be dmatch (X, Y) := sup min mi dB (DgXL , DgYL ), L

i

where DgXL and DgYL are the persistence diagrams of the d-parameter persistence modules X and Y restricted along line L. Our result extends naturally to this linear relationship between these two parameters. Indeed, if we restrict both d-parameter persistence modules to a region I = I1 × · · · × Id , denoted XI and YI , where each Ii is an interval of the real line, then Theorem 1 implies the following corollary. Formally, XI is the module X restricted only to vector spaces in {Xu }u∈I (similarly for Y).

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Corollary 5 dmatch (XI , Y I ) ≤ dmatch (X, Y ) where dmatch (XI , Y I ) is computed by restricting DgX L and DgY L to the subinterval of the line L passing through the region I. Proof For a fixed line L with direction m, consider a region I restricted to L, denoted IL ⊂ I ∩ L. Recall that DgXL and DgYL are the persistence diagrams of the d-parameter persistence modules X and Y restricted along the line L. Based on Theorem 1, dB (DgXILL , DgYILL ) ≤ dB (DgXL , DgYL ).

(4)

From the definition of supremum, we know that ∀ > 0, there is a line L such that I I dmatch (XI , YI ) −  < min mi dB (DgXLL , DgYLL ). i

Using observation (4) above, we see that dmatch (XI , YI ) −  < min mi dB (DgXL , DgYL ). i

The right-hand side is, of course, less than the supremum over all lines L, the definition of dmatch (X, Y). Hence, for every  > 0, we have dmatch (XI , YI ) −  < dmatch (X, Y); in other words, dmatch (XI , YI ) ≤ dmatch (X, Y), as desired.   The matching distance between two d-parameter persistence modules requires factoring through all possible 1-parameter persistence modules by restricting to lines. A more direct way to compare two d-parameter persistence modules is the interleaving distance which does not rely on persistence diagrams (which are only available for 1-parameter persistence modules). Our definition of the interleaving distance, dI , for d-parameter persistence modules follows the treatment in [16, Sect. 12.2]. We first need the notion of a δ-interleaving for two such d-parameter − → persistence modules X and Y. For a given δ ≥ 0, let δ = (δ, . . . , δ) ∈ Rd ; we − → → to represent the module X shifted diagonally by δ . Also, use the notation X− δ X := Xu≤v . A δ-interleaving between X and Y consists of two families of ρu→v →} − →} linear maps {ϕu : Xu → Yu+− d and {ψu : Yu → X d satisfying: δ u∈R u+ δ u∈R → ◦ ϕu and ρ Y 1. (Triangular commutativity) ∀u ∈ Rd , ρ X − → = ψu+− − → = δ u→u+2 δ u→u+2 δ → ◦ ψu . ϕu+− δ

X 2. (Rectangular commutativity) ∀u ≤ v ∈ Rd , ϕv ◦ ρu→v = ρY Y ψv ◦ ρu→v

=

ρX − → − → u+ δ →v+ δ

◦ ψu .

See Fig. 2 for illustrations of both types of commutativity.

− → − → u+ δ →v+ δ

◦ ϕu and

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→ u+ δ

u

→ u→u+2 δ

ϕu ψu

Y

→ u+2δ

ρX

X

u

→ u+ δ

ψu+→ δ ρY → u→u+2 δ → → u+ δ u+2δ

→ → u+ δ →v+ δ

→ v+ δ ψv

X

ϕu+→ δ

ρX

ϕu

ρX u→v

ϕv

ρY

→ → u+ δ →v+ δ

Y

ψu u

v ρY u→v

Fig. 2 Triangular commutativity (left) and rectangular commutativity (right) for a δ-interleaving

Then the interleaving distance between X and Y is given by dI (X, Y) = inf{X and Y are δ-interleaved}. δ

If no such δ ∈ R+ exists, X and Y are said to be ∞-interleaved with dI (X, Y) = ∞. Since the interleaving distance is based directly on the d-dimensional modules, and not on zigzag modules or their corresponding persistence diagrams, our Theorem 1 does not apply. However, a restricted distance inequality still holds through a different argument. Theorem 2 dI (XI , Y I ) ≤ dI (X, Y ). Proof Let us say that dI (X, Y) = δ so that X and Y are δ-interleaved. Then, if we restrict the modules to the region I the same interleaving maps used for X and Y are still interleaving maps for XI and YI . But since there are fewer spaces in XI and YI there may be additional interleaving maps in the restricted case. Therefore, {δ : X and Y are δ-interleaved} ⊆ {δ : XI and YI are δ-interleaved}, and so we have the desired result, dI (X, Y) = inf{X and Y are δ-interleaved} δ

≥ inf{XI and YI are δ-interleaved} = dI (XI , YI ). δ

  In the case where X and Y are 1-parameter persistence modules, Theorem 2 (together with the fact that dI (X, Y) = dB (DgX, DgY) [10]) is equivalent to Theorem 1. However, this is only in the case of traditional persistence modules where all linear maps go forwards, not in the more general case of zigzag modules

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where maps are allowed to go forwards and backwards. So, while special cases of Theorems 1 and 2 are equivalent neither theorem implies the other.

5 Discussion Theorem 1 and its corollaries, as well as Theorem 2, provide explicit relationships between distances between persistence modules computed locally and globally, and the resulting inequalities are broadly applicable. For instance, the fact that the local bottleneck distance is bounded above by the global bottleneck distance allows for a single global computation to potentially rule out local differences if the global distance is low. If looking for local differences, starting with a global computation may save computational time if there are too many local comparisons to make. On the other hand, the global bottleneck distance being bounded below by the local version allows smaller computations to approach the global truth, while perhaps being more computationally tractable. In future work, we would like to extend these ideas to generalized persistence, where instead of a linear sequence of topological spaces one considers topological spaces and transformations that form a poset. In contrast to zigzag persistence, this generalized persistence does not have the notion of a persistence diagram. Theorem 2 is the first step in this direction for the case of d-parameter persistence using interleaving distance. We expect similar techniques to be needed in the poset case. However, a notion of “local” would have to be defined in the poset setting. Acknowledgments We are grateful for the Women in Computational Topology (WinCompTop) workshop for initiating our research collaboration. In particular, participant travel support was made possible through the grant NSF-DMS-1619908. Our group was also supported by the American Institute of Mathematics (AIM) Structured Quartet Research Ensembles (SQuaRE) program. E.P. was supported by the High Performance Data Analytics (HPDA) program at Pacific Northwest National Laboratory. RS was partially supported by NSF grant DMS-1854705 and the SIMONS Collaboration grant 318086. B.W. was partially funded by NSF-IIS-1513616, NSF-DBI1661375, and NIH-R01-1R01EB022876-01. Y.W. was supported by NSF grants CCF-1740761 and DMS-1547357. L.Z. was supported by NSF grant CDS&E-MSS-1854703.

References 1. Aanjaneya, M., Chazal, F., Chen, D., Glisse, M., Guibas, L., Morozov, D.: Metric graph reconstruction from noisy data. Int. J. Comput. Geom. Appl. 22(4), 305–325 (2012) 2. Ahmed, M., Fasy, B.T., Wenk, C.: Local persistent homology based distance between maps. In: Proceedings of the 22nd ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pp. 43–52 (2014) 3. Bendich, P., Cohen-Steiner, D., Edelsbrunner, H., Harer, J., Morozov, D.: Inferring local homology from sampled stratified spaces. In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, pp. 536–546 (2007)

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4. Bendich, P., Gasparovic, E., Harer, J., Izmailov, R., Ness, L.: Multi-scale local shape analysis and feature selection in machine learning applications. In: International Joint Conference on Neural Networks, pp. 1–8 (2015) 5. Bendich, P., Wang, B., Mukherjee, S.: Local homology transfer and stratification learning. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 1355–1370 (2012) 6. Bille, P.: A survey on tree edit distance and related problems. Theor. Comput. Sci. 337(1–3), 217–239 (2005) 7. Carlsson, G., de Silva, V.: Zigzag persistence. Found. Comput. Math. 10(4), 367–405 (2010) 8. Carlsson, G., de Silva, V., Morozov, D.: Zigzag persistent homology and real-valued functions. In: Proceedings of the 25th Annual Symposium on Computational Geometry, pp. 247–256 (2009) 9. Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. Discrete Comput. Geom. 42(1), 71–93 (2009) 10. Chazal, F., De Silva, V., Glisse, M., Oudot, S.: The Structure and Stability of Persistence Modules. Springer, Berlin (2016) 11. Chernov, A., Kurlin, V.: Reconstructing persistent graph structures from noisy images. Image-a 3(5), 19–22 (2013) 12. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37, 103–120 (2007) 13. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Extending persistence using poincaré and lefschetz duality. Found. Comput. Math. 9(1), 79–103 (2009) 14. Cohen-Steiner, D., Edelsbrunner, H., Harer, J., Mileyko, Y.: Lipschitz functions have lp -stable persistence. Found. Comput. Math. 10(2), 127–139 (2010) 15. Dey, T.K., Shi, D., Wang, Y.: Comparing graphs via persistence distortion. In: Proceedings of the 31st International Symposium on Computational Geometry, vol. 34, pp. 491–506 (2015) 16. Dey, T.K., Wang, Y.: Computational Topology for Data Analysis. Cambridge University Press (forthcoming) 17. Edelsbrunner, H., Harer, J.: Persistent homology—a survey. Contemp. Math. 453, 257–282 (2008) 18. Edelsbrunner, H., Morozov, D.: Persistent homology. In: Goodman, J.E., O’Rourke, J., Tóth, C.D. (eds.) Handbook of Discrete and Computational Geometry, chap. 24. CRC Press, Boca Raton (2017) 19. Harshaw, C.R., Bridges, R.A., Lannacone, M.D., Reed, J.W., Goodall, J.R.: Graphprints: towards a graph analytic method for network anomaly detection. In: Proceedings of the 11th Annual Cyber and Information Security Research Conference, p. 15 (2016) 20. Landi, C.: The rank invariant stability via interleavings. In: Chambers, E.W., Fasy, B.T., Ziegelmeier, L. (eds.) Research in Computational Topology, pp. 1–10. Springer, Cham (2018) 21. Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., Alon, U.: Network motifs: Simple building blocks of complex networks. Science 298(5594), 824–827 (2002) 22. Milosavljevi´c, N., Morozov, D., Skraba, P.: Zigzag persistent homology in matrix multiplication time. In: Proceedings of the 20th Annual Symposium on Computational Geometry, pp. 216–225 (2011) 23. Shen-Orr, S.S., Milo, R., Mangan, S., Alon, U.: Network motifs in the transcriptional regulation network of Escherichia coli. Nat. Genet. 31(1), 64 (2002) 24. Wang, B., Summa, B., Pascucci, V., Vejdemo-Johansson, M.: Branching and circular features in high dimensional data. IEEE Trans. Visualization Comput. Graph. 17(12), 1902–1911 (2011)

Tile-Transitive Tilings of the Euclidean and Hyperbolic Planes by Ribbons Benedikt Kolbe and Vanessa Robins

Abstract We present a method to enumerate tile-transitive crystallographic tilings of the Euclidean and hyperbolic planes by unbounded ribbon tiles up to equivariant equivalence. The hyperbolic case is relevant to self-assembly of branched polymers. Our result is achieved by combining and extending known methods for enumerating crystallographic disk-like tilings. We obtain a natural way of describing all possible stabiliser subgroups of tile-transitive tilings using a topological viewpoint of the tile edges as a graph embedded in an orbifold, and a group theoretical one derived from the structure of fundamental domains for discrete groups of planar isometries.

1 Introduction Tilings from repeating motifs appear in all cultures and have long been studied in mathematics, art, engineering and science. The bulk of the mathematical work has focussed on patterns in the Euclidean plane (the book “Tilings and Patterns” by Grünbaum and Shephard [14] contains a comprehensive survey of the field up to the mid 1980s) but the importance of hyperbolic geometry as a model for natural forms is increasingly recognised [20, 22, 32, 39]. A motivating discovery that inspires the work in this paper is the observation that star co-polymer systems consisting of three mutually immiscible arms can self-assemble into structures modelled by stripes on the gyroid triply periodic minimal surface [5, 23]. A natural question is whether there is a way to describe all possible different types of stripes that can arise on surfaces like the gyroid. The gyroid surface has genus three in its smallest side-preserving translational unit cell, and therefore has the hyperbolic plane as its

B. Kolbe Université de Lorraine, CNRS, Inria, LORIA, Nancy, France e-mail: [email protected] V. Robins () Research School of Physics, Australian National University, Canberra, ACT, Australia e-mail: [email protected] © The Author(s) and the Association for Women in Mathematics 2022 E. Gasparovic et al. (eds.), Research in Computational Topology 2, Association for Women in Mathematics Series 30, https://doi.org/10.1007/978-3-030-95519-9_4

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Fig. 1 Nets in R3 on the gyroid from ribbon tilings with 22222 symmetry in H2

simply-connected Riemann covering space. Its 3D space-group symmetries induce a non-Euclidean crystallographic group generated by hyperbolic isometries that are known explicitly [36, 37]. Stripe patterns on the gyroid lift via the covering map to tilings of the hyperbolic plane by infinitely long strips, or ribbons. This paper deals primarily with tilings of the hyperbolic plane by ribbons. The defining property of a ribbon tile is the existence of a translation isometry that maps a given tile back onto itself, along with the restriction that the tile is simply connected. See Fig. 1 for some ribbon tilings and their projections to the gyroid triply periodic minimal surface. The Euclidean case of striped patterns is described in Section 6.5 of [14], citing earlier work by Wollny [43]; there are 26 distinct types of crystallographic ribbon tilings of the Euclidean plane, a result that can readily be proven using the enumerative methods we present here. Deducing similar results for the hyperbolic plane requires different mathematical techniques to those of [14]. Our methods are based on the classification of 2D discrete groups of isometries up to isomorphism via their marked quotient spaces [3, 30, 41, 42]; from the enumeration of classes of polygonal fundamental domains for these groups via graphs on their quotient spaces [26, 28]; and from the results of Dress et al. who developed the field now known as combinatorial tiling theory [8, 9]. See [17] for a detailed introduction to combinatorial tiling theory for the Euclidean and hyperbolic planes, [6] for algorithms and [44] for a very recent implementation leading to a database of tilings of the sphere, and the Euclidean and hyperbolic planes. Any investigation of tilings relies on a notion of equivalence among tilings. The notion of equivalence we consider here is purely topological. However, we also fix the symmetry group of the tiling in question, thereby extending the definition of equivalence of combinatorial tiling theory (see Sect. 3). Combinatorial tiling theory treats periodic (crystallographic) tilings of a simply connected manifold where the tiles are compact topological disks and defines an invariant called the Delaney-Dress symbol or D-symbol as a weighted and coloured graph. From the D-symbol it is

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possible to reconstruct the tile shapes and adjacencies and the isomorphism class of the symmetry group for the tiling. In the 2D setting, a D-symbol encodes a finite triangulation derived from the tiling by barycentric subdivision. The underlying space of the triangulation is the quotient of the plane by a discrete group of isometries that preserve the tiling. This quotient space is a 2-orbifold and can be viewed as a compact surface with (possibly) a finite number of boundary components and a finite number of isolated marked points. When the orbifold is an orientable manifold with no boundary or singular points, then the D-symbol encodes the same information as the rotation system for a 2-cell embedding of a graph [31]. Although the full theory of D-symbols does not directly generalise to our setting with unbounded ribbon tiles, the correspondence between crystallographic patterns and graphs on 2-orbifolds does (see Sect. 3). By relying on results of combinatorial tiling theory and related work characterising tile-transitive tilings by compact disks, the main issues to overcome are 1. characterising the possible stabiliser subgroups for ribbon tiles, 2. listing the tile-transitive unbounded ribbon tilings compatible with a given symmetry group, and 3. determining which of the ribbon tilings listed in the previous step are equivalent. Solutions to the above challenges are the main contributions of this paper. To achieve these goals, we combine the D-symbol point of view with another approach to combinatorial tiling theory, described in [26–28]. For Euclidean tilings by ribbons the possible stabiliser groups are the seven frieze groups; for the hyperbolic case infinitely many such stabiliser groups, i.e., nonEuclidean frieze groups are possible. These are described further in Sect. 2. Definitions and notation for combinatorial tiling theory are covered in Sect. 3. The enumeration of tile-transitive ribbon tilings is achieved by deleting edges from a fundamental domain tiling. The results needed to characterise the existence and structure of ribbon tiles are detailed in Sect. 4. The last step is described in Sect. 5, together with an enumeration algorithm for different classes of tile-transitive ribbon tilings. Previous work treating unbounded tiles and related to this paper includes Huson’s papers on tile-transitive partial tilings of the Euclidean plane and of infinitely long strips [18, 19]. A different approach to enumerating uniform locally finite tilings of the hyperbolic plane (i.e. vertex-transitive tilings by regular polygons) was introduced in [35]. The exploration of crystallographic line and tree patterns in the hyperbolic plane goes back to Hyde et al. [10–12, 21]. An important aspect of our approach to enumerations using D-symbols is that it allows for an adaptation of the methods of isotopic tiling theory [24, 25] to systematically enumerate the distinct ways in which tilings fit on compact surfaces.

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2 Groups of Isometries and the Orbifold Fundamental Group In this section, we introduce the concepts and results in geometric group theory that make possible the methods we use in subsequent sections. Throughout this paper, let X be either the Euclidean (E2 ) or hyperbolic (H2 ) plane, and let  be a discrete group of isometries of X with compact quotient space. If X = E2 then  is one of the 17 wallpaper groups of crystallography [3]. If X = H2 , then  is known as a NEC group (non-Euclidean crystallographic group). Note that our results naturally extend to cofinite-area symmetry groups, but we focus on the cocompact case, for simplicity and because it represents the setting most relevant for the envisaged applications. A key result in geometric group theory is that these groups of isometries are completely classified up to isomorphism by an associated orbifold. The homeomorphism class of the orbifold is in turn specified by Conway’s orbifold symbol [4], a highly-readable version of Macbeath’s group signature [30], as described below. For more detailed definitions of the concepts involved, refer to [34, Chapter 13]. Definition 1 Let  be a wallpaper or NEC group. A geometric (good) 2-orbifold, O = X/ , is a quotient space obtained by identifying points of X under the action of . The orbifold retains the metric information carried by the particular isometries of  by specifying an atlas of charts compatible with the  action on the topological space and the branch points of the projection map p : X → X/ . It is well-known [41] that 2-orbifolds have the topology of a finite-area 2manifold with a finite number of boundary components. Boundaries in a 2-orbifold arise from the fixed lines of reflection isometries. Other special points arise as the fixed points of rotational isometries; these are called corner points if they lie on a boundary and cone points if they lie in the interior of the orbifold. The branching number, N , of a cone or corner point is the maximal order of a rotational isometry, σ , that fixes that point, i.e. σ N = id. The boundaries, corner and cone points are collectively referred to as the singular locus, , of the orbifold. The topology of a 2-orbifold (O) is therefore specified by a symbol as follows: 1. The number of handles, h, if the orbifold is orientable, or the number of crosscaps, k, if non-orientable. Handles are denoted by ◦ at the beginning of the orbifold symbol. Cross-caps are denoted by × at the end of the orbifold symbol. 2. The branching number for each cone point, listed in arbitrary order after any handles. 3. The number of boundary components, q. Each boundary component is represented by a ∗ in the symbol. Branching numbers for the corner points lying on each boundary component are listed in cyclic order, such that each boundary component has a consistent orientation for the manifold. The ordering of the boundary components is arbitrary.

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For example, an orbifold symbol for hyperbolic tilings in Fig. 1 is 22222, or 25 , telling us that there are five distinct conjugacy classes of 2-fold rotations in this symmetry group. It is known [4, 41] that any orbifold symbol will correspond to a group of isometries of either E2 , H2 , or S2 , except for the symbols A, ∗A, AB, and ∗AB, with A = B, which do not represent a good orbifold in the sense that it is covered by a manifold. Moreover, the plane geometry associated with an orbifold is determined by computing a curvature-related quantity (the orbifold Euler characteristic) directly from the group symbol. The symmetry group associated with a geometric orbifold also has an interpretation as a type of fundamental group defined by equivalence classes of loops. Intuitively, orbifold loops are piecewise lifts to X of based closed curves in O. The general definition of orbifold loops in O and their homotopies is quite involved, but it is sufficient for our purposes to view them as closed curves in O, disjoint from the singular locus except possibly at isolated points of mirror boundaries. In addition to being a based homotopy of the underlying topological space, we also require that homotopies of simple curves in O do not change the incidence relation of a curve to any singular point or section of mirror boundary; that the group  is fixed; and that paths in O meeting a mirror boundary transversally will lift to paths that cross the mirror line in X, to eliminate the ambiguity of the lift in this situation. This means that curves are allowed to slide along boundary components, as long as they do not cross a cone point. See [34, Chapter 13] or [4]. The orbifold fundamental group π1orb (O, x), x ∈ O \ , is defined as the set of orbifold loops based at x0 ∈ X up to homotopy equivalence, where x0 ∈ p−1 (x). It is known that for the geometric 2-orbifolds considered in this paper,  % π1orb (X/ ) in the natural way [34, Theorem 13.3.2]. In other words, π1orb (X/ ) is the group of deck transformations for the branched covering map p : X → O. Other discrete groups of isometries  will be important when we discuss the internal symmetries of a tile T . For a bounded tile in X, T homeomorphic to a closed disk, the possible symmetry groups include those that fix a single point, namely the cyclic CN and dihedral DN groups for N ≥ 2, and D1 the symmetry group generated by a single reflection. When these groups are viewed as acting on T , we can form the quotient space T /CN or T /DN and describe the topology of these quotient spaces similarly to the orbifold symbol above, using signatures for rosette patterns: N• and ∗N• for the cyclic and dihedral groups respectively [3]. The symbol • represents for us a section of tile boundary in the quotient space. Note that the group D1 is abstractly isomorphic to C2 , but their associated quotient spaces have different singular loci and different signatures. We next consider isometries of a tile T that is homeomorphic to [0, 1] × R. The possible discrete symmetry groups, with compact quotient space, for such a tile are D1 , D2 , C2 or one of the seven frieze groups. Again, we specify the quotient space structure of T /  using a descriptive signature or symbol for the symmetry group Γ . In Table 1 we give the signatures (from [3]) and Hermann-Mauguin (IUCr) name of each of the frieze groups. We specify that in these signatures we use the ∗ symbol to represent a single boundary component of T /  due to fixed points of

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Table 1 Signature of T /  and IUCr name for the seven frieze groups, the index of the translation isometry in , and a presentation that makes the translation isometry t explicit in each case Group ∞∞ ∞x ∞∗ ∗∞∞ 22∞ 2∗∞ ∗22∞

Name p1 p11g p11m p1m1 p2 p2mg p2mm

Index 1 2 2 2 2 4 4

Group presentation &t'   g, t | g 2 = t   r, t | r 2 , rtr = t   r , r , t | r12 , r22 , r1 r2 = t 1 2  q , q , t | q2, q2, q q = t  1 2 2 1 2 2 12 2  r, q, t | r , q , (qr) = t   r1 , r2 , r3 , t | r12 , r22 , r32 , (r1 r2 )2 , (r2 r3 )2 , r1 r3 = t

reflection isometries. Furthermore, we use the ∞ symbol to represent a segment of tile boundary in T / . So the signature ∗∞∞ implies that T /  is a disk with a mirror boundary interrupted by two tile boundary segments and so combinatorially it is a quadrilateral. Note that in other contexts, the ∞ symbol can represent an orbifold puncture as might be generated by a parabolic isometry of H2 . The different contexts are just other ways of obtaining a geometric representation of the same abstract group. It is important to note that the above rosette and frieze groups can be realised using isometries of either the Euclidean or hyperbolic plane. This is in contrast to the geometric 2-orbifolds which can be realised by isometries in exactly one of the three plane geometries. Figure 1, top row left and right, for which each ribbon tile has symmetry group 22∞. Figure 1, middle, shows an example with symmetry group ∞∞. If a rosette or frieze group occurs as the subgroup of a wallpaper or NEC group then it has infinite index. The general question of characterising the subgroups of infinite index in NEC groups is covered in [15, 16] where it is shown, in particular, that if the subgroup has no reflection elements then it must be a free product of cyclic groups (of finite or infinite order). So, for example, 22∞ is simply the free product of two copies of C2 . In the hyperbolic plane we will naturally encounter unbounded simply connected tiles with branching structure homeomorphic to a neighbourhood of an infinite tree embedded in H2 . See Fig. 8c for an example. We call such tiles branched ribbons. We shall see in Sect. 4 that the isometries of such a tile are isomorphic to a group action on a tree, which are covered by the theory of Bass-Serre [38]. The simplest examples of group actions on trees are those that have a line segment as fundamental domain. These groups are a free product with amalgamation of the subgroups that fix the vertices, amalgamated via the subgroup that fixes the edge (viewed as an oriented line segment). For example, if the line segment generating the tree has end points on rotation centres of order A and B, and the edge  group is trivial, then the group is  = CA ∗ CB = q1 , q2 , t | q1A , q2B , q1 q2 = t and T /  has the signature AB∞.

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3 Combinatorial Tiling Theory In this Section, we recall the results from the two different points of combinatorial tiling theory we rely on in the later stages of the paper. However, we give more general definitions to set the stage for the material of the following sections, as we point out along the way. Standard or classical combinatorial tiling theory describes the adjacency structure of periodic tilings of a simply connected metric space for which each tile is homeomorphic to a closed bounded disk [9, 17]. In this paper, we employ a more general definition of tiling. Definition 2 A crystallographic tiling of X is a locally finite, countable set T of connected closed domains, with cl(int(T )) = T for all tiles T , such that every point x ∈ X belongs to at least one tile, all tiles have pairwise disjoint interiors, and such that T is invariant under a discrete group of isometries of X. For emphasis, we call a simply connected compact tile a disk, and a tiling where every tile is a disk a disk-like tiling. Similarly, a tiling in which every tile is a ribbon or branched ribbon is called a ribbon tiling. A simple example of a ribbon tiling is a covering of E2 by vertical strips, Tk = [k, k + 1] × R, for k ∈ Z. The trivial ribbon tiling consists of one tile that is all of X. Other examples are the partial tilings of the Euclidean plane enumerated by Huson in [18], if we treat the complementary regions of types 2 and 3 in that paper as tiles. As in [17], we define the vertices and edges of a tile topologically rather than using the geometry of straight lines and corners. A vertex is a point that is contained in at least three tiles, and an edge is a connected segment of the intersection of two tiles. In the classical setting, an edge is a compact simple curve joining two vertices, but in the general setting an edge can be an unbounded 1-dimensional sub-manifold embedded in X, possibly with no vertices. Definition 3 Let T be a tiling of X and let  be a discrete group of isometries. If T = γ T := {γ T | T ∈ T} for all γ ∈  then we call the pair (T, ) an equivariant tiling and  the symmetry group of T. Note that this definition does not require  to be the maximal symmetry group for the tiling T. Two tiles T1 , T2 ∈ T are equivalent if there exists γ ∈  such that γ T1 = T2 . The orbit of a tile T (or an edge or a vertex) is the subset of T given by images of T : .T = {γ T for γ ∈ }. Given a particular tile T ∈ T, the stabiliser subgroup T is the subgroup of  that fixes T , i.e. T = {γ ∈  | γ T = T }. A tile is called fundamental if T is trivial and we call the whole tiling fundamental if this is true for all tiles. An equivariant tiling is called tile-k-transitive, when k is the number of distinct orbits of tiles under the action of . A fundamental tile-1transitive equivariant tiling (T, ) has a single type of tile that is a fundamental domain for . Conversely, any fundamental domain for  homeomorphic to a bounded disk also gives rise to such a tiling. We will call such tilings fundamental domain tilings in the following. The following is a central definition for combinatorial tiling theory, and is also the notion of equivalence among tilings we consider.

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Definition 4 Two equivariant tilings (T1 , 1 ) and (T2 , 2 ) of a simply connected space X are equivariantly equivalent if there is a homeomorphism, φ, of X and a group isomorphism h : 1 → 2 , such that φ(T ) ∈ T2 for all T ∈ T1 and h(γ )[φ(T )] = φ(γ T ) for all γ ∈ 1 . Intuitively, this definition means that two tilings are considered to be equivalent if there is a self-map of X that preserves the tile adjacencies and their symmetries. A natural question is whether there is an invariant that detects when two tilings are equivariantly equivalent. Dress et al., [8, 9] show that a complete invariant, the D-symbol, is indeed possible for disk-like tilings of simply connected manifolds. First, recall that a flag in this context is a triple of (tile, edge, vertex) where each lower-dimensional element is a face of the higher dimensional ones, and that two flags are adjacent if they differ by a single one of their elements. We picture each flag as a triangle in the (-compatible) barycentric subdivision of the tiling, so that each triangle spans a tile centre-point, edge mid-point and tiling vertex. The D-symbol consists of a graph whose nodes are the -equivalence classes of flags. Each node has three edges, one each of three colours to record adjacencies of the flag, and two weights, corresponding to the number of edges of the tile and the degree of the vertex the flag represents [6]. Theorem 1 ([17, Lemma 1],[8]) Two equivariant disk-like tilings of X are equivariantly equivalent if and only if their D-symbols are isomorphic as weighted and coloured graphs. The properties of D-symbols are exploited in [6, 17, 44] to achieve a fully algorithmic approach to the enumeration and identification up to equivariant equivalence of disk-like tilings of S2 , E2 and H2 . In particular, it is established that one can enumerate equivariant equivalence classes of fundamental domains in a purely combinatorial way, by producing the D-symbols that represent them. This enumeration proceeds by first specifying the number of edges and vertex degrees of a tile; the isomorphism class of the group (i.e. its orbifold symbol) is then computed from the D-symbol. An alternative approach [26–28] starts with the orbifold symbol and finds all possible combinatorial types of fundamental domain, as described in the next paragraphs. The edges and vertices of any tiling (T, ) map onto a graph G = (V , E) embedded in the orbifold under the covering map p : X → X/ . In general, vertices of G come from vertices of T, except in the case that an edge midpoint in T is a cone point of order 2. Such an edge of T maps onto an edge of G with a vertex of degree 1 on the cone point. Lucic et al. [26, 27] have characterised the graphs associated with all possible combinatorial types of polygonal fundamental domain, and present an algorithm for enumerating these from the group signature (i.e. orbifold symbol) in [28]. Theorem 2 ([27, Theorem 4.1]) A topological polygon F is a fundamental domain for a wallpaper or NEC group  if and only if the boundary of F ⊂ X maps under p : X → X/  = O to a connected graph G embedded in O with the following properties:

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1. O \ G is an open disk. 2. Each cone point is a vertex of G with at least one incident edge in G. 3. Let  O be the closed surface of genus g obtained from O by capping each boundary component of O with a disk and forgetting about the cone points. Then G is ˆ with one vertex and 2g loops if  contractible in  O to the graph G O is orientable  and g loops if O is non-orientable. 4. Each (mirror) boundary component of O lies in a subgraph Gi of G. This means each corner point is a vertex of G with at least two incident edges. Moreover, when Gi is contracted in  O, it becomes a vertex with at least one incident edge in G \ Gi . 5. Any vertex of G that does not lie on a cone point, corner point or boundary must have at least three incident edges in G. See [27, 28] for general figures, and Fig. 3 for a simple Euclidean example. From now on, we restrict our attention to tile-1-transitive tilings. The definition of combinatorial equivalence in [26, 27] is made for fundamental domain tilings, but also applies to the case of more general tile-1-transitive tilings as per Definition 2. Definition 5 Two tiles are combinatorially equivalent if there is a bijection mapping one onto the other that preserves the incidence relations of vertices and edges, their cyclic order, the -equivalence of vertices, directed edges, and the isomorphism class of the stabiliser subgroups of each of these and also of the tile itself. Two tile-1-transitive tilings are combinatorially equivalent if their representative tiles are combinatorially equivalent. We will show in Sect. 4 that two tiles in a tile-1-transitive tiling are combinatorially equivalent if and only if the tilings are equivariantly equivalent. Note that it follows straight from the definition that equivariant equivalence implies combinatorial equivalence. The final result from geometric group theory that we require in order to enumerate ribbon tilings is a classical method for obtaining a group presentation from the intersection pattern of a tessellation of X by the -orbit of fundamental domains [29, 42]. Theorem 3 Let (T, ) be a fundamental tile-1-transitive disk-like tiling and T0 a tile. Then the adjacency between tiles defines a presentation for , where each edge of T0 corresponds to a generator and each vertex to a relation. If ei = T0 ∩ Ti is an edge, then the edge-generator is an element, [ei ] ∈ , such that Ti = [ei ]T0 . The relation at vertex v is found by listing the generators associated with each edge crossing when making a clockwise circuit around v. We call the group elements associated with edges the Wilkie generators (although according to Macbeath [29], the idea goes back to Fricke and Klein’s enumeration of Fuchsian groups). Wilkie’s proof is given for polygonal Dirichlet (Voronoi) fundamental domains [42], but it holds more generally as detailed in [29].

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4 Orbifold Paths and Tile Glueing In this section, we develop a relationship between closed paths in the orbifold O = X/ , their lifts in X, and equivariant tilings (T, ) that contain unbounded tiles. Furthermore, we study how the topological structure of tiles relates to their stabiliser subgroup and how to construct the possible stabiliser subgroups of tiles by edge deletions from fundamental tilings. Recall that a non-fundamental tile is one that has a non-trivial stabiliser. In the D-symbol approach to the enumeration of disk-like tilings, it is shown in [17] that any non-fundamental disk-like tiling (T, ) is obtained from a fundamental one (F, ) by performing tile glueing. Since the tiles are homeomorphic to a closed bounded disk, their stabiliser group must be a finite cyclic or dihedral group. We characterise the possible tile glueing operations as erasing edges from the graph G on the orbifold O associated to (F, ). Given a tile-1-transitive fundamental tiling by disks (F, ), let G be its corresponding graph on O. By Theorem 2, O \ G is a disk. Tile glueing erases at most two edges from G to get G so that G is connected, O \ G is still a disk and contains some part of the singular locus of O, and the corresponding tiling (T, ) has one class of non-fundamental tile. The edges of G that can be erased are of three types: 1. An edge of G that has a vertex of degree 1 at a cone point of order N. This glues N copies of the fundamental tile into one new one with stabiliser group N•. 2. A pair of edges of G that lie in a mirror boundary and meet at a vertex of degree 2 on a corner point of order N. This glues 2N copies of a fundamental tile into one new one with stabiliser group ∗N•. 3. A single edge of G that is a segment of mirror boundary and has vertices of degree at least 3 (or degree 2 on a corner point). This glues two copies of a tile together into one with stabiliser group ∗•. We now work through the possibilities for tile gluing to obtain tile-transitive ribbon tilings. Lemma 1 The stabiliser subgroup, H , for a non-fundamental tile T is infinite if and only if it contains a translation isometry. Proof First note that if H contains a translation (or glide) isometry then it must be infinite. Now assume H is an infinite group of isometries of H2 . If H is abelian, then it must be generated by a single isometry of infinite order, i.e., a translation, glide or parabolic rotation (an isometry with a single fixed point at infinity) [13, Section 4.5]. This last case can be ruled out as our tiling group  is assumed to have a compact orbifold. If H is non-abelian, then it is known that such a group must contain a translation [13, Section 4.5]. Finally, we consider the case that H is an infinite discrete subgroup of isometries of E2 : H ⊂ O(2)  R2 . If H consists entirely of rotations and/or reflection

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isometries and is discrete, it must be a discrete subgroup of O(2), and therefore closed, since discrete subgroups of topological groups are well-known to be closed. By compactness this would imply H is finite. But we assumed H to be infinite so not all its elements can be rotations and reflections and we see that it must contain a translation or glide.   We project the edges of a tiling onto an embedded graph G = (V , E) in O. Let S = {e1 , . . . , ek } ⊂ E be the edges to be erased and R = E \ S be the edges that remain. Let GR = (VR , R) ⊂ G be the subgraph obtained from G by erasing S and any vertices left isolated. We also avoid “dangling ends”, so for all e ∈ R with a vertex v of degree 1 in GR with v ∈ /  we add e to S. This will ensure that the tilings stay within the class of closed domains tilings. Theorem 4 Let X = E2 or H2 and suppose  is a wallpaper or NEC group. Suppose we are given a fundamental tile-1-transitive tiling (F, ) whose edges map onto a graph G embedded in O = X/ . Let S = {e1 , . . . , ek } be a subset of edges of G whose removal avoids dangling ends. Then erasing all preimages of these edges from F results in a non-fundamental tile-1-transitive tiling (T, ), such that the stabiliser group of each tile T ∈ T is isomorphic to the subgroup of  generated by the erased edges. Proof We use the correspondence between edges of a fundamental domain, F ∈ F, and generators for  as described in Theorem 3. If f is an edge of F , then f = F ∩ γ (F ) for an element γ ∈ , and we use the notation [f ] for this Wilkie generator. We also know that each edge f is mapped to another edge f ∈ F (possibly itself) by the element [f ], so that [f ] = [f ]−1 . The image of f in O is an edge e ∈ G, and p(f ) = p(f ) = e. Choose a point x0 ∈ int(F ), and for each ei ∈ G choose a single tile edge fi ∈ p−1 (ei ) ∩ F . Then for each fi , there is a simple orbifold loop αei based at x = p(x0 ), with αei (0) = x0 and αei (1) = [fi ](x0 ), such that αei ([0, 1]) is a connected curve in X that intersects the boundary of F in a single point of fi . This is possible because F is path-connected. In the deck-transformation correspondence between π1orb (O, x) and , we then have that [αei ] ∼ [fi ]. Now let H be the subgroup of   generated by the group elements associated with edges ei ∈ S and let T = η∈H η(F ). By this definition H is the stabiliser subgroup of the tile T . T is path connected by the following argument. Each orbifold loop αei has a connected representative from x ∈ F to [fi ](x) in X, and any other such connected representative of αei has the end-points γ (x) and γ [fi ](x) for some γ ∈ . Therefore, for each η ∈ H , there is a path in X from x to η(x) that lies entirely within T , obtained by writing η as a word in [f1 ], . . . , [fk ], and forming the corresponding concatenation of the lifts of the αei loops. If H is infinite, T must be unbounded as it is the union of infinitely many distinct copies of F . If H is finite, then T will be bounded. Next consider the action of an isometry, γ ∈  on the tile T . We use the facts that the construction of T began with a particular choice of fundamental domain tile F ∈ F, and that  acts transitively on the tiling F. If γ ∈ H , then γ F ⊂ T ,

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Fig. 2 Fundamental tile-1-transitive tilings in H2 are drawn with blue edges, and the full set of lifts of orbifold loops are drawn in green. A set of generators of each symmetry group is labelled 1, 2, 3, 4 according to their order in the given Conway symbol. The symbols given in each case are the orbifold of the fundamental tiling followed by the ribbon tile stabiliser group. (a) 2223; 23∞. (b) 2224; ∞∞. (c) 2224; 24∞

γ η ∈ H for all η ∈ H and so γ T = T . If γ ∈ / H , then γ F ⊂ T , and in particular, γ ηF ⊂ T for any η ∈ H , so that γ (int(T )) ∩ int(T ) = ∅. It follows that for any two γ , γ ∈ / H , that either γ T = γ T or γ (int(T )) ∩ γ int(T ) = ∅. Let T be the union of all distinct images of T . It then follows from the tile-transitivity of (F, ), that (T, ) is also a tile-transitive tiling of X.   Theorem 5 Let (T, ) be a tile-1-transitive tiling, possibly obtained via edge deletion from a fundamental tile-1-transitive tiling as in Theorem 4. If the stabiliser group H is infinite, then the tile T is simply connected: it is a ribbon or branched ribbon. If H is finite, then T is a disk. Proof Assume first that H is infinite. From Theorem 4, it follows that X is the union of path-connected, unbounded tiles of the form γ T for γ ∈  and that all of these tiles have disjoint interiors. By Lemma 1, H contains a translation. If T is not simply connected, then its complement includes a bounded component of X that is covered by isometric copies of T , which is clearly a contradiction. If H is finite, then T will be bounded. The tile T must be simply connected: If it were not, then it contains a boundary component that is homeomorphic to a compact circle C bounding a disk D. Now, (T, ) is tile-transitive, so D must contain isometric copies of T , which is clearly impossible, since any such copy would itself contain yet more copies of T , which would eventually contradict the local finiteness of the tiling T. In particular, in this case, the erased edges must be one of the three cases discussed in Sect. 3.   Figure 2 illustrates how orbifold loops are associated with unbounded tiles. The tile edges are drawn in blue and the lifts of the loops in green. Edges crossed by an orbifold loop are deleted to obtain ribbon tiles, and the green graphs collapse to trees embedded in these tiles. Isometries of the ribbon tiles are isomorphic to a group action on this tree.

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Proposition 1 Let (T, ) be a tile-1-transitive tiling. Let T be the tile produced via edge deletion of two non-adjacent edges and their orbits under , avoiding dangling ends, of a fundamental tile F in T. Then the produced tile T is a ribbon tile. Proof The stabiliser subgroup H of T is generated by the deleted edges by Theorem 4. If H is finite, then by Theorem 5, it must have resulted from a classical gluing procedure as described in Sect. 3. This does not include deleting two nonadjacent edges from a tiling. Therefore, H must be infinite and contain a translation by Lemma 1.   This is not a complete characterisation of edge deletions that create ribbon tiles. For example, deleting two neighbouring edges from a fundamental domain tiling can result in either a disk-like or a ribbon tiling, see Fig. 9 for an example.

5 Enumeration and Classification of Tile-Transitive Tilings In this section we discuss the enumeration of equivariant equivalence classes of tile-1-transitive tilings, prove the equivalence of combinatorial equivalence and equivariant equivalence, and give a Euclidean example. We first establish the following observation that facilitates our enumerative approach. Proposition 2 By successively deleting edges, as in Theorem 4, from all fundamental tile-1-transitive tilings, one can generate all combinatorial equivalence classes of tile-1-transitive tilings. For non-fundamental disk-like tilings, the statement of the proposition is discussed in [17, Section 4] and [7]. Proof First note that it is clear that by adding edges to any tiling by ribbons, it is possible to turn it into a fundamental disk-like tiling. What we need to show is that it is possible to turn the ribbon tiling into a fundamental disk-like tiling with one tile orbit. The edges of any tile-1-transitive equivariant tiling (T, ) project to the edges of a graph G on the quotient space O = X/ . If T is a ribbon tiling, then O \ G must contain a non-contractible loop or points of the singular locus. We find a finite set C of simple curves based at b ∈ O \ G such that O \ (G ∪ C) is a single disk without any elements of the singular locus in its interior. Then, the lift of G ∪ C to X delineates a fundamental domain tiling for  such that edge C yield the original tiling T.   Before discussing the enumeration algorithm, we first prove the equivalence of combinatorial equivalence and equivariant equivalence of tilings. Proposition 3 Let (Ti , i ) (i = 1, 2) be two equivariant tile-1-transitive tilings. Then a tile T1 ∈ T1 is combinatorially equivalent to a tile T2 ∈ T2 if and only if (T1 , 1 ) and (T2 , 2 ) are equivariantly equivalent.

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Proof As mentioned before, it is clear that equivariant equivalence implies combinatorial equivalence. We note that even in the case that both T1 and T2 are trivial tilings the conclusion depends on a deep result [30, Theorem 3] on the classification of NEC groups (and the Bieberbach theorems [1, 2, 40] in the Euclidean case) that states that the algebraic isomorphism between the stabiliser subgroups of the two trivial tilings is induced by a homeomorphism of the universal covering space X. See also [45, Theorems 4.6.3], from which the statement of the proposition follows for tile-1-transitive disk-like tilings. For general ribbon tilings, Proposition 2 implies that inserting edges appropriately cuts Ti into copies of some fundamental domains T˜i ⊂ X for i (i = 1, 2). Furthermore, each edge ei of T˜i has an associated Wilkie generator corresponding to a loop λei in X/ i . This loop has a one-to-one lift that intersects exactly one edge of T˜i once, and no copies of any other edges. Note that combinatorial equivalence implies that the subgroups of i generated by the Wilkie generators associated to inserted edges are isomorphic. Moreover, the equivalence of all other edges along with their stabilisers and those of vertices means that 1 and 2 are generated by similar elements induced by edge identifications of T˜i . Thus, 1 and 2 are isomorphic. As a result, there is a way to insert the edges in both X/ 1 and X/ 2 so that the presentation of 1 and 2 w.r.t. the Wilkie generators is the same. We assume that this is the case for T˜1 and T2 . Then, since algebraic isomorphisms of symmetry groups are induced by homeomorphisms of X [45, Theorem 4.6.4], any correspondance between the Wilkie generators of the two tilings by the fundamental domain T˜1 , resp. T˜2 is induced by a homeomorphism that maps the generators accordingly. Now, Wilkie generators of a fundamental domain tiling naturally give rise to the dual tiling by choosing a base point in the fundamental domain associated to a set of Wilkie generators and connecting this point to its image under each generator by a simple path that intersects exactly one edge of the tiling. Therefore, we see that, after possibly applying a 2 -equivariant isotopy (a lifted isotopy of the orbifold X/ 2 ), that there exists a homeomorphism that sends T˜1 to T˜2 sends the edges accounted for in T1 to edges in T2 .   While the definition of equivariant equivalence relies on the notion of the tiling and symmetry groups, the definition of combinatorial equivalence is cast in terms of the image of the projection of the tile, its edges and vertices to the quotient space. Proposition 3 shows that we can equivalently view equivalence classes of tile-1-transitive tilings as combinatorial classes of graphs on the (labelled) quotient space associated to an orbifold. We are now in a position to enumerate equivariant equivalence classes of crystallographic tile-transitive tilings obtained via the generalised glue operation described in the previous section. The steps are as follows: 1. Select a symmetry group of interest, and construct its orbifold. 2. Enumerate the finitely many possible tile-1-transitive fundamental tilings with methods described in [27, 28] or [17], and represent these as graphs embedded in the orbifold.

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Fig. 3 Left to right: A Euclidean orbifold quotient space with symbol 22∗. A slightly deformed standard fundamental domain (the vertical lines of this tiling lie along the mirrors and the two inequivalent centres of rotation are marked). The corresponding edge graph (in black) for this tiling

Fig. 4 The other combinatorial types of fundamental domain for 22∗ depicted as tilings and their edge graphs below. From left to right, the tilings are equivariantly equivalent to triangular, quadrilateral, and pentagonal polygonal regions

3. Systematically delete subsets of edges from the embedded graphs as described in the section above to derive all tile-1-transitive non-fundamental tilings, both regular ones with bounded tiles and general ones with unbounded ribbon or branched-ribbon tiles. 4. Keep only one representative of any set of equivariantly equivalent tilings. Steps 1–3 are illustrated for the wallpaper group pmg with Euclidean orbifold 22∗ in Figs. 3, 4, 5, and 6, reproducing the corresponding results from [14, Table 6.5.1] with our methods. For step 4, we rely mainly on Proposition 3 to check for equivalence among graphs on the orbifold that represent tilings and leave an algorithmic approach for future endeavours. The resulting list of tilings from the above enumeration procedure up until the last step will naturally contain equivariantly equivalent duplicates. For the tile-1transitive cases where the orbifold topology and singular locus is not too complex, it is possible to determine the equivalence classes by eye because combinatorial and equivariant equivalence are the same. For a more systematic approach, it is

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Fig. 5 By deleting subsets of edges from the four fundamental domain edge graphs shown in Figs. 3 and 4, we obtain five possible equivariant equivalence classes of non-fundamental tilings of 22∗. The edges that remain are shown in black. The red dotted line represents a mirror boundary. The frieze group stabilisers are given by their quotient space symbol. There are also five possible non-fundamental disk-like tilings, not illustrated. (a) ∞∞. (b) ∞∗. (c) 22∞. (d) ∗∞∞. (e) 2 ∗ ∞

Fig. 6 These ribbon tiling patterns correspond directly to the quotient diagrams of Fig. 5; the ribbon tile boundaries map to the black edge-graphs. Each drawing shows 2 by 2 translational unit cells in the symmetry group 22∗. The symmetry of the ribbon in (c) can only be geometrically depicted by marking the tiles, e.g. with an ‘L’ motif. The STSN labels refer to [14, Table 6.5.1]. (a) ∞∞. STS8. (b) ∞∗. STS14. (c) 22∞. STS22. (d) ∗∞∞. STS17. (e) 2 ∗ ∞. STS23

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desirable to have a computable invariant. As already stated, the D-symbol provides such an invariant for disk-like tilings. By Proposition 2, by introducing edges, any tile-1-transitive tiling can be made into a disk-like tiling. On the other hand, there are many ways to introduce these extra edges. A proper treatment of how to find a canonical choice could help in the classification of more general tilings. One idea would be to start with the graph G of tile edges on O and insert some number of coloured edges. The algorithm for enumerating the possible ways of doing this adapts that for fundamental domains as given in [28]. We compare the resulting disk-like tilings using an ordering for D-symbols described in [6, 44]. This ordering allows us to find a unique, minimally complex representative for the tiling that we call the coloured D-symbol. However, a comprehensive treatment of this procedure lies outside the scope of this paper. Remark 1 The enumeration of non-fundamental tile-1-transitive tilings, in both the disk-like and more general tiling cases effectively constructs those subgroups that are generated by a subset of the parent group Wilkie generators for a fundamental tiling. The subgroups are the stabiliser groups of the non-fundamental tile while the parent group generators are dictated by the particular fundamental domain that we start with. We observe that some stabiliser subgroups are derivable from each fundamental domain (examples (c) and (e) in Figs. 5, 6), but this does not hold in all cases. In fact, there are some non-fundamental tilings that arise from edge deletion in just one combinatorial class of fundamental domain (Fig. 5d for example).

6 Hyperbolic Tiling Examples In this section we look at hyperbolic examples illustrating the enumeration in Sect. 5 and the theory developed above for the symmetry group 2224. There are nine combinatorially distinct fundamental domain tilings with this symmetry, shown in Fig. 7. More information about these tilings, coloured pictures and their D-symbols can be found in [33]. Non-fundamental disk-like tilings are built from these fundamental domains by deleting a single edge that terminates at a cone point. This yields five combinatorially distinct tilings, each with stabiliser group either 2• = C2 or 4• = C4 . To construct the tile-1-transitive ribbon tilings, we need to delete at least two edges from one of the nine fundamental domains. After performing all possible combinations, we find that just three combinatorially distinct ribbon tilings are possible, shown in Fig. 8. The example in Fig. 8a can be generated by deleting suitable subsets of edges from six fundamental domains, namely those of Fig. 7c and e–i. The one in Fig. 8b can be built from each of the nine fundamental domain tilings with multiple distinct edge deletions from some domains giving the same tile-class. The branched ribbon tiling in Fig. 8c can be found in eight of the fundamental domain tilings; only the minimal triangular fundamental domain outlined in Fig. 7a cannot support this branched ribbon. Figure 9 illustrates that this

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Fig. 7 Fundamental tilings with symmetry group 2224 with the 4-fold rotation located at vertex 4 in each case. The naming QSn is that used in the epinet database [33]. (a) QS20. (b) QS21. (c) QS22. (d) QS23. (e) QS24. (f) QS25. (g) QS26. (h) QS27. (i) QS28

triangular fundamental domain supports only one type of ribbon tile, the one with stabiliser 22∞ shown in Fig. 8b. Unlike the Euclidean 22∗ example, all three classes of ribbon tilings in Fig. 8 can be created by erasing edges from single fundamental domain tiling; both the fundamental domains in Fig. 7c and f. The combinatorial classification of fundamental domains and ribbon tilings is ultimately obtained by graphs embedded in an orbifold quotient space. This means symmetry groups with orbifold symbols of the same form have the same number of possible ribbon tilings. For example, any group of the form 222a, for a > 2,

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Fig. 8 The three distinct classes of ribbon tilings with symmetry group 2224 with blue edges. (a) Ribbon tiling with stabiliser group ∞∞. (b) Ribbon tiling with stabiliser group 22∞. (c) Branched ribbon tiling with stabiliser group 24∞. The medial axis in red shows the branching structure

Fig. 9 Deleting any pair of edges from the triangular fundamental domain of Fig. 7a leads to a ribbon tiling with stabiliser 22∞. Each of these three tilings is equivalent to that in Fig. 8b

will have just as many equivariantly inequivalent fundamental domains and tile-1transitive ribbon tilings as 2224.

7 Summary and Outlook In this paper, we showed how to enumerate periodic, locally-finite tile-transitive tilings of the Euclidean or hyperbolic plane X, by unbounded ribbon tiles. Returning to the initial inspiration of this work—the question of how to enumerate stripe patterns on the gyroid—this can now be partially achieved by finding the (branched) ribbon tilings in the symmetry groups compatible with the covering map that wraps the hyperbolic plane onto this periodic surface. Moreover, since we fix the symmetry group of the tilings under investigation, the methods of [24, 25] give a natural approach to using the theory developed here for enumerations and investigations into isotopy classes of ribbon tilings on non-simply connected surfaces. The next step in this work is to enumerate tile-k-transitive tilings to include tiles with an infinite stabiliser group. One of the main technical difficulties in extending

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our results to this case concerns the correct definition of the class of tilings under consideration. The standard split and glue methods to enumerate tile-k-transitive Dsymbols can lead to tiles for which cl(int(T )) = T , and tile-intersections T1 ∩ T2 that are not connected. While these technicalities are not insurmountable, they make the definition of the coloured D-symbol and proofs of results substantially more involved. Acknowledgments Both authors would like to thank Myfanwy Evans from the University of Potsdam, and Stephen Hyde, Stuart Ramsden, and Olaf Delgado-Friedrichs from the Australian National University for fruitful discussions and enlivening talks about the problem of classifying crystallographic tilings over many years. BK is supported by grant ANR-17-CE40-0033 of the French National Research Agency ANR (project SoS) (https://members.loria.fr/Monique.Teillaud/ collab/SoS/). VR was supported by ARC Future Fellowship FT140100604.

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16. Hoare, A.H.M., Karrass, A., Solitar, D.: Subgroups of NEC groups. Commun. Pure and Appl. Math. XXVI (1973) 17. Huson, D.H.: The generation and classification of tile-k-transitive tilings of the Euclidean plane, the sphere and the hyperbolic plane. Geom. Dedicata 47(3), 269–296 (1993). https:// doi.org/10.1007/BF01263661 18. Huson, D.H.: Tile-transitive partial tilings of the plane. Contrib. Algebra Geom. 34(1), 87–118 (1993) 19. Huson, D.H.: Ribbon tilings from spherical ones. Geom. Dedicata 63, 147–152 (1996) 20. Hyde, S.T., Landh, T., Lidin, S., Ninham, B., Larsson, K., Andersson, S.: The Language of Shape. Elsevier Science, Amsterdam (1996) 21. Hyde, S.T., Oguey, C.: From 2D hyperbolic forests to 3D Euclidean entangled thickets. Eur. Phys. J. B 16(4), 613–630 (2000). https://doi.org/10.1007/PL00011063 22. Kapfer, S.C., Hyde, S.T., Mecke, K., Arns, C.H., Schröder-Turk, G.E.: Minimal surface scaffold designs for tissue engineering. Biomaterials 32(29), 6875–6882 (2011). http://www. sciencedirect.com/science/article/pii/S0142961211006776 23. Kirkensgaard, J.J.K., Evans, M.E., de Campo, L., Hyde, S.T.: Hierarchical self-assembly of a striped gyroid formed by threaded chiral mesoscale networks. Proc. Nat. Acad. Sci. USA 111(4), 1271–6 (2014) 24. Kolbe, B., Evans, M.E.: Enumerating isotopy classes of tilings guided by the symmetry of triply-periodic minimal surfaces. SIAM Journal on Applied Algebra and Geometry. (2022) 25. Kolbe, B., Evans, M.E.: Enumerating isotopy classes of tilings guided by the symmetry of triply-periodic minimal surfaces. SIAM Journal on Applied Algebra and Geometry. (2022) 26. Luˇci´c, Z., Molnár, E.: Combinatorial classification of fundamental domains of finite area for planar discontinuous isometry groups. Arch. Math. 54(5), 511–520 (1990). https://doi.org/10. 1007/BF01188679 27. Luˇci´c, Z., Molnár, E.: Fundamental domains for planar discontinuous groups and uniform tilings. Geom. Dedicata 40(2), 125–143 (1991). https://doi.org/10.1007/BF00145910 28. Luˇci´c, Z., Molnár, E., Vasiljevi´c, N.: An algorithm for classification of fundamental polygons for a plane discontinuous group. In: Conder, M.D.E., Deza, A., Weiss, A.I. (eds.) Discrete Geometry and Symmetry, pp. 257–278. Springer, Cham (2018) 29. Macbeath, A.M.: Groups of homeomorphisms of a simply connected space. Ann. Math. 79(3), 473 (1964). https://doi.org/10.2307/1970405 30. Macbeath, A.M.: The classification of non-Euclidean plane crystallographic groups. Can. J. Math. 19, 1192–1205 (1967). http://www.cms.math.ca/10.4153/CJM-1967-108-5 31. Mohar, B., Thomassen, C.: Graphs on Surfaces. The Johns Hopkins University Press, Baltimore (2001) 32. Pedersen, M.C., Hyde, S.T.: Polyhedra and packings from hyperbolic honeycombs. Proc. Nat. Acad. Sci. 115(27), 6905–6910 (2018). http://www.pnas.org/lookup/doi/10.1073/pnas. 1720307115 33. Ramsden, S., Robins, V., Hyde, S.: Euclidean patterns in non-Euclidean tilings. http://epinet. anu.edu.au/. Accessed 20 March 2020 34. Ratcliffe, J.: Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics. Springer, New York (2006). 35. Renault, D.: The uniform locally finite tilings of the plane. J. Comb. Theory Ser. B 98(4), 651–671 (2008). https://doi.org/10.1016/j.jctb.2007.10.003 36. Robins, V., Ramsden, S.J., Hyde, S.T.: A note on the two symmetry-preserving covering maps of the gyroid minimal surface. Eur. Phys. J. B 48(1), 107–111 (2005) 37. Sadoc, J., Charvolin, J.: Infinite periodic minimal surfaces and their crystallography in the hyperbolic plane. Acta Crystallogr. A 45(1), 10–20 (1989) 38. Serre, J.: Trees. Springer Monographs in Mathematics. Springer, Berlin (2002) 39. Sharon, E., Roman, B., Marder, M., Shin, G.S., Swinney, H.L.: Buckling cascades in free sheets. Nature 419(6907), 579–579 (2002). https://doi.org/10.1038/419579a 40. Szczepa´nski, A.: Geometry of Crystallographic Groups. Algebra and Discrete Mathematics. World Scientific, Singapore (2012)

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Graph Pseudometrics from a Topological Point of View Ana Lucia Garcia-Pulido, Kathryn Hess, Jane Tan, Katharine Turner, Bei Wang, and Naya Yerolemou

Abstract We explore pseudometrics for directed graphs in order to better understand their topological properties. The directed flag complex associated to a directed graph provides a useful bridge between network science and topology. Indeed, it has often been observed that phenomena exhibited by real-world networks reflect the topology of their flag complexes, as measured, for example, by Betti numbers or simplex counts. As it is often computationally expensive (or even unfeasible) to determine such topological features exactly, it would be extremely valuable to have pseudometrics on the set of directed graphs that can both detect the topological differences and be computed efficiently. To facilitate work in this direction, we introduce methods to measure how well a graph pseudometric captures the topology of a directed graph. We then use these methods to evaluate some well-established pseudometrics, using test data drawn from several families of random graphs.

A. L. Garcia-Pulido Department of Computer Science, University of Liverpool, Liverpool, UK e-mail: [email protected] K. Hess Laboratory for Topology and Neuroscience, Brain Mind Institute, Ecole polytechnique fédérale de Lausanne (EPFL), Lausanne, Switzerland e-mail: [email protected] J. Tan () · N. Yerolemou Mathematical Institute, University of Oxford, Oxford, UK e-mail: [email protected]; [email protected] K. Turner Mathematical Sciences Institute, Australian National University, Canberra, Australia e-mail: [email protected] B. Wang School of Computing, University of Utah, Salt Lake City, UT, USA e-mail: [email protected] © The Author(s) and the Association for Women in Mathematics 2022 E. Gasparovic et al. (eds.), Research in Computational Topology 2, Association for Women in Mathematics Series 30, https://doi.org/10.1007/978-3-030-95519-9_5

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1 Introduction A typical strategy for studying complex networks is to extract features (i.e., parameters, properties, etc.) of the networks that are simpler to analyze and compare than the networks themselves, yet still capture their essential structures. Depending on the context, these features can be local or global and include, for instance, the number of connected components, the number of cycles, the lengths of shortest paths/cycles, graphlet counts, and the graph spectrum, as well as many nonnumerical properties. One method for extracting properties of a directed network begins with the construction of its associated directed flag complex, which is a topological space built from the directed cliques in the graph. Topological features of the directed flag complex provide revealing properties of the original network. For instance, the homology groups of the directed flag complex reflect how cliques assemble to form the graph globally, enabling us to distinguish between some graphs: if the homology groups of two graphs are different, then the two graphs are not isomorphic. From the homology groups one can extract Betti numbers, which measure the rank of the homology groups. While the 0-th Betti number is simply the number of connected components of the graph, higher Betti numbers measure how intricately higher dimensional cliques intersect. In contrast to classical graph parameters, these invariants capture higher order structures. Applications of topological methods to the analysis of networks have been motivated, in particular, by the desire to understand the relation between function and structure of biological networks (see e.g. [8, 9, 18, 24, 28, 29]). The work in these articles strongly suggests that biological function reflects topological structure, as measured by Betti numbers for example. A significant barrier to confirming and exploiting this observation, however, is the computational complexity of determining these features for real-world networks. For instance, directly comparing the homology groups or the Betti numbers of two flag complexes is an NP-hard problem [1]. In this paper, we explore the hypothesis that there are pseudometrics on the set of directed graphs that detect differences in their topological features. As a first step to identifying or constructing such a pseudometric, we introduce methods to measure how well a given pseudometric reflects differences in the topological features of two graphs. We implement our methods of comparison and apply them to existing candidate pseudometrics on the set of directed graphs. There are already many high-performing pseudometrics on the space of graphs that take into account onedimensional structural features (see, e.g., [14, 21, 31]), but there is a dearth of literature on whether they also capture any high-dimensional structure. We note that these methods can also be applied to undirected graphs by considering instead the usual flag complex. However, the analysis in the present paper is restricted to the directed case, since many real-world networks (particularly biological networks)

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are naturally directed. Our test data set is drawn from several families of directed random graphs for which we can control the parameters and behavior. We present here an experimental study, comparing topological pseudometrics based on Betti numbers and simplex counts with several well-established pseudometrics for directed graphs. Our analysis is four-fold. First, we study the similarities between clusterings based on these pseudometrics by computing both their FowlkesMallows indices and the distance correlations between them. Almost all of the pseudometrics tested are shown to be closely related with high Fowlkes-Mallows indices and high distance correlations. Second, we apply k-nearest neighbors (kNN) classification as a measure of classification accuracy for three models of random graphs. We find that all pseudometrics achieve near perfect classification accuracy. Third, we test for relationships between each pseudometric and the various random graph parameters by performing permutation tests with distance correlation and the Fowlkes-Mallows indices. Using the permutation tests, we can reject the null hypothesis of independence between all the different pseudometrics when considered over pooled sets of directed graphs with multiple parameter values. However, we cannot in general reject the null hypothesis of independence once we restrict to a specific model and parameter value. This indicates that the latent variable of the parameter of the model is important. Finally, we apply k-NN regression to our pseudometrics to try to predict the topological feature vectors of given graphs. Outline We begin by reviewing the requisite topological and combinatorial background in Sect. 2. This includes the definitions of our topological feature vectors together with the topological pseudometrics that they induce, as well as a brief introduction to each of the existing pseudometrics that we compare to our topologybased ones: TriadEuclid, TriadEMD, and Portrait Divergence. We present two methods for comparing pseudometrics in Sect. 3, one based on clustering and the other on distance correlation. Here, we also describe the permutation test and kNN regression, as well as the classification methods that we use. Technical details of our experiments can be found in Sect. 4. The final results of the comparison are presented in Sect. 5.

2 Directed Graph Pseudometrics A directed graph G consists of a set of vertices V together with a set of edges E, which are ordered pairs of vertices. All graphs in this paper are finite, meaning V and E are both finite sets. The direction of an edge (u, v) ∈ E is taken to be from u, the origin, to v, the destination. We require that G not contain any self-loops, that is, for each (u, v) ∈ E, u = v. Secondly, for each pair of vertices (u, v) ∈ V × V , there is at most one directed edge from u to v. Note, however, that we do allow both

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(u, v) ∈ E and (v, u) ∈ E. In other words, the directed graphs we consider are simple except for bigons, and we shall simply refer to them as graphs or digraphs. We use several standard definitions associated with digraphs. The out-degree of a vertex v is the number of edges having v as the origin, while its in-degree is the number of edges having v as the destination. A vertex v is a sink if its out-degree is zero and its in-degree is at least one and a source if its in-degree is zero and its outdegree is at least one. A path in G is a list of distinct edges such that the destination of the i-th edge is the same as the origin of the (i + 1)-st edge and such that no vertex is traversed more than once, except the path may end at the vertex where it started. If the first and last vertices of a path are the same, we call it a cycle. Let G denote the set of all finite directed graphs. We will view G as a space endowed with several natural pseudometrics. First, recall that a pseudometric on a set X is a function dX : X × X → [0, ∞) such that for all x, y, z ∈ X, 1. 2. 3.

dX (x, x) = 0; dX (x, y) = dX (y, x) (symmetry); and dX (x, z) ≤ dX (x, y) + dX (y, z) (triangle inequality).

Importantly, points need not all be distinguishable by a pseudometric: it is possible that x = y, even though dX (x, y) = 0. The pair (X, dX ) is a pseudometric space. We will describe three well-established pseudometrics on G: TriadEuclid in Sect. 2.2, TriadEMD in Sect. 2.3, and portrait divergence in Sect. 2.4. Before this though, we define two topological summaries for elements in G based on Betti numbers and simplex counts in Sect. 2.1, which provide a crucial point of comparison.

2.1 Betti Numbers and Simplex Counts The key construction we study in this paper is the directed flag complex of a directed graph. For a more detailed account we refer to the work of Luetgehetmann et al. [16] and Reimann et al. [24]. We assume familiarity with standard notions of abstract simplicial complexes and simplicial homology, introduced for instance in [11, 19]. Definition 1 (Abstract Directed Simplicial Complex) An abstract directed simplicial complex on a vertex set V is a collection K of lists (i.e., totally ordered sets) of elements of V such that for every sequence σ ∈ K, every subsequence τ of σ belongs to K. An element σ ∈ K is called a (directed) simplex. If σ is of length p + 1, then we call it an p-simplex. The collection of p-simplices of K is denoted Kp . If σ ∈ K and τ ⊂ σ , then τ is called a face of σ . The i-th face of an p-simplex σ = (v0 , . . . , vp ) is the (p − 1)-simplex obtained by removing the vi from the list σ .

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The notion of an abstract directed simplicial complex is a variant of the more common notion of abstract simplicial complex. Henceforth, we always mean abstract directed simplicial complexes when we say simplicial complexes. The following definition illustrates how directed simplicial complexes arise naturally from directed graphs. Definition 2 (Directed Flag Complex) Given a directed graph G = (V , E), the directed flag complex of G, denoted K = K(G), is defined as follows. • Take K0 = V . • For p ≥ 1, a directed p-simplex σ in Kn is an (p + 1)-tuple of vertices (v0 , . . . , vp ) such that there is a directed edge from vi to vj for every pair of vertices vi , vj with 0 ≤ i < j ≤ p. For σ = (v0 , . . . , vp ) ∈ K, we call v0 the source of the simplex, since there is a directed edge from v0 to vi for every i > 0. Similarly, we call vp the sink of the simplex, since there is a directed edge from vi to vp for every i < p. Note that these are consistent with the equivalent digraph notions since a p-simplex is characterised by the ordered sequence of vertices and not by the underlying set of vertices. Throughout this paper, we consider simplicial homology of directed flag complexes with F2 -coefficients, where homology is defined in the usual way. For a simplicial complex K, let Hp (K) denote its p-th homology group and βp (K) = dim Hp (K) its p-th Betti number. Given G ∈ G, we work with two simple topology-based feature vectors on G, defined as follows. Let p ∈ Z≥0 denote the maximum homological dimension of interest, which is context-dependent; in our experiments, p = 6. For 0 ≤ k ≤ p, let bk (G) = max{0, log (βk (K (G)))} and   b(G) = b0 (G), . . . , bp (G) . Let γk (G) be the number of directed k-simplices of K(G), ck (G) max{0, log (γk (K (G)))}, and

=

  c(G) = c0 (G), . . . , cp (G) . The vectors b(G) and c(G) consist of the logarithms of the Betti numbers and simplex counts of K(G), respectively. Let $ · $2 denote the Euclidean norm on Rn . Definition 3 (Topological Pseudometrics) The pseudometrics dβ and d on G are specified by dβ (G, G ) = $b(G) − b(G )$2 ;

(1)

d (G, G ) = $c(G) − c(G )$2 .

(2)

for any pair of directed graphs G, G ∈ G.

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2.2 TriadEuclid First introduced by Przulj et al. [23], the term graphlet is often used to mean a small connected graph up to a fixed size. A graphlet-based pseudometric compares the graphlet counts in pairs of graphs. Graphlet-based pseudometrics are a well established tool in network analysis, as they capture local structural similarity of two graphs. Xu and Reinert [35] introduced two graphlet-based pseudometrics that outperform the best previously defined directed graphlet methods in graph classification tasks: TriadEuclid and TriadEMD. Both of these consider only directed graphlets on three vertices. In this section we focus on TriadEuclid, which measures the difference between 3-graphlet counts of two directed graphs in terms of their Euclidean distance. We remark that there are exactly 13 isomorphism classes of connected directed graphlets with 3 vertices (referred to as 3-graphlets). A complete list can be found in [35]. Let G = (V , E) a directed graph. If the induced subgraph determined by three vertices of G is connected, it must be isomorphic to one of the 13 3-graphlets. Let ni (G) be the number of induced subgraphs of G that are isomorphic to the i-th graphlet, i ∈ I = {1, . . . , 13}. Define φ(G) ∈ N13 by φ(G)j = 

nj (G) i∈I ni (G)

(3)

for j ∈ I . Definition 4 The TriadEuclid pseudometric on the set G of finite directed graphs is defined by TriadEuclid(G, G ) = $φ(G) − φ(G )$2 ,

(4)

for all G, H ∈ G, where $ · $2 denotes the Euclidean norm. The complexity of this algorithm is O(nd 2 ), where d is the maximum degree of vertices in G and G , n is the maximum number of vertices in G and G .

2.3 TriadEMD TriadEMD is another graphlet-based pseudometric defined by Xu and Reinert [35], computed in terms of the earth mover distance between generalized degree distributions of two directed graphs. Definition 5 Given a directed graph H = (V , E), an automorphism of H is a graph isomorphism h : H → H . The set of automorphisms of H forms a group under composition, denoted Aut(H ), which acts on H .

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1

Fig. 1 A graph with two vertex orbits {1} and {2, 3}

3

2 Fig. 2 Graph G with  Orb {1} -degree distribution given in Table 1

1 2 3

  Table 1 Orb {1} -degree distribution of G Fig. 2

k = degree P{1} (k)

0 0.5

4

1 0.25

2 0

3 0.25

Let v ∈ V . The orbit of v under the action of Aut(H ) is defined by Orb(v) = {h(v)|h ∈ Aut(H )}.

(5)

Projecting to orbits, one considers the genuinely different positions that a vertex takes in a fixed graphlet. Figure 1 shows an example of a three-vertex graph with only two distinct orbits: Orb(1) = {1},

Orb(2) = Orb(3) = {2, 3}.

A complete list of the 30 orbits of 3-graphlets can be found in [35]. A list of graphlets with between two and four vertices is given in [26]. Degree distributions can be used to study hubs in a network, i.e., vertices of high degree. The degree distribution P of an undirected graph is defined so that P (k) is the proportion of vertices in G with degree k: P (k) =

#{v ∈ V | deg(v) = k} . #V

(6)

When G is directed, there are analogous in-degree and out-degree distributions. TriadEMD is defined in terms of the degree distributions of orbits in triads. For any orbit i of a fixed graphlet, the orbit-i-degree of a vertex v of a digraph G is the number of copies of orbit i in G of which v is a vertex [35]. The orbit-i-degree distribution Pi of G is defined so that Pi (k) is the proportion of vertices in G with orbit-i-degree k. Consider, for example, the G on four vertices shown in Fig. 2, for which  graph  we now calculate the Orb {1} -degree distribution of the graphlet in Fig. 1. The   Orb {1} -degrees of vertices 1, 2, 3, and 4 are 1, 3, 0, and 0 respectively; Table 1   gives the Orb {1} -degree distribution.

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The original definitions of the orbit degrees and orbit graphlet metrics for undirected and directed graphs, together with additional examples can be found in [22, 26, 34, 35]. Definition 6 Let  be the set of 30 orbits of 3-graphlets. The TriadEMD pseudometric on the set G of directed graphs is given by TriadEMD(G, G ) =

1  EMD(Pi , Pi ), 30

(7)

i∈

where Pi (resp. Pi ) denotes the orbit-i-degree distribution of G (resp. G ), and EMD denotes the earth mover distance (EMD) between the distributions. Recall that the EMD between two probability distributions P and Q on the real line is defined by  EMD(P , Q) =

∞ −∞

|FP (x) − FQ (x)|dx,

(8)

where FP and FQ are the cumulative distribution functions of P and Q respectively. The complexity of this algorithm is O(n log n), where n is the maximum between the numbers of vertices of G and of G .

2.4 Portrait Divergence First introduced by Bagrow and Bollt [2], Network Portrait Divergence is a pseudometric that compares the distributions of the shortest-path lengths of two graphs. Recall that the diameter of a finite directed graph is the maximum over all pairs of vertices of the length of the shortest (directed) path from the first vertex to the second. Definition 7 Let G = (V , E) be a directed graph of diameter d, with n vertices, and denote by dsp (v, u) the shortest path distance from v to u. For every vertex v ∈ V and 0 ≤ l ≤ d, define svl = #{u ∈ V |dsp (v, u) = l}.

(9)

The network portrait [3] of G is the matrix B = (Blk ) with entries Blk = #{v ∈ V |svl = k},

(10)

where 0 ≤ l ≤ d and 1 ≤ k ≤ n − 1. That is, Blk is the number of vertices that are at distance l from exactly k other vertices.

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The first row of the network portrait B of G is precisely the degree distribution of G. The second row is the degree distribution of next-nearest neighbors, and so on. The network portrait B also captures structural features of G such as the number of edges, the diameter of G, and the distribution of shortest paths. It has been shown to be a graph invariant [2]. The complexity of computing an unweighted portrait is O(mN + N 2 ), due to the procedure of finding minimum-length paths, where m is the number of edges and N the number of vertices in G. Let G = (V , E) be a directed graph. Let P (k, l) be the probability of choosing two vertices (u, v) ∈ V ×V uniformly at random such that dsp (u, v) = l and sul = k. If sul = k, then there exist k vertices u1 , . . . , uk with dsp (u, ui ) = l. Therefore, #{(u, v) ∈ V × V : dsp (u, v) = l, sul = k} = k · #{v ∈ V : svl = k} = k · Blk , and in consequence, P (k, l) =

kBlk . n2

(11)

Given graphs G and G , we define (joint) distributions P and P for all rows of their portraits B and B . The KL-divergence between P and P is then defined by KL(P ||P ) =

max(d,d N  ) l=0

k=0

P (k, l) log

P (k, l) . P (k, l)

(12)

Definition 8 The Portrait Divergence (PD) pseudometric on the set G of finite directed graphs is the Jensen-Shannon divergence, P D(G, G ) =

1 1 KL(P ||M) + KL(P ||M), 2 2

(13)

for all G, G ∈ G, where M = (P + P )/2 is the mixture distribution of P and P .

3 Statistical Tools We are interested in comparing known pseudometrics on G (e.g., TriadEuclid, TriadEMD, and PD) to dβ and d , which requires some care. A natural first idea would be to use Gromov-Hausdorff distance, but since it measures how close two spaces are to being isometric, it is too rigid for practical purposes. For example, changing even just one Betti number of a single graph can result in a very large Gromov-Hausdorff distance. Multiplying a topological pseudometric by a constant has a similar effect as well.

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Instead, we employ two comparison methods that are invariant under rescaling either pseudometric and more robust under perturbations of points. The first method is based on the distance correlation between pseudometrics (Sect. 3.1), while the second is based on the Fowlkes-Mallows index of the clusterings of the two pseudometrics (Sect. 3.2). Both rely on taking finite samples S ⊂ G and performing an analysis on S. We remark that similar comparison methods can also be applied to undirected graphs by replacing the directed flag complex with the flag complex, though we restrict to the directed case in this paper.

3.1 Distance Correlation The concept of distance correlation was first introduced in by Szekely et al.[30] for two paired random vectors in Euclidean space and generalized to metric spaces by Lyons [17]. Distance correlation measures linear and non linear relationships between two distributions lying in possibly different metric spaces. We follow the definition of the sample distance correlation of a paired sample as formulated by Turner and Spreemann [33]. We use the sample distance correlation as an estimation of the distance correlation and refer to the sample distance correlation simply as the distance correlation. Let (X, dX ) and (Y, dY ) be pseudometric spaces, and let (X, Y ) =   (xi , yi ) 1≤i≤l ⊂ X × Y be paired samples. For 1 ≤ i, j ≤ l, let ai,j = dX (xi , xj ) and bi,j = dY (yi , yj ), so that a = (ai,j ) and b = (bi,j ) denote matrices of pairwise i

distances in X, and Y, respectively. Let a i and b denote the row means and a j and bj the column means of the matrices a and b. Let a and b denote the total matrix means. Define doubly centred matrices (Ak,l ) and (Bk,l ) by Ak,l = ak,l −a k −a l +a k and Bk,l = bk,l − b − bl + b. Definition 9 The sample distance covariance of the paired sample (X, Y ) is dcov(X, Y ) =

n 1  Ak,l Bk,l . n2

(14)

k,l=1

When dcov(X, Y ) ≥ 0, let dCov(X, Y ) = (dcov(X, Y ))1/2 . The sample variance of the sample X is defined to be ⎛

⎞1/2 n  1 dVar(X) = ⎝ 2 A2k,l ⎠ . n

(15)

k,l=1

If dVar(X) dVar(Y ) = 0 and dcov(X, Y ) ≥ 0, the sample distance correlation is given by

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dCor(X, Y ) =

dCov(X, Y ) . (dVar(X) dVar(Y ))1/2

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If dVar(X) dVar(Y ) = 0, then we set dCor(X, Y ) = 0. For strongly negative metrics spaces, which includes Euclidean space, the sample distance covariance between X and Y is always non-negative, and X and Y are independent if and only if dCov(X, Y ) = 0, see [17]. This is not true for all metric spaces, but in all of our cases the sample covariance is non-negative, and therefore the sample distance correlation is well-defined. Notice that sample distance correlation is invariant under scalar multiplication of either of the two metrics. Cauchy-Schwarz implies that | dCor(X, Y )| ≤ 1 and that equality is attained when the doubly centred matrices are scalar multiples of each other. Thus, high correlation measures the (possibly non-linear) relationship between X and Y arising from a linear relationship between their corresponding metrics. In our case we have Y = X, and we consider the sample distance correlation on paired samples (X, X).

3.2 Fowlkes-Mallows Index Our second method of comparing pseudometric spaces is based on clustering. Suppose we have pseudometric spaces (X, d0 ) and (X, d). Given a finite sample X ⊂ X, we can compute hierarchical clusterings A0 and A corresponding to both pseudometrics and hence apply standard techniques of cluster evaluation. In particular, we use the Fowlkes-Mallows index [7], which provides a measure of similarity between two clusterings. Making such comparisons across many choices of sample X provides a means of comparing (X, d) to (X, d0 ). We employ a standard agglomerative hierarchical clustering algorithm (see documentation of [12] for details), initialised with every point in a separate cluster. In each step, two clusters of minimum distance to each other are chosen and merged into one to move up the hierarchy. We use complete linkage, so the distance between clusters is the maximum distance between any two points of those clusters. The algorithm terminates when all points are in a single cluster. In some applications, the number of expected clusters is known from the outset, allowing early termination of the algorithm. As this does not apply to our case, we first run the clustering algorithm and then use silhouette analysis to choose the most natural number of clusters with respect to d0 . This method, introduced by Rousseeuw [25], assesses how well a clustering captures the structure of the data using only internal distance information. Informally, we judge how well a single point x has been placed within a cluster based on a silhouette value, which quantifies how close x is to the other points in its own cluster in contrast to points in other clusters, and then extend to a numerical score for the clustering.

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Definition 10 Suppose that we have points in a pseudometric space (X, d) partitioned into clusters C1 , . . . , Ck . Given a point u ∈ Ci , let a(u) be the mean distance between u and other points in Ci , that is a(u) =

1 |Ci | − 1



d(u, v).

(17)

v=u,v∈Ci

For every j = i, let aj (u) be the mean distance between u and points in cluster Cj , aj (u) =

1  d(u, v). |Cj |

(18)

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Set b(u) = minj =i aj (u). If |Ci | > 1, define the silhouette value of u by s(u) =

b(u) − a(u) , max{a(u), b(u)}

(19)

and set s(u) = 0 if |Ci | = 1. It follows immediately from the definition that s(u) ∈ [−1, 1] and that • s(u) ≈ 1 indicates that u is appropriately clustered; • s(u) ≈ −1 means that u is closer to points from a different cluster, • if s(u) ≈ 0, then u is almost equidistant between Ci and at least one other cluster Cj . Based on silhouette values of the points in each of the clusters, one can define a score for the entire clustering. Definition 11 The silhouette coefficient of the clustering C1 , . . . , Ck of l points u1 , .., ul is their average silhouette value: SC =

1  s(ui ). l

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1≤i≤l

The definition above enables us to determine if the clustering C1 , . . . , Ck reflects a natural structure present in our samples or if instead it seems forced. The actual comparative element in our method based on the silhouette coefficient comes from the use of the Fowlkes-Mallows index [7], which measures either the similarity between two clusterings with k clusters or the similarity between one clustering and a benchmark classification. We use the former, but both proceed by considering whether pairs of points are consistently assigned to clusters or classified. Specifically, let X be a set with n elements, and let A and A be two clusterings of X with k clusters each. Say that a pair of points are together in a clustering if and only if they are assigned to the same cluster. Let T P (true positives) be the number of pairs of points that are together in both A and A , F P (false positives) the number of

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pairs together in A but not in A , F N (false negatives) the number of pairs together in A but not in A, and T N (true negatives) the number of pairs that are not together in either A or A . Definition 12 The Fowlkes-Mallows index F Mk of the pair A, A (for clusterings with k clusters) is given by ! F Mk :=

TP TP · . T P + FP T P + FN

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Notice that F Mk ∈ [0, 1] and that the clusterings A and A are identical when F Mk = 1 and the most dissimilar when F Mk = 0. In our experiments, given a directed graph, we apply silhouette analysis to determine the number of clusters k that gives the highest silhouette score for dβ (respectively, d ). We then compare a known pseudometric d (triadEuclid, triadEMD, and PD) to dβ (respectively, d ) in terms of the Fowlkes-Mallows index of their associated clusterings. Remark A particular challenge in comparing clusterings is the absence of benchmark datasets. It is also difficult to determine the optimal number of clusters for a given dataset. In order to circumvent this issue, we use silhouette analysis to determine the most appropriate number of clusters. In our experiments, we obtain a unique number of clusters that realizes the maximal silhouette coefficient; however, this is not guaranteed in general. In the case where multiple numbers of clusters attain the same maximum silhouette score, one could easily adapt the present method by computing Fowlkes-Mallows indices for every viable number of clusters and then taking, for instance, the mean score.

3.3 Permutation Tests for Paired Data The permutation test has become a default method for testing independence. It is a provably valid and consistent test for any consistent dependency measure, such as distance correlation. For more details concerning permutation tests with distance correlation we refer to [27]. A permutation test for paired data involves computing some summary statistic (in our case, either distance correlation or the Fowlkes-Mallows index) on the original pairing and then comparing this summary statistic to that of a shuffled pairing. If the two pseudometrics are truly independent, then the percentile of the summary statistic amongst all those with permuted pairings is drawn uniformly over possible percentiles. In particular, we can test for a null-hypothesis of independence for two pseudometrics with a permutation test where the sampled p-value is the percentile of the original distance correlation (resp. Fowlkes-Mallows index) among those for the permuted pairings.

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Since we compute many p-values for comparing various pairs of pseudometrics over many different sample sets of directed graphs, we must correct for multiple hypothesis testing. The family-wise error rate is the probability of making one or more false discoveries when performing multiple hypotheses tests. The simplest, most conservative method to bound the family-wise error rate is the Bonferroni correction, which can be applied to any collection of hypothesis tests, regardless of any dependency structure among the variables. In the Bonferroni correction for N tests, we simply divide the target significance level by N ; to bound the family-wise error rate by α we use α/N as the threshold for rejecting the null hypothesis. In order to apply the Bonferroni correction, we consider in our analyses each table of p-values as a family. To consider all the permutation tests in all of the tables of this paper simultaneously, we must instead bound the false discovery rate: the number of discoveries (tests where we reject the null hypothesis) that may be false positives as a proportion of all discoveries made, including both true and false discoveries. To affirm that the expected number of discoveries that are false is at most α, we apply the conservative approach by Benjamini and Yekutieli [4] formulated below, since we do not know what dependency structures there may be among the variables.  1 Theorem 1 [4] Let C(N ) = N j =1 j . Let p(1) ≤ p(2) ≤ . . . ≤ p(N ) be the ordered observed p-values. Define " k := max i : p(i) ≤

# i α . mC(N)

(22)

If no such i exists, reject no hypothesis. If an i exists, and we reject the null hypotheses H(1) , H(2) . . . H(k) , then we have controlled the false discovery rate at a level less than or equal to α.

3.4 k-NN Classification and Regression Given a function f on a data set and a set of training data for which the function value is known, one intuitive approach to estimating f on a test point is to extrapolate from the function values on the training points closest to it. This is the method of k-nearest neighbors (k-NN). There are two slightly different algorithms for this method, depending on whether the unknown function is categorical or real/vector-valued. For a categorical functions, one applies k-NN classification. For a given test point, one considers the k nearest training points and assigns to the test point the category that appears the most frequently among them. We measure success in terms of classification rate, i.e., the proportion of classifications that are correct. For real- or vector-valued functions, k-NN regression is more appropriate. For a given test point, one considers the k nearest training points and assigns to the

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test point the average of their function values. The difference between the function estimate and the true function value is called the residual. The goal is for the residuals to be small as possible. We therefore measure success via the mean squared error (MSE), i.e., the mean of the squares of the errors/residuals. There are many ways to split data into training and testing sets. In this paper, we use leave-one-out, a form of cross-validation, where at each stage we pick a single data point as the testing set and use everything else as the training set. We then repeat the process so that every data point is considered as the test sample at some point.

4 Experimental Setup In this section, we provide the technical and implementation details of our experimental setup. First, we describe the random graph models used to generate our test graphs and the associated parameter selection. We then compute topological pseudometrics dβ and d for the collections of test graphs in order to perform a comparative study.

4.1 Random Graph Models and Parameters In our experiments, we analyze directed graphs from families based on three random graph models: directed Erdös–Rényi random graphs (ER), directed geometric random graphs (GR), and random k-out graphs with preferential attachment (PA). There are numerous standard references for these models (see, for instance, [20]). We focused on these three random models to initialise the study of pseudometrics from a topological perspective. In particular, GR and PA are used to model real world networks, such as mobile ad hoc networks, the World Wide Web, and social networks. We include a brief description with the purpose of describing the specifications used to generate our test data. Random Graph Models A directed ER random graph is generated by starting with a fixed set of n vertices and adding a directed edge between each ordered pair of vertices independently with probability ρ. Note that the edges u → v and v → u are also chosen independently of each other, and in particular it is possible for both to be present. A classic geometric random graph is generated by placing vertices uniformly at random in the unit square, and then adding an edge between two vertices whenever the (Euclidean) distance between the vertices is at most equal to a fixed parameter r. We consider (biased) oriented GR random graphs obtained by taking an undirected graph generated in the classical sense with vertex set {1, . . . , n}, and then for each edge uv (with u < v) choosing the direction u → v with probability 1/3 and v → u

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with probability 2/3. The directions are chosen independently for each edge in the undirected graph, but in this collection it is not possible to have both directed edges between a single pair of vertices. A PA random graph with parameter k is generated as follows: give each vertex an initial weight of 1, and select a vertex u with out-degree less than k, uniformly at random. Choose another vertex v with probability proportional to its weight, add a directed edge from u to v and increase the weight of v by 1. This process terminates when every vertex has out-degree k. The initial output may have parallel (repeated) directed edges, which we subsequently reduce to a single directed edge leaving at most one edge in each direction between any pair of vertices. Test Graphs Our test graphs consist of two collections of directed graphs, all on 500 vertices, generated according to the preceding descriptions. The first collection consists of 120 graphs, with 10 for each of the following parameters: ER with ρ ∈ {0.03, 0.06, 0.1, 0.15, 0.2, 0.25}, GR with r ∈ {0.1, 0.175, 0.3}, and PA with k ∈ {20, 40, 70}. We refer to this collection as the point-drawn collection, since the parameters are chosen from a discrete set of values. For the second collection, we generated 300 directed graphs with 100 for each of the three random graph models. The parameters for a fixed model were obtained by generating 25 values independently uniformly at random from a set of four predefined intervals: ρ values are chosen from intervals in {(0, 0.01), (0.02, 0.03), (0.05, 0.07), (0.09, 0.1)} for the ER graphs, r values from {(0, 0.02), (0.04, 0.05), (0.08, 0.12), (0.15, 0.175)} for GR graphs, and k values from {(4, 7), (12, 18), (22, 25), (27, 30)} for PA graphs. We refer to this collection as the interval-drawn collection. With the point-drawn collection, we aim to understand the relationships of the topological pseudometrics to the model parameter and to be able to determine if strong relationships between a given pseudometric and a topological metric may have originated from the latent parameters of the models. The interval point-drawn collection allows us to study how well a pseudometric predicts topological features. We chose model parameters that ensure that our datasets include graphs with genuinely different topologies and graphs with non-trivial 6th Betti numbers.

4.2 Computing Topological Pseudometrics dβ and d For each test graph, we run our experiments by comparing well-established pseudometrics (TriadEuclid, TriadEMD, and PD) against the topological pseudometrics dβ and d defined in Sect. 2. Computing d β and d  To compute Betti numbers of directed graphs, we use the Flagser [16] software available via a Python implementation pyflagser [32]. For homology computations, Flagser comes with an approximation option, which speeds up computation time while maintaining remarkable accuracy [16]. In particular, it can be used to approximate the homology of a directed flag complex

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(associated with a test graph), by skipping columns that require more reduction steps than a user-chosen approximation parameter (denoted ). We use Flagser to compute the Betti numbers of the directed flag complex of each test graph up to dimension 6. We also compute approximate Betti numbers with  = 100 , 101 , 102 , 103 , and 104 to construct an approximate dβ pseudometric. The corresponding errors of the logarithm of the Betti numbers (i.e., vectors of max{0, log(βk )}) across our full data set is 36.1%, 7.72%, 1.53%, 0.22% and 0.04%, respectively. Since there is no theoretical error estimate for those approximate parameters, we calculated exact Betti numbers and then compared the approximated Betti numbers in order to calculate the errors above. We remark that similar calculations with  = 105 produced the exact Betti numbers. As explained in Sect. 2, we then compute the distance dβ (G, G ) between two directed graphs G and G as the Euclidean distance between the vector of (possibly approximated) Betti numbers associated to the graphs G and G . The pseudometric d is defined similarly, with Betti numbers replaced by simplex counts. A Parameter Distance dp In addition to dβ and d , we compute an additional pseudometric dp for comparison, defined in terms of the parameters associated with our random graph models. For test graphs G and G generated by a fixed random graph model (ER, GR, or PA), we define dp (G, G ) to be the absolute value of the difference of the parameters used to generate them. For instance, if G and G are generated with a ER random graph model with ρ = 0.03 and 0.06 respectively, then dp (G, G ) = |0.03 − 0.06| = 0.03. Details of Implementation All calculations are performed using Python 3.9. The source code for our experiments can be found in our github repository https://github. com/winsy17/graph_pseudo_top_view. Each ER/GR/PA graph is generated using the Python package NetworkX. In the case of the point-drawn collection, we calculate the exact Betti numbers for all the parameters, except for ρ ∈ {0.15, 0.2, 0.25}, r = 0.3 and k = 70 for which we use the approximation option of pyflagser with  = 105 because they are not computationally feasible to calculate exactly. For the interval-drawn collection, we restrict the parameters to values that are computationally feasible and explore in pyflagser. We perform hierarchical clustering with complete linkage as our clustering technique. As a control, we also generate a random positive definite matrix with zeros on the diagonal that we include as a “random” pseudometric, denoted as “random”. Distributions of Betti Numbers As illustrated in Fig. 3, the Betti numbers seem to be strongly related to the parameters used to generate the test graphs. An analogous figure appears in [13] for the undirected case. This is well aligned with the observations in [15], where Lasalle studied the behavior of Betti numbers for directed ER graphs.

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Fig. 3 The logarithm of Betti numbers across a range of parameters for Erd˝os–Rényi random graphs, each with 100 vertices. The x-axis corresponds to the parameter ρ ranging from 0 to 0.39, while the y-axis corresponds to the maximum of the logarithm of Betti numbers and 0. Similar results are observed for geometric random and preferential attachment random graphs

5 Experimental Results In this section, we first study the similarities between clusterings based on dβ and d and those based on the three well-established pseudometrics by computing the Fowlkes-Mallows indices and distance correlations between them (Sect. 5.1), first for each of the three random graph models individually, then for all three combined. This analysis shows that when we consider them model by model, the pseudometrics we study are closely related to one another, with a few exceptions, which vary between models. When all three models are considered simultaneously, the differences between the pseudometrics are substantially more stark. We next apply k-NN classification and report the classification accuracy for each of the three models of random graphs by model parameter and for the full collection of all of our random graphs by model type (Sect. 5.2). We achieve 100% accuracy in almost all cases for the point-drawn collection. In the interval-drawn case, MSE is very low for the ER and GR models and rather high for the PA model. We then explore the relationship of each of our pseudometrics to the various random graph parameters by performing permutation tests with distance correlation

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between a fixed pseudometric and dβ or d , for datasets from the interval-drawn collections (Sect. 5.3). When we consider all graph models together, the very small p-value enables us to reject the null hypothesis of independence of the pseudometrics. The results are more nuanced when we instead study the models individually, one parameter at a time, where we can reject the null hypothesis of independence only rarely and then almost only for the relationship between dβ and d . Finally, we use k-NN regression on our pseudometrics in the interval-drawn case to predict the topological feature vectors b(G) or c(G) and report the MSE (Sect. 5.4). Both dβ and d perform well in predicting b(G) and c(G), better than any other pseudometric for the GR and PA models. The case of the ER model is more complex, as (unsurprisingly) d and dβ best predict c(G) and b(G), respectively, but the other pseudometrics perform quite well, too.

5.1 Fowlkes-Mallows Indices and Distance Correlation Between Pseudometrics We compute the Fowlkes-Mallows (FM) indices and distance correlations of various pairs of pseudometrics for the interval-drawn collection of random graphs; recall that this contains 100 graphs for each of the three models, ER, GR, and PA. We first treat each set of 100 graphs separately and report the resulting FM indices and distance correlations for ER, GR, and PA in Figs. 4, 5, and 6, respectively. We then combine all 300 graphs and report the results in Fig. 7. Recall that a higher value for the FM index indicates greater similarity between the clusterings induced by two pseudometrics. On the other hand, a higher value of distance correlation measures a higher level of dependence between the two

Fig. 4 Interval-drawn collection: Erdös–Rényi (ER) random graphs

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Fig. 5 Interval-drawn collection: geometric random (GR) graphs

Fig. 6 Interval-drawn collection: preferential attachment (PA) random graphs

Fig. 7 Interval-drawn collection with all three random graph models

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pseudometrics. Both the FM index and distance correlation take values between 0 and 1. Starting from Fig. 4, we study various pseudometrics for a collection of 100 ER random graphs. The first column lists the pseudometrics we consider: from top to bottom, the topological pseudometrics based on Betti numbers (dβ ) and simplex counts (d ); the pseudometric dp based on model parameters; well-established (previously known) pseudometrics including PD (Portrait Divergence), TriadEuclid, and TriadEMD; dβ using approximated Betti numbers with approximating parameters ( = 100 , 101 , 102 , 103 , 104 ); and a “random” pseudometric generated with random positive definite matrix with zeros in the diagonal. The column labeled “FM-d β ” computes the FM index between dβ and the pseudometric of each row (treated as the benchmark classification). For instance, the FM index between dβ and P D is 0.8145. The number of clusters used for the FM index computation is chosen by a silhouette analysis of the column pseudometric. Notice that this way of choosing the number of clusters may result in different FM indices of the same two pseudometrics. For example, in Fig. 4 the FM index of d with respect to dβ differs to the FM index of dβ with respect to d . Similarly, the column “dCor-d  ” computes the sample distance correlation between d and a pseudometric of each row. For instance, the sample distance correlation between d and PD is 0.86. Considering Fig. 4 closely, we notice that almost all the pseudometrics (except the random metric in the last row) are closely related, since the table contains high FM indices and high distance correlations. For simplicity, from now on, we exclude the last row (random) from our discussion. In particular, the FM-d β column of Fig. 4 indicates that the clusters based on dβ are very similar to those based on d (treated as the benchmark classification), with an FM index of 0.982. Moreover, dβ is highly correlated with d , with a distance correlation of 0.968 in the dCor-d β column. In addition, dβ is also highly correlated with dp where dCor(dβ , dp ) = 0.9687. However, as the TriadEMD entries in the FM-d β and dCor-d β columns show, high distance correlation between metrics (dβ vs. TriadEMD, 0.9518) does not imply similar clusterings: the clusters given by dβ are less similar to those given by TriadEMD than for any other pseudometric, with an FM index of 0.6907. We remark that it is not too surprising that some of the numbers of Fig. 4 associated to the Fawlkes-Mallows index coincide. For example, this occurs for FM-d β of the pseudometrics dp , PD and TriadEuclid. This phenomenon is due to identical clustering of the pseudometrics dp , PD and TriadEuclid at the number of clusters chosen for dβ . This phenomenon can also be observed in subsequent figures/tables. Careful inspection of Figs. 5 and 6 leads to a similar observation that the pseudometrics we study are highly related to one another (with a few exceptions— with FM index less than 0.7). In particular, dβ and d consistently have high FM indices and high distance correlations across all three models. There are also some marked differences among the three models. For instance, the FM index between dβ and PD for PA random graphs is quite low—0.6887 in the FM-d β column of Fig. 6—in comparison with the other two models.

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We combine all 300 random graphs to study FM indices and distance correlations in Fig. 7. Both dβ and d remain reasonably similar to and dependent on one another, with an FM index of 0.7327 and a distance correlation of 0.9154. We also see a sharp fall in the FM indices and distance correlations between the topological (dβ , d ) and well-established pseudometrics (PD, TriadEuclid, TriadEMD). One potential explanation for these low FM indices and distance correlations is that the relationship observed model by model is due to a latent variable of the chosen model of random graph in addition to the parameter of that model.

5.2 Classification Accuracy Recall that our point-drawn collection of random graphs consists of 10 ER random graphs generated for each of six parameters (six labels), 10 GR graphs generated for each of three parameters (three labels), and 10 PA graphs generated for each of three parameters (three labels). We apply k-NN classification and report the classification accuracy for each of the three random graph models, as presented in Tables 2, 3, and 4. As shown in Table 2, using the PD, dβ , or d pseudometrics, we achieve 100% accuracy for all three random graph models; the accuracy for TriadEuclid and TriadEMD is slightly lower. If we treat all 120 random graphs in the point-drawn collection together and try to classify them into 3 classes (ER, GR and PA), we achieve 100% classification accuracy for all pseudometrics (excluding the random metric), as shown in Table 3. For the interval-drawn collection, we apply k-NN regression and report the MSE in predicting the model parameters. As shown in Table 4, for both ER and Table 2 Point-drawn collection: parameter classification rate within each set of random graphs

Table 3 Point-drawn collection: model classification rate for all three sets of random graphs combined

Pseudometrics PD TriadEuclid TriadEMD dβ d Random

ER 1 0.9167 0.9167 1 1 0.1833

Pseudometrics PD TriadEuclid TriadEMD dβ d Random

GR 1 1 1 1 1 0.2667

PA 1 1 1 1 1 0.2

Classification rate 1 1 1 1 1 0.3917

Graph Pseudometrics from a Topological Point of View Table 4 Interval-drawn collection: MSE with k-NN regression in predicting model parameters

Pseudometric PD TriadEuclid TriadEMD dβ d Random

ER (×e − 06) 0.9787 0.8367 2.207 2.956 1.653 1373

121 GR (×e − 06) 7.269 447.6 11.83 6.665 9.03 4047

PA 0.1488 0.1656 0.6636 0.4048 0.4084 98.3

GR random graphs, the regression using both well-established and topological pseudometrics achieves very low MSE (excluding the random metric); while for the PA random graphs, none of these pseudometrics performs well.

5.3 Permutation Tests We are interested in the statistical significance of relationships between wellestablished pseudometrics and dβ (respectively, d ). Permutation tests provide a method to test statistical significance of high distance correlation or FowlkesMallows (FM) index as demonstrating dependence between two pseudometrics. Our null hypothesis is that the two pseudometrics being compared are independent. For small p-values we can reject the null hypothesis and deduce dependency. For insufficiently small p-value we cannot reject the null hypothesis of independence. This does not imply the two pseudometrics are independent, but rather that we cannot rule out the independence. We perform permutation tests across all four datasets from the interval-drawn collection, based on either FM index or distance correlation as the dependency measures. One of our null-hypotheses is thus the independence of pseudometrics from our topological metrics (dβ , d or dp ) as measured by the FM index. We also consider the analogous null-hypothesis of independence with respect to distance correlation. With exception of the random pseudometric, all of the calculated p-values equal 0.0006667. Hence we can reject both of our null-hypotheses with p-value equal to 0.0006667. Except for the random control, the FM indices (respectively, distance correlations) calculated in Sect. 5.1 are so high that for every single random permutation the result is lower than for the original (unpermuted) FM indices (respectively, distance correlations). With 1500 permutations used in these tests, we compute a p-value of 0.000667. In each case we have performed at most 66 tests. Since 0.000667 < 0.05/66 we reject the null-hypothesis (except the random control) with a family-wise error of 0.05. If we were to consider all of the permutation tests combined, then the number of tests (264) would require a prohibitive number of permutations to make the

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Bonferroni correction useful. We would require p-values of at most 0.000189 to be able to establish any test as significant, which would require a minimum of 5280 permutations for each test. We use instead the Benjamini-Hochberg procedure to combine all the tests. Since 0.000667
0, and 0 < ε < δ+1 . CTLNs were first defined in [15], δ where the ε < δ+1 condition was motivated by the desired property that subgraphs consisting of a single directed edge i → j should not be allowed to support stable fixed points. Note that the upper bound on ε implies ε < 1, rendering the W matrix effectively inhibitory. We think of the graph edges as excitatory connections in a

1A

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Fig. 1 (a) A neural network with excitatory pyramidal neurons (triangles) and a background network of inhibitory interneurons (gray circles) that produces a global inhibition. The corresponding graph (right) retains only the excitatory neurons and their connections. (b) TLN dynamics. (c) A graph that is a 3-cycle (left), and a solution for the corresponding CTLN showing that network activity follows the arrows in the graph (right). Peak activity occurs sequentially in the cyclic order 123. Unless otherwise noted, all simulations have parameters ε = 0.25, δ = 0.5, and θ = 1

sea of inhibition (Fig. 1a). Figure 1c shows an example solution for a CTLN whose graph is a 3-cycle. TLNs are high-dimensional nonlinear systems whose dynamics are still poorly understood. However, in the special case of CTLNs, there appears to be a strong connection between the attractors of the network and the pattern of stable and unstable fixed points [14, 17].2 Moreover, these fixed points can often be completely determined by the structure of the underlying graph. In prior work, a series of graph rules were proven that can be used to determine fixed points of the CTLN by analyzing G, irrespective of the choice of parameters ε, δ, and θ [8, 9]. A key observation is that for a given network, there can be at most one fixed point per support, σ ⊆ [n], where the support of a fixed point is the subset of active neurons (i.e., supp x = {i | xi > 0}). For a given choice of parameters, we use the notation def

FP(G) = {σ ⊆ [n] | σ is a fixed point support of W (G, ε, δ)}, def

where [n] = {1, . . . , n}. For many graphs, we find that the fixed point supports in FP(G) are confined to a subset of the neurons. In other words, there is a partition {ω, τ } of the vertices of G such that, for every σ ∈ FP(G), we have σ ⊆ τ (see Fig. 2a). In these cases, we observe that solutions of the network activity x(t) tend to converge to a region of the state space where the most active neurons are in τ , and those in ω are either silent or have very low firing (see Fig. 2b). In other words, the attractors live where the fixed points live. This motivates us to define directional graphs. A directional graph G is a graph with a proper subset of neurons τ such that FP(G) ⊆ FP(G|τ ), where G|τ is the induced subgraph obtained by restricting to the vertices of τ . For example, the graph 2A

fixed point, x ∗ , of a TLN is a solution that satisfies dxi /dt|x ∗ = 0 for each i ∈ [n].

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Fig. 2 Directional graphs and covers. (a) A directional graph, with fixed points supported in the subset of nodes τ = {3, 4}. (b) A solution of a CTLN with the graph in (a). The network was initialized with the activity concentrated on the neurons in ω = {1, 2}, but the activity flows from ω → τ . (c) A graph with a partition of the nodes, each component in a different color. (d) A directional cover of G. Subsets of nodes, νi , νj , νk , and νl , correspond to the partition in (c). The four induced subgraphs within each oval, of the form Gij = G|νi ∪νj , are all directional graphs with direction νi → νj , as given by the arrows. (e) The nerve associated to the directional cover in (d). The corresponding network can be viewed as a dimensional reduction of the one in (c)

in Fig. 2a is directional with a single fixed point supported in τ = {3, 4}. We also require an additional technical condition that allows us to prove that certain natural compositions, like chaining directional graphs together, produce a new directional graph (see Definition 3.1 for the full definition). In simulations, such as the one in Fig. 2b, we have seen that directional graphs display feedforward dynamics, even if their architecture does not follow a feedforward structure. Activity that is initially concentrated on ω flows towards τ , giving the dynamics an ω → τ directionality. Thus, from a bird’s eye view, directional graphs behave like a single directed edge, where the activity flows from the source to the sink. These observations prompted us to ask the following question: if we cover a graph G with a collection of directional graphs, what can we say about FP(G) from the combinatorial structure of the cover? In this paper we develop tools to answer this question, inspired by the construction of the nerve of a cover of a topological space. We define a directional cover of G as a set of directional subgraphs that cover G and have well-behaved intersections.3 Effectively, such a cover is entirely determined by a partition of the

3 This is analogous to the definition of a “good” cover of a topological space, which also requires well-behaved intersections. Nerves of good covers reflect the topology of the underlying space [5, 13].

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vertices of G, denoted {νi }, that satisfies special properties. (See Definition 4.3 for a precise definition.) We define the nerve of a directional cover as a new graph N that has a directed edge for each directional graph in the cover, and a vertex for each component νi of the partition. Figure 2c,d depicts a graph with a partition of the nodes (indicated by the colors), and its corresponding directional cover. The edges of the nerve reflect the local dynamics of G, and the nerve itself encodes the combinatorics of the intersection pattern of the cover: the directional graphs overlap precisely at vertices of the nerve where their edges meet. The partition of the vertices of G induces a canonical quotient map, π : VG → VN := {νi }, that simply identifies all the vertices in each component νi . Figure 2e is the nerve of the directional node in d, and the quotient map π sends each color in c to the corresponding node with the same color in e -within the nerve N, we often label the node νi simply as i. As an illustration of directional covers and nerves, consider the graph in Fig. 3a. This graph is a chain of ten 5-cliques where the edges between adjacent cliques all follow the pattern shown in panel c: there are edges forward from every node in the first clique to every node in the second clique; every node in the second clique (except for the top node) sends edges back to every node in the first clique. Most edges are thus bidirectional arrows (in black), while the edges that only go

Fig. 3 Example graph with a directional cover, its nerve, and network activity that flows along the nerve. (a) A chain of ten 5-cliques where the edges between adjacent cliques all follow the pattern shown in panel (c). (b) The nerve of the graph in A induced by the partition of the vertices as ν1 , . . . , ν10 . (c) The graph G restricted to a pair of adjacent cliques. All edges in black are bidirectional, while those in green are unidirectional from the green clique to the pink clique. This restricted graph is directional, and all the graphs in the directional cover of G have this form. (d) A solution to the CTLN defined by G (with ε = 0.25 and δ = 0.5), where the activity is initialized on the nodes in the first clique ν1 . The transient dynamics slowly activate each clique in sequence, following the path of the nerve, until the solution converges to the stable fixed point supported on the nodes in ν10

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forward from clique i to clique i+1 are in color. The induced subgraphs G|νi ∪νi+1 are all directional with direction νi → νi+1 , despite all the back edges from right to left. This means that FP(G|νi ∪νi+1 ) ⊆ FP(G|νi+1 ), and we expect the activity of the neurons to flow from νi to νi+1 (left to right). Figure 3b depicts the nerve of G. Figure 3d shows the solution to a CTLN defined by G, where we have initialized all the activity on the nodes in the first clique ν1 . We see that the activity eventually converges to the final component, shown in purple, where the fixed points of the network are concentrated. The transient dynamics, however, are rather slow, with each clique activated in a sequence that follows the path-like structure of the nerve. Note that this network behaves similarly to a synfire chain [1, 2, 12], despite numerous backward edges between components that completely destroy the feedforward architecture (Fig. 3c). The sequential dynamics are maintained because these backward edges do not disrupt the directionality of the graphs in the cover. The main goal of this paper is to prove nerve theorems for CTLNs. Broadly speaking, such a nerve theorem is a result that gives information about the dynamics of a network from properties of the nerve. Specifically, we are interested in results that allow us to constrain the fixed points of G by analyzing structural properties of N. Ideally, we would like to prove the following kind of result: If G has a directional cover with nerve N, then σ ∈ FP(G) ⇒ π(σ ) ∈ FP(N),

(3)

where π is the canonical quotient map from VG to VN . This can be quite powerful in cases where N is a much smaller and simpler graph. Unfortunately, the statement (3) is not in general true. However, we do find that this holds whenever the nerve N is a directed acyclic graph (DAG) or N is a cycle (see Theorems 4.8 and 4.9). More generally, whenever N admits a DAG decomposition (see Definition 2.16), Theorem 4.7 gives a result similar in spirit to (3) and allows us to greatly constrain FP(G). Our nerve theorems can be used to simplify a complex network by finding a nontrivial directional cover and studying its nerve. Finding such covers is an art, however, and we do not yet have a systematic way of doing it. On the other hand, nerve theorems can also be used to engineer complex networks with prescribed dynamic properties. This is how we constructed the example in Fig. 3. We explore both kinds of applications in the last section of the paper. The organization of this paper is as follows. In Sect. 2, we review some graph theory terminology and basic background and notation for CTLNs. We also introduce the DAG decomposition of a graph. In Sect. 3, we define directional graphs, prove that certain graph structures are always directional, and provide some other families of examples. In Sect. 4, we introduce directional covers and their associated nerves. Here we also state and prove our main results, Theorems 4.7, 4.8 and 4.9. Finally, in Sect. 5, we illustrate the power of our theorems with some applications.

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2 Preliminaries In this section we review some useful terminology from graph theory and summarize essential background and prior results about fixed points of CTLNs. We also introduce the DAG decomposition of a graph, a notion that will appear in our main nerve theorems.

2.1 Graph Theory Terminology Definition 2.1 A directed graph G can be described as a tuple G = (VG , EG ), where VG is a finite set called the set of vertices and EG ⊆ VG × VG is the set of (directed) edges, where (i, j ) ∈ EG means there is a directed edge i → j from i to j in G. If (i, j ) ∈ / EG , we write i → j . A directed graph is simple if it has no self-loops, so that (i, i) ∈ / EG for all i ∈ VG . A directed graph is oriented if it has no bidirectional edges. In this paper, we restrict ourselves to simple directed graphs. Unless otherwise noted, we will use the word graph to refer to simple directed graphs. Notation 2.2 Let G be a graph with vertex set VG and edge set EG . For any subset of vertices σ ⊆ VG , denote by G|σ the induced subgraph obtained by restricting to the vertices σ . More precisely, G|σ = (σ, E|σ ) where E|σ = {(i, j ) ∈ EG | i, j ∈ σ }. Let σ1 , σ2 ⊆ VG be two subsets of the vertices of G. We denote by EG (σ1 , σ2 ) ⊆ EG the set of directed edges from vertices in σ1 to vertices in σ2 in G. Next, we define some basic notions relevant to graphs. Definition 2.3 Let G be a graph and v ∈ VG be a vertex in G. The in-degree of v is the number of incoming edges to v. The out-degree of v is the number of outgoing edges from v. We say v is a source if v has no incoming edges, and we say v is a proper source if it is a source that has at least one outgoing edge. We say v is a sink if v has no outgoing edges. Note that a source that is not a proper source is an isolated vertex, and thus it is also a sink. Definition 2.4 We say that a graph G has uniform in-degree if every vertex v ∈ VG has the same in-degree d. Note that an independent set is a graph with uniform in-degree d = 0. A k-clique is an all-to-all bidirectionally connected graph with uniform in-degree d = k − 1. And an n-cycle is a graph with n edges, 1 → 2 → · · · → n → 1, which has uniform in-degree d = 1.

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2.2 Background on Fixed Points of CTLNs In this subsection we recall the results from [8] that are relevant for this work and include simple proofs to some of these to provide intuition to the reader. A fixed point of a CTLN is simply a fixed point of the network equations (1). In dxi other words, it is a vector x ∗ ∈ Rn≥0 such that |x=x ∗ = 0 for all i ∈ [n]. As dt explained in [8], fixed points of CTLNs can be labelled by their supports (i.e. the subset of active neurons), and for a given G the set of all fixed point supports is denoted FP(G). Lemma 2.5 ([8]) Let G be a graph on n vertices, and suppose G has uniform indegree. Then G has a full-support fixed point, σ = [n] ∈ FP(G). In particular, this lemma says that cliques, cycles, and independent sets all have a full-support fixed point. In fact, this fixed point is symmetric, with xi∗ = xj∗ for all i, j ∈ [n]. This is true even for uniform in-degree graphs that are not symmetric. More generally, fixed points can have very different values across neurons. However, there is some level of “graphical balance” that is required of G|σ for any fixed point support σ . For example, if σ contains a pair of neurons j, k that have the property that all neurons mapping to j are also mapping to k, and j → k but k → j , then σ cannot be a fixed point support. This is because k is receiving strictly more inputs than j , and this imbalance rules out their ability to coexist in the same fixed point support. To see this more rigorously, we have the following lemma. Lemma 2.6 Let G be a CTLN and σ ⊆ VG . Suppose there exist vertices j, k ∈ σ such that for each i ∈ σ \ {j, k}, if i → j then i → k. Furthermore, suppose j → k but k → j . Then σ ∈ / FP(G). Proof To obtain a contradiction, assume σ ∈ FP(G). The corresponding fixed point x satisfies xi > 0 for all i ∈ σ , and dxi /dt = 0. In particular, setting dxj /dt = 0 and dxk /dt = 0 (and recalling Wjj = Wkk = 0) we obtain: xj =



Wj i xi + Wj k xk + θ,

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Wki xi + Wkj xj + θ.

i∈σ \{j,k}

Now observe that for each i ∈ σ \ {j, k}, the fact that i → j implies i → k tells us that Wj i ≤ Wki , (see Eq. (2)). This means the summation term in the xj equation above is less than or equal to the analogous term in the xk equation. Using this fact, we see that xj − Wj k xk ≤ xk − Wkj xj , which can be rearranged as, (1 + Wkj )xj ≤ (1 + Wj k )xk .

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Now recall that j → k but k → j , so Wkj = −1 + ε and Wj k = −1 − δ. The above inequality thus says that εxj ≤ −δxk . But this is a contradiction, because εxj > 0 and −δxk < 0. And so no fixed point supported on σ can exist.   The conditions on j, k ∈ σ used in the above lemma is an example of so-called graphical domination. This notion was first defined in [8], and provides a useful tool for ruling in and ruling out fixed points of CTLNs purely based on the graph structure, and independently of the ε, δ and θ parameters. Definition 2.7 (graphical domination) Let G be a graph, σ ⊆ VG a subset of the vertices, and j, k ∈ VG such that {j, k} ∩ σ = ∅. We say that k graphically dominates j with respect to σ , and write k >σ j , if the following three conditions hold: 1. For all i ∈ σ \ {j, k} if i → j , then i → k. 2. If j ∈ σ , then j → k. 3. If k ∈ σ , then k  j . This definition of domination covers more cases than what we saw in Lemma 2.6. This greater generality is reflected in the main theorem about domination, which appeared as Theorem 4 in [8]. We cite a special case of this theorem below (also illustrated in Fig. 4). Theorem 2.8 (graphical domination [8]) Let σ ⊆ VG be a subset of the vertices of a graph G. If there is a j ∈ σ and a k ∈ VG such that k >σ j (k graphically dominates j with respect to σ ), then σ ∈ / FP(G). We will furthermore use the following useful equivalence, which states that σ can only be a fixed point support if σ ∈ FP(G|σ ) and the fixed point survives the addition of each individual k ∈ / σ.

Fig. 4 The two cases of graphical domination in Theorem 2.8. In each panel, k graphically dominates j with respect to σ (the outermost shaded region). The inner shaded regions illustrate the subsets of nodes that send edges to j and k. Note that the vertices sending edges to j are a subset of those sending edges to k, but this containment need not be strict. The dashed arrow indicates an optional edge between j and k

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Lemma 2.9 ([8, Corollary 2]) Consider a CTLN determined by a graph G on a set of vertices VG , and let σ ⊆ VG . Then σ ∈ FP(G) ⇔ σ ∈ FP(G|σ ∪{k} ) for all k ∈ VG . In particular, σ ∈ FP(G) ⇒ σ ∈ FP(G|σ ). Moreover, σ ∈ FP(G) ⇒ σ ∈ FP(G|τ ) for any τ with σ ⊆ τ . One simple case where graphical domination can be used to rule out a fixed point support is whenever a graph contains a proper source. This is Rule 6 in [8]. Lemma 2.10 (sources [8]) Let G be a graph and σ ⊆ VG . If there exists a j ∈ σ such that j is a proper source in G|σ or j is a proper source in G|σ ∪{} for some  ∈ VG , then σ ∈ / FP(G). Proof If j is a proper source in G|σ , then there exists k ∈ σ such that j → k. Since j has no other inputs in σ , clearly k >σ j. If j is not a proper source in G|σ but is a proper source in G|σ ∪ , then j → , and hence  >σ j . In either case, by Theorem 2.8 we have that σ ∈ / FP(G).   The lemma above allows us to rule out fixed points of cycles that are not full support. Lemma 2.11 (cycles) If G is a cycle on n vertices, then G has a unique fixed point, which has full support. In other words, FP(G) = {[n]}. Proof First observe that [n] ∈ FP(G) by Lemma 2.5 because a cycle has uniform in-degree 1. To see that this is the only fixed point support of G, consider any proper subset σ  VG . Since G is a cycle, G|σ either contains a path or is an independent set. If it contains a path, then the source of that path is a proper source in G|σ . If it is an independent set, then for any i ∈ σ , we have i →  in G for  = i + 1. Then i is a proper source in G|σ ∪ . Thus by Lemma 2.10, σ ∈ / FP(G). Thus, FP(G) = {[n]}.   Another simple case when graphical domination can be used to rule out fixed points is whenever σ has a target in G. For k ∈ VG , we say that k is a target of σ if i → k for every i ∈ σ \ {k}. Lemma 2.12 (targets [8]) Let G be a graph and σ ⊆ VG . Suppose k ∈ VG is a target of σ . 1. If k ∈ VG \ σ , then σ ∈ / FP(G). 2. If k ∈ σ and there exists a j ∈ σ such that k → j , then σ ∈ / FP(G). Proof In case 1, it is straightforward to see that k >σ j for any j ∈ σ . In case 2, we see that for the particular j such that k → j , we have k >σ j . In either case, by Theorem 2.8 we have that σ ∈ / FP(G).  

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The target lemma allows us to rule out fixed points of cliques that are not full support. Lemma 2.13 (cliques) If G is a clique on n vertices, then G has a unique fixed point, which has full support. In other words, FP(G) = {[n]}. Proof First observe that [n] ∈ FP(G) by Lemma 2.5 because a clique has uniform in-degree n − 1. To see that this is the only fixed point support of G, consider any proper subset σ  VG and let k ∈ VG \ σ . Then k is a target of σ , and so by Lemma 2.12, σ ∈ / FP(G). Thus, FP(G) = {[n]}.   Finally, using a more general form of domination defined in [8], we obtain the following survival rule telling us precisely when a uniform in-degree fixed point survives as a fixed point of a larger network (Theorem 5 of [8]): Theorem 2.14 (uniform in-degree [8]) Let G be a graph and σ ⊆ VG such that def

G|σ has uniform in-degree d. For k ∈ VG \ σ , let dk = |{i ∈ σ | i → k}| be the number of edges k receives from σ . Then σ ∈ FP(G) ⇔ dk ≤ d for every k ∈ VG \ σ. Other than this theorem we will not use the more general form of domination. Therefore, in the remainder of this work when we say domination we mean graphical domination.

2.3 The DAG Decomposition Two of our main results are nerve lemmas involving directed acyclic graphs (DAGs). Recall that a DAG is a graph that has no directed cycles. There is a well known characterization of DAGs in terms of a topological ordering of their vertices. In particular, G is a DAG if and only if there exists an ordering of the vertices such that edges in G only go from lower numbered to higher numbered vertices. In other words, if i → j then i < j ; equivalently if i > j then i → j . Lemma 2.15 (DAGs) Let G be a DAG and let τ = {sinks of G}. Then the fixed point supports of G are all the nonempty subsets of τ , i.e. FP(G) = P(τ ) \ {∅}, where P(τ ) denotes the power set of τ . Proof First to see that P(τ ) \ {∅} ⊆ FP(G), notice that any non-empty subset of τ is an independent set of sinks. An independent set has uniform in-degree 0, and thus

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by Theorem 2.14, an independent set produces a fixed point when it has no outgoing edges. Since all the nodes in τ are sinks, every subset of τ has no outgoing edges, and so every subset produces a fixed point support in FP(G). Next, to see that no other sets can produce fixed points of G, consider σ ⊆ VG such that σ ⊆ τ . Let j be the lowest number vertex in σ \ τ according to some topological ordering of G. Then j has no incoming edges from other nodes in σ since edges in a DAG can only go from lower numbered vertices to higher number vertices. Moreover, there exists some  ∈ VG such that j →  since otherwise j would be a sink, but j ∈ / τ by design, which contains all the sinks of G. Thus j is a proper source in G|σ ∪ , and so σ ∈ / FP(G) by Lemma 2.10.   Many graphs that are not DAGs nevertheless have a DAG-like structure on a subgraph. This will also be a useful concept for our nerve theorems. Definition 2.16 (DAG decomposition) Let G be a graph. For ω, τ ⊆ VG , we say ˙ is a partition of the vertices VG that (ω, τ ) is a DAG decomposition of G if ω∪τ such that: 1. G|ω is a DAG, 2. G|τ contains all sinks of G, 3. there are no edges from τ back to ω, i.e., EG (τ, ω) = ∅. We say a DAG decomposition is non-trivial if ω = ∅. We say a DAG decomposition is maximal if ω is as large as possible. More precisely, (ω, τ ) is a maximal DAG decomposition if there is no other DAG decomposition (ω , τ ) with ω  ω . Every graph G that has at least one proper source j has a DAG decomposition with ω = {j } and τ = VG \ {j }. But DAG decompositions are most valuable when τ is as small as possible. To minimize the size of τ , we’d like to “grow” ω as much as possible, as in a maximal DAG decomposition. It turns out that there is straightforward procedure for generating a maximal DAG decomposition of a graph, and moreover, the maximal DAG decomposition is in fact unique. Specifically, one can iteratively refine a DAG decomposition by moving any nodes that are proper sources in G|τ to ω (see Fig. 5). This process will maintain the property that G|ω is a DAG (each node that is moved to ω will be at the end of the “topological ordering” of the DAG) while also guaranteeing that there are no edges from nodes in τ back to nodes in ω. Finally, the process terminates when there are no nodes in τ that are proper sources in G|τ . It turns out that the τ satisfying G|τ has no proper sources is both minimal, in the sense that |τ | is smallest and τ ⊆ τ for any other DAG decomposition (ω , τ ), and unique. As a result, this process yields the unique maximal DAG decomposition. Note in particular that in any maximal DAG decomposition (ω, τ ), τ cannot have proper sources, because if it did one could move such a vertex to ω, contradicting maximality. Lemma 2.17 Suppose that G contains a proper source. Then the DAG decomposition (ω, τ ) of G satisfying G|τ has no proper sources is a maximal DAG decomposition. In particular, G has a unique maximal DAG decomposition.

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Fig. 5 Iterative construction of DAG decompositions. (a) A DAG decomposition of a graph where ω contains only a single source. The gray highlighted node 2 is a proper source in G|τ , but not a source in the full graph. (b) A second DAG decomposition is obtained by moving node 2 to ω. Now node 5 has become a proper source in the new G|τ . (c) A third DAG decomposition is obtained by moving 5 to ω. In this decomposition, G|τ has no proper sources. Notice that node 7 is a source, but because it has no outgoing edges, it is not a proper source so will not be moved to ω. In fact, node 7 is a sink, and thus is required to be in τ by condition 2 of DAG decompositions. We have thus arrived at the unique DAG decomposition with minimal τ and maximal DAG ω

Proof Let (ω, τ ) be a DAG decomposition of G satisfying G|τ has no proper sources, and let (ω , τ ) be any other DAG decomposition of G. Suppose ω ⊆ ω. Then since each DAG decomposition is a partition of the vertices, this condition on ω implies that τ ⊆ τ . Then there exists a node i0 ∈ τ \ τ . Since i0 ∈ τ and G|τ has no proper sources, there exists some i1 ∈ τ such that i1 → i0 . Since i1 also is not a proper source in G|τ , there exists some i2 ∈ τ such that i2 → i1 → i0 . Again i2 is not a proper source, and so there exists i3 ∈ τ such that i3 → i2 → i1 → i0 . Note that in the other DAG decomposition (ω , τ ), since i0 ∈ / τ , we must have i0 ∈ ω . Moreover, by the definition of DAG decomposition, there are no edges from nodes in τ to nodes in ω , and so all nodes in the path i3 → i2 → i1 → i0 must also be in ω . We can continue to trace the path backwards in this way through G|τ for arbitrarily many steps since it has no proper sources, but since τ is finite, at some point some node must appear twice in this path. Thus this sequence of nodes must contain a bidirectional edges and/or a directed cycle. But all the nodes in this sequence must be in ω , and by definition, G|ω must be a DAG, thus it cannot contain any bidirectional edges or directed cycles. Thus we have a contradiction, and so τ ⊆ τ , and thus ω ⊆ ω. Since any DAG decomposition (ω, τ ) of G satisfying G|τ has no proper sources must have maximal ω, it follows that there must be a unique decomposition satisfying this property. Finally, since any maximal DAG decomposition must satisfy this property, it follows there is a unique one and it is this one.  

3 Directional Graphs In this section, we focus on a special class of graphs known as directional graphs, first defined in [16]. The motivating heuristic behind directional graphs is that they

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are graphs whose vertices can be partitioned into two sets ω and τ such that when the neural activity is initialized on nodes in ω, it flows to the nodes in τ . In simulations, we have seen that this flow of activity occurs whenever the fixed points of G are confined to live in τ , so that FP(G) ⊆ FP(G|τ ). In order to guarantee nice properties when we union together directional graphs, we require something slightly stronger in our definition of directional graphs, namely that the collapse of the fixed points onto the subnetwork G|τ be the result of graphical domination. Definition 3.1 (directional graph) We say that a graph G is directional, with ˙ = VG is a nontrivial partition of the vertices (ω, τ = ∅, direction ω → τ , if ω∪τ ω ∩ τ = ∅) such that FP(G) ⊆ FP(G|τ ) by way of graphical domination. Specifically, we require the following property: for every σ ⊆ τ , there exists some j ∈ σ ∩ ω and k ∈ VG such that k graphically dominates j with respect to σ , i.e. k >σ j . When this is the case we say σ dies by (graphical) domination. As mentioned above, we predict that directional graphs will have feedforward dynamics, so that activity that is initially concentrated on G|ω should flow towards G|τ , giving the dynamics an ω → τ directionality. The most natural examples of directional graphs are those where G has an explicit feedforward architecture in G|ω , for example when G|ω is a DAG, and there are no edges from τ back to ω. In this case, it seems intuitive that the dynamics will flow along this feedforward structure in ω and end up concentrated in τ . It turns out that any DAG decomposition of a graph G immediately yields a directional partition as intuitively predicted. Lemma 3.2 If (ω, τ ) is a DAG decomposition of G, then G is directional with direction ω → τ . The key to the proof of Lemma 3.2 is the well known characterization of DAGs in terms of a topological ordering of their vertices. Recall that G is a DAG if and only if there exists an ordering of the vertices such that edges in G only go from lower numbered to higher numbered vertices, i.e., if i > j , then i → j . Proof To show that G is directional, we must show that any σ ⊆ VG that intersects ω dies by graphical domination. Suppose σ ∩ ω = ∅, and let j be the lowest numbered vertex in σ ∩ ω with respect to some topological ordering of the DAG G|ω . Since all the sinks in G are contained in τ , j ∈ ω must have at least one outgoing edge in G, so j → k for some k ∈ VG . Moreover, j has no incoming edges in G|σ ∪k because of its numbering in the topological ordering. Thus, j is a proper source in G|σ ∪k and by Lemma 2.10, σ ∈ / FP(G) because k >σ j . Thus every σ with σ ∩ ω = ∅ dies by graphical domination, and so G is directional with direction ω → τ .   DAG decompositions are a very special case of directional graphs ω → τ where there are no back edges from τ to ω, and the ω component of the graph is a DAG. Neither condition needs to hold for more general directional graphs. Figure 6 shows several types of directional graphs. In panel a, there are only edges from ω → τ as in a DAG decomposition. In panel b, the existence of a target in τ that receives edges

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Fig. 6 Three types of directional graphs. (a) A nontrivial DAG decomposition (ω, τ ) is a directional graph with direction ω → τ . (b) If τ contains a target node of ω, and there are no back edges τ → ω, then G is directional irrespective of the structure of G|ω . (c) A more general directional graph can have a variety of forward and backward edges

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from all nodes in ω guarantees that G is directional irrespective of the structure of G|ω . Finally, in panel c we see a schematic of a directional graph with both forward edges from ω to τ and backward edges from τ to ω. In fact, directional graphs can have a surprisingly large number of back edges while still preserving their “forward” directionality. All the graphs in Fig. 7a are directional with ω → τ , and each of the graphs in a3–a6 actually has as many back edges from τ to ω as it does forward edges. The dynamics for a3 and a6 are shown on the right, and we see that even if we initialize the activity purely on nodes in ω, the activity flows ω → τ as predicted by the directionality. Note that none of the graphs in panel a has a proper source, and thus none has a nontrivial DAG decomposition.

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Panel b in Fig. 7 shows some example graphs that are not directional for any partition of the vertices. This is because every vertex is involved in at least one fixed point support, so there cannot be a collapse FP(G) ⊆ FP(G|τ ) for any τ  VG . The graph in b2 is particularly surprising since it only has edges forward from the 2-clique 12 to 34, so we might expect this to yield a directional decomposition. But the forward edges are not sufficient to kill the 2-clique 12, and so we see from the dynamics on the right, that we are not guaranteed a directionality of flow. Instead, 12 supports a stable fixed point, and thus when we initialize activity on those nodes, it remains there, and never flows to the other stable fixed point 34. Remark 3.3 If G is a directional graph, its directional decomposition is not unique. For example, as long as ω has more than one vertex, then vertices can always be moved from it to τ and maintain directionality. However, directional decompositions are most useful when the τ component is as small as possible, since this gives the strongest restrictions on the possible fixed point supports of the whole network. One candidate τ for such a decomposition is τ := ∪σ ∈FP(G) σ . However, this set does not guarantee a directional decomposition since we have not guaranteed that all subsets of VG that intersect ω := VG \ τ die by graphical domination. In order to satisfy this property, it may be necessary to add some additional vertices to τ , and doing this in a minimal way may not be unique. It is an open question if every directional graph has a unique directional decomposition with minimal τ .

4 Directional Covers and Nerve Theorems In this section, we aim to characterize the fixed points of more complex graphs by covering the graph with directional graphs, and then analyzing a simpler associated object known as the nerve of the cover. The intuition is as follows. As described in Sect. 3, if G is a directional graph with direction ω → τ , the activity of the network flows from ω to τ . Thus, from a bird’s eye view, the flow of activity of such a graph can be represented by the flow of activity along a single directed edge from source to sink. Moreover, this flow of activity reflects restrictions imposed on the fixed point supports as well. With this in mind, we will take any graph G and aim to cover it with directional graphs that have appropriate pairwise intersections. From this cover, we construct a nerve, which is a simplified graph where subsets of vertices are collapsed to single points, and each directional graph of the cover is now represented by a single directed edge. These edges are glued to one another in a way representative of the intersection pattern of the cover. We will see that, with this construction, we are able to deduce certain restrictions on the fixed point supports of the original graph G by studying the fixed points of the nerve of the cover, which is in general a simpler graph.

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4.1 Directional Covers and Nerves We begin by making the notion of directional cover and its nerve precise. Definition 4.1 (graph cover) Let G be a graph. A graph cover of G is a collection of induced subgraphs U = {Gi := G|Vi | for some Vi ⊆ VG } such that G is the union of the Gi . In other words, VG = ∪i∈I Vi and EG = ∪i∈I EGi . Remark 4.2 Note that every vertex and every edge of G must live in at least one Gi , but often they live in multiple Gi within the cover. In particular, since the covering graphs are induced subgraphs of G, if u, v ∈ Vi and u, v ∈ Vj , then any edges between u and v will be in both Gi and Gj . Next we turn to a special type of graph cover which we call rigid directional cover. In a rigid directional cover, we require that all the graphs of the cover are directional and that they overlap in prescribed ways that will facilitate associating a nerve to the cover and ensure that this nerve captures constraints on FP(G). The rigid condition can be informally described as follows. Consider a graph cover U of G, where all the covering graphs are directional. Let G1 , G2 ∈ U be a pair of graphs in the cover, with directional decompositions ω1 → τ1 and ω2 → τ2 , respectively. The graph cover U is rigid if for any pair G1 and G2 that have nontrivial intersection, their overlap is of one of the following three types: 1. The τ component of the first graph acts as the ω component of the second, i.e., VG1 ∩ VG2 = τ1 = ω2 . In this case we say the graphs have a chaining overlap. (See Fig. 8a.) 2. The two covering graphs intersect exactly at their τ component, i.e., VG1 ∩ VG2 = τ1 = τ2 . In this case we say the graphs have merging overlap. (See Fig. 8b.) 3. The two covering graphs intersect exactly at their ω component, i.e., VG1 ∩ VG2 = ω1 = ω2 , and have the additional property that there are no back edges from vertices in τ to vertices in ω in either graph, i.e., EG1 (τ1 , ω1 ) = EG2 (τ2 , ω2 ) = ∅. In this case we say the graphs have a splitting overlap. (See Fig. 8c.)

Fig. 8 A pair of graphs G1 and G2 that have (a) a chaining overlap (b) a merging overlap and (c) a splitting overlap

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Effectively, a rigid directional cover is always induced by a partition of the vertices of the underlying graph and this partition encodes all the information of the cover itself. Therefore, we formally define a rigid graph cover as follows. Definition 4.3 (directional cover and its nerve) Let G be a graph. Given a partition of the vertices, ν = {ν1 , . . . , νn }, let E = E(G, ν) := {(i, j ) ∈ [n]×[n] | G|νi ∪νj is directional with direction νi → νj }, I = I(G, ν) := {i ∈ [n] | G|νi is disconnected from the rest of the graph}. We say that the partition {ν1 , . . . , νn } induces a rigid directional cover of G if: 1. For every pair (νi , νj ) either (i, j ) ∈ E or (j, i) ∈ E or there are no edges between νi and νj . In other words, the set U = {Gij := G|νi ∪νj | (i, j ) ∈ E} ∪ {G|νi | i ∈ I} is a graph cover of G. 2. Whenever Gij , Gik ∈ U, they have “splitting overlap”, meaning there are no edges from νj to νi and no edges from νk to νi , i.e., EG (νj , νi ) = EG (νk , νi ) = ∅. We define the nerve of the cover, denoted by N = N(G, U), to be the graph with vertex set VN := [n] and edge set EN := E. The partition ν induces a canonical quotient map π : VG → VN that identifies all the vertices of a component νi , so that π(νi ) = {i} for each i ∈ VN . Note that an arbitrary partition will not typically induce a rigid directional cover because there will be pairs (νi , νj ) with edges between them, but the induced subgraphs G|νi ∪νj will not be directional. In contrast, partitions that do induce a rigid directional cover must have G|νi ∪νj directional whenever there are edges between νi and νj . Note that we do not require the G|νi for i ∈ I to be directional; these graphs are included in U simply to ensure that isolated components of the graph are still covered. In this paper we will only work with rigid directional covers. Therefore, in the remainder of this work we will use the term directional cover to refer to a rigid directional cover. Figure 9 gives an illustration of a graph G with vertex partition ν = {νi } that induces a collection of covering graphs {Gij } that are all directional. In the nerve, N = N(G, U), we see a vertex i for each component νi from G, and an edge i → j corresponding to each covering graph Gij , which has direction νi → νj . Lemma 4.4 Let G be a graph with nerve N = N(G, U) for some directional cover U. Then the nerve N is a simple directed graph that is oriented. Proof Recall that given a partition ν = {ν1 , . . . , νn } of the vertices of G, the nerve N is defined as a graph on n vertices with edge set E given in Definition 4.3. The edge set E is straightforward to determine from the partition ν. Specifically, if there are no edges between νi and νj in G, then neither (i, j ) nor (j, i) are in E, since G|νi ∪νj is a disjoint union, which can never be directional. If there are any edges

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Fig. 9 A graph with a directional cover and the corresponding nerve. (Left) A graph G. (Middle) A directional cover U of G. The sets νi , νj , νk , νl , νm partition the vertices of G. All the edges of G are contained within some of the covering graphs {Gij } shown, each of which is directional with direction νi → νj . (Right) The nerve N = N(G, U) with a vertex for each component of the partition and a directed edge for each directional graph of the cover

between νi and νj , we must have either (i, j ) ∈ E or (j, i) ∈ E in order for the {Gij } to cover G. Moreover, we can only have one of (i, j ) or (j, i) in E since G|νi ∪νj can never be directional with both νi → νj and νj → νi . Thus, N is oriented. Finally, to see that N is simple, notice that (i, i) ∈ / E since Gii can never be directional with ω = τ = νi .   Given the complexity of the requirements of a directional cover, specifically that every covering graph of the form G|νi ∪νj be directional, it is natural to ask when a graph actually has such a cover. Of course, every graph has a trivial directional cover induced by the trivial partition ν1 = VG ; in this case, the nerve of the cover is just a single point. At the other extreme, whenever G is an oriented graph, the partition of singletons νi = {i} will induce a directional cover, whose nerve is precisely the original graph G. While these two trivial covers exist, they clearly do not provide any insight into the structure or expected dynamics of G. There is an art to finding a partition of VG from which a cover with an informative nerve can be obtained. It is important to note that not every graph has a directional cover induced by a nontrivial partition. For example, if G is a clique, then there is no nontrivial partition of VG that can admit a directional cover since every Gij will be a clique, and thus not directional. At the other extreme, there are graphs with multiple nontrivial partitions of VG which induce directional covers. See for example, Fig. 10c,d, which shows two different covers of the same graph, and where the nerve of the first one is directional while the nerve of the second is a cycle. It is an open question which graphs have at least one nontrivial partition that admits a directional cover. And unfortunately, there is currently no efficient way to find all the partitions of a graph that do induce directional covers. However, when we have a directional cover of G, obtained either by brute force search or from intuition into the original construction of the graph, the nerve of the cover can give significant insight into the collection of fixed points and consequently into the predicted dynamics of the underlying network. In particular, nerve theorems ensure that there is a provable connection between FP(G) and the structure of the nerve under certain conditions on G and/or N(G, U).

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Fig. 10 Counterexamples to general nerve theorems. (a) and (b) give two different graphs with partitions that induce directional covers. The nerve of each cover is shown to the right. (c) and (d) give two different partitions for the same graph. In (c), the partition induces a nerve that is directional, while in (d), the partition induces a nerve that is a cycle

4.2 Nerve Theorems Ideally, we would hope for a nerve theorem that provides a strong connection between the fixed point supports of the original graph and those of the nerve. For example, we might hope that for any graph G that admits a directional cover with nerve N we can guarantee a condition such as σ ∈ FP(G) ⇒ π(σ ) ∈ FP(N).

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Unfortunately, though, this strong restriction on FP(G) does not hold for all graphs and all directional covers. Example 4.5 For the graph in Fig. 10a, we see that the partition {ν1 , . . . , ν4 } shown induces a directional cover of G. The nerve N = N(G, U) of this cover is shown on the right. Recall that a cycle is uniform in-degree 1 and thus it supports a fixed point precisely when no external vertex receives more than one edge from it (see Theorem 2.14 in Sect. 2 ). Thus we have 123 ∈ FP(G)4 since the external vertices 4 and 5 each receive only one edge from the cycle. However,

4 Recall

that we write 123 to denote the fixed point support {1, 2, 3}.

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π(123) = 123 ∈ / FP(N) since in the nerve, the cycle 123 has two outgoing edges to vertex 4. Thus, σ ∈ FP(G) ⇒ π(σ ) ∈ FP(N) for this example graph. An alternative style of nerve theorem would enable us to at least restrict the candidate fixed point supports of G based on the structure of the nerve. For example, we might hope that whenever the nerve N(G, U) is directional with direction W → T that the directionality would pullback to guarantee G is directional with direction ω → τ for ω = π −1 (W) and τ = π −1 (T). If this held, then we could guarantee that the fixed point supports of G were confined to τ = π −1 (T), and thus FP(G) ⊆ FP(G|τ ) by Lemma 2.9. Such a result would be somewhat weaker than (4) in that the restrictions on σ ∈ FP(G) would not be as strong. On other hand, it would also be somewhat stronger in a different direction, since a result like this would guarantee the presence of graphical domination relationships for ruling out fixed point supports, which is not something guaranteed by (4). Unfortunately, this alternative nerve theorem does not hold in general either, and the same graph from Fig. 10a provides a counterexample, as do the other graphs in Fig. 10. Example 4.6 It is straightforward to see that the nerve N in Fig. 10a is directional for W := {1, 2, 3} and T := {4}: every subset S ⊆ VN that intersects W either has a proper source in G|S , and thus dies from domination by Lemma 2.10, or contains 123, in which case we have 4 >S 1. However, G is not directional since 12345 ∈ FP(G), so there is no collapse of the fixed point supports of G onto a proper subset τ . Figure 10c gives another counterexample where G has a full support fixed point but a directional cover whose nerve is directional. We have seen that in general the existence of a directional relationship W → T of the nerve does not guarantee directionality of G. But are there certain conditions under which this holds? It turns out that we can pullback such a directionality relationship in the special case when the nerve has a nontrivial DAG decomposition (see Definition 2.16). Theorem 4.7 (DAG decomposition of the nerve) Let G be a graph with nerve N = N(G, U) where U is a directional cover induced by a partition {ν1 , . . . , νn }, and let π : VG → VN = [n] be the canonical quotient map of the partition. Then for any DAG decomposition (W, T) of the nerve N, we have that G is directional with direction ω → τ for ω = π −1 (W) and τ = π −1 (T). In particular, FP(G) ⊆ FP(G|τ ), and so for all σ ∈ FP(G), we have π(σ ) ⊆ T. Notice that in Theorem 4.7, we can only conclude that σ ∈ FP(G) ⇒ π(σ ) ⊆ T, and so we do not quite have the ideal nerve theorem result that π(σ ) ∈ FP(N) as in (4). The conclusion in Theorem 4.7 is weaker than (4) because although the directionality of N guarantees that every element of FP(N) is contained in T as is each π(σ ), we cannot guarantee that π(σ ) is actually a fixed point support of N.

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Fig. 11 Example directional covers and nerves. (a1–c1) Graphs with simple directional covers in which every pair of covering graphs have the same type of overlap (chaining overlap in (a), merging overlap in (b), and splitting overlap in (c)). (a2–c2) Nerves for the simple directional covers above

Next we consider when the nerve N is itself a DAG so that, in the maximal DAG decomposition, T is precisely the sinks of N. We can immediately apply Theorem 4.7 to see that the directionality of N pulls back to G, but in fact we can say something stronger: the ideal nerve theorem conditions of (4) hold in this case. Moreover, it turns out that there are further restrictions on the fixed point supports of G in terms of the fixed points of the component subgraphs G|νi , which are prescribed by the partition. Theorem 4.8 (DAG nerve) Let G be a graph with nerve N = N(G, U) where U is a directional cover induced by a partition {ν1 , . . . , νn }, and let π : VG → VN = [n] be the canonical quotient map of the partition. Suppose that N is a DAG, and let T = {sinks of N} and W = VN \ T. Then G is directional with direction ω → τ for ω = π −1 (W) and τ = π −1 (T). Moreover, 1. σ ∈ FP(G) ⇒ π(σ ) ∈ FP(N) = P(T) \ {∅}, where P(T) denotes the power set of T. 2. σ ∈ FP(G) ⇒ σ ∩ νi ∈ FP(G|νi ) ∪ {∅} for all i ∈ T and σ ∩ νj = ∅ for all j ∈ W. Figure 11 illustrates three special cases of directional covers U of a graph G with their corresponding nerves shown below. These directional covers have pairwise overlaps that are either: only chainings (Fig. 11a), only mergings (Fig. 11b), or only splittings (Fig. 11c). We refer to these types of overlaps as n-chaining, n-merging and n-splitting, respectively. We see that their corresponding nerves are DAGs where the set of sinks T is either a single sink, T = {n}, as in panels a2 and b2, or an independent set of sinks, T = {2, 3, . . . , n}, as in panel c2. Theorem 4.7 tells us that in each case the underlying graph G is directional with direction ω → τ for τ = π −1 (T). Additionally, in the case of n-splitting, Theorem 4.8 gives a stronger result. Namely,

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σ ∈ FP(G) ⇒ π(σ ) ∈ FP(N) and σ ∩ νi ∈ FP(G|νi ) ∪ {∅} for i ∈ {2, 3, . . . , n}. That is: any fixed point support σ of G gets pushed forward to a fixed point support of the nerve N, and for any 2 ≤ i ≤ n we have that if σ intersects νi then this intersection is also a fixed point support of the induced subgraph on νi . Thus far, we have only seen nerve theorems in the case when N has a nontrivial DAG component, but it turns out that a similar nerve result holds in the case when N is a cycle (and thus has no DAG component). Theorem 4.9 (cycle nerve) Let G be a graph with nerve N = N(G, U) where U is a directional cover induced by a partition {ν1 , . . . , νn }, and let π :VG →VN = [n] be the canonical quotient map. Suppose that N is a cycle on n vertices. Then 1. σ ∈ FP(G) ⇒ π(σ ) ∈ FP(N) = {[n]} 2. If {ν1 , . . . , νn } is a simply-embedded partition σ ∈ FP(G) ⇒ σ ∩ νi ∈ FP(G|νi ) for all i ∈ [n].

of

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Theorem 4.9 is a repackaging of results from [16], which explores graphs known as directional cycles; in the terminology of this paper, these are precisely graphs with a directional cover whose nerve is a cycle. In [16, Theorem 1.2], it was shown that for this family of graphs, every fixed point support must nontrivially intersect every νi , and so for every σ ∈ FP(G), we have π(σ ) = [n]. Recall that when N is a cycle, FP(N) = {[n]} by Lemma 2.11, and thus, combining these results, we are guaranteed that π(σ ) ∈ FP(N). In [16, Theorem 1.5], it was also shown that when the partition {ν1 , . . . , νn } has a special property, known as simply-embedded,5 then every fixed point support must restrict to a fixed point in each of the component subgraphs G|νi , yielding the second part of Theorem 4.9. It is worth noting that another special family of graphs with directional covers was previously studied in [8, Section 5]. That work focused on composite graphs, which are graphs where all the vertices in a component behave identically with respect to the rest of the graph. Consequently, the only directional covering graphs Gij used in the cover are those that have all possible edges forward from νi to νj and no backward edges. In this context, the components of a composite graph correspond to the partition {ν1 , . . . , νn } that induces the directional cover, and the skeleton of the composite graph is its nerve. With this perspective, many of the results of [8, Section 5] can be reinterpreted as nerve theorems for the special family of composite graphs.

say that {ν1 , . . . , νn } is a simply-embedded partition if every vertex in νi is treated identically by the rest of the graph. In other words, for every j ∈ VG \ νi if j → k for some k ∈ νi , then j →  for all  ∈ νi .

5 We

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4.3 Proofs of Nerve Theorems Throughout this subsection we fix the following notation: G is a graph with a partition {ν1 , . . . , νn } that induces a directional cover U. The nerve N(G, U) is denoted by N, and π : VG → VN = [n] is the canonical quotient map induced by the partition. Before proving the nerve theorems we give an overview of the structure of the proofs. To prove Theorem 4.7 (DAG decomposition of the nerve), we first show that whenever N has a proper source s, we can guarantee that G is directional for ω = π −1 (s) and τ = VG \ω (see Lemma 4.11). This gives a rather coarse directional decomposition of G. We will then consider the general case when we have a DAG decomposition (W, T) of the nerve N. We will use the previous result and show inductively that G is directional with direction π −1 (W) → π −1 (T). For this proof, we will use three ingredients we briefly describe now. First, we will use a topological ordering on W, which guarantees that the only possible edges in N|W are from lower numbered vertices to higher number vertices. With respect to this topological ordering, vertex 1 in N is a proper source; vertex 2 is a proper source in N|VN \{1} ; vertex 3 is a proper source in N|VN \{1,2} , and so on. This ordering will allow us to induct on |W|. The second ingredient is Lemma 4.12. This result will allow us to refine a directional decomposition of a graph in order to grow ω, and consequently shrink τ by looking at directional decompositions of the subgraph induced on the vertices in τ . The third and final ingredient is Lemma 4.13. This result will show that the construction of a directional cover and its nerve is compatible with taking subgraphs of the underlying graph corresponding to only some of the components of the partition inducing the cover. At the core of the proofs is the process of “extending domination”. To see why this idea is central, consider a directional cover of G and σ ⊆ VG a subset of the vertices of G. Let μ be the intersection of σ with the vertices of one of the covering graphs such that μ is not contained in the τ component of that covering graph. Since μ is completely contained within a directional graph then it must die by domination. Under certain conditions, one can show that σ dies by domination by extending the domination relationship on μ to all of σ . Therefore, all the proofs hinge on the following technical result that determines when domination can be extended to a superset. Lemma 4.10 (restricting and extending domination) Let G be a graph, and let α ⊂ μ ⊂ σ ⊆ VG be subsets of vertices with α possibly empty (see Fig. 12). Then the following hold: (a) Restriction: Suppose k >σ j where j ∈ μ and k ∈ VG . Then k >μ j. (b) Extension: Suppose k >μ j where j ∈ μ \ α and k ∈ VG . If there are no edges from σ \ μ to μ \ α (i.e., EG (σ \ μ, μ \ α) = ∅), then k >σ j. Proof Recall from Definition 2.7 that k >σ j if the following three conditions hold:

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Fig. 12 Figure for Lemma 4.10. A graph G with subsets of the vertices α ⊂ μ ⊆ σ ⊂ VG satisfying the conditions of Lemma 4.10. Specifically, the vertex j ∈ μ \ α, the vertex k ∈ VG , and there are no edges from vertices in σ to those in μ \ α

1. For all i ∈ σ \ {j, k} if i → j , then i → k. 2. If j ∈ σ , then j → k. 3. If k ∈ σ , then k  j . For (a), Restriction, we see that k >σ j immediately implies that k >μ j since if condition 1 holds for all of σ , then it holds for the subset μ as well, and conditions 2 and 3 go through directly. For (b), Extension, we see condition 2 goes through immediately since j ∈ μ ⊂ σ . For condition 3, observe that if k ∈ σ then either k ∈ μ or k ∈ σ \ μ, and in both cases k  j as required. Finally, for condition 1, notice that for i ∈ μ this condition holds because k >μ j , while for i ∈ σ \ μ, this condition holds trivially because j ∈ μ \ α and there are no edges from σ \ μ to μ \ α by hypothesis. Thus condition 1 holds as well, and so k >σ j .   We can now prove that it is possible to pull back directionality from the nerve N to G whenever N has a proper source. Lemma 4.11 If s is a proper source in N, then G is directional with ω = π −1 (s) and τ = VG \ ω. Proof Let s be a proper source in N, ω := π −1 (s), and τ := VG \ ω. To show that G is directional with direction ω → τ , consider σ ⊆ VG such that σ ∩ ω = ∅. We need to show that σ dies by domination, i.e., that there exists a j ∈ σ ∩ ω and a k ∈ VG such that k >σ j . The organization of the proof is as follows. We first find a covering graph Gs1 ∈ U such that ω ⊆ VGs1 . Then we set μ ⊆ σ to be the restriction of σ to Gs1 . Since Gs1 is directional, μ dies by domination. We will show that by setting α ⊂ μ to be the intersection of μ with the τ component of Gs1 , the conditions of Lemma 4.10 hold (see Fig. 13). Thus, we can extend the domination relationship to all of σ . To find Gs1 , notice that since s is a proper source in N there exists at least one vertex in N that s sends an edge to; without loss of generality, label the vertices of N that s sends edges to as 1, . . . , m. Since s → 1 in N, the covering graph

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Fig. 13 Figure for proof of Lemma 4.11. Vertex s is a proper source in N (right) with outgoing edges to vertices 1,2, and 3. In G (left), we see the corresponding components νs and {ν1 , ν2 , ν3 }. We consider a subset σ ⊆ VG (outlined in red), and let μ := σ ∩(νs ∪ν1 ) (shaded in purple). Arrows with an x through them indicate that no edges are allowed in the specified direction between the relevant components. Specifically, there are no edges from {ν1 , ν2 , ν3 } into νs because the graphs Gs1 , Gs2 , and Gs3 must have “splitting overlap” by the definition of directional cover. There are no edges from any other vertices in G into νs because there are no edges in the nerve between s and any of the other vertices besides 1, 2, and 3

Gs1 := G|νs ∪ν1 must be directional with direction νs → ν1 . Let μ := σ ∩ (νs ∪ ν1 ) (see Fig. 13). Since Gs1 is directional, there exists a j ∈ μ ∩ νs and k ∈ VGs1 such that k >μ j . Following the notation of Lemma 4.10, let α := μ \ νs = μ ∩ ν1 . We will show that there are no edges in G from vertices in σ \ μ to vertices in μ \ α = μ ∩ νs , enabling us to extend the domination relationship from μ to all of σ . Note that by definition of directional cover, there can only be edges between νs and ν in G if there is an edge between s and  in N (see Definition 4.3). Thus the only candidate vertices in G that could send edges into μ \ α = μ ∩ νs are those in {ν1 , . . . , νm } since the only edges in N that involve s are those from s to 1, . . . , m. But since (s, 1), . . . , (s, m) ∈ EN , condition (2) of the definition of directional cover requires that the covering graphs Gs1 , . . . , Gsm have splitting overlap, so there are no edges from ν to νs for any 1 ≤  ≤ m. Thus, there are no edges from σ \ μ into μ ∩ νs = μ \ α, and so by Lemma 4.10, the domination relationship k >μ j extends to give k >σ j . Hence G is directional with direction ω → τ .   We now give the two lemmas that allow us to inductively use the result above. First, we show that one can refine a directional decomposition of a graph by looking at possible directional decompositions of the subgraph induced on the τ component. Lemma 4.12 Suppose that G is a directional graph with direction ω1 → τ1 and that G|τ1 is also directional with direction ω2 → τ2 (see Fig. 14). Then G is directional with direction ω1 ∪ ω2 → τ2 .

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Fig. 14 Figure for Lemma 4.12. A directional graph G with an initial directional partition ω1 → τ1 . Additionally, G|τ1 is directional with direction ω2 → τ2 . These two partitions can then be combined to show that G is directional for a larger set ω = ω1 ∪ ω2 (outlined in gray) and smaller set τ = τ2

Proof Let ω = ω1 ∪ ω2 and τ = τ2 . To show that G is directional with direction ω → τ , consider σ ⊆ VG such that σ ∩ ω = ∅. We will show that σ dies by domination. Case 1: σ ∩ ω1 = ∅. Then since G is directional with ω1 → τ1 , σ dies by domination. Case 2: σ ∩ω1 = ∅. Then σ ⊆ τ1 . Since σ ∩ω = ∅, we have σ ∩ω2 = ∅. Then since G|τ1 is directional with ω2 → τ2 , there exists j ∈ σ ∩ ω2 and k ∈ VG|τ1 = τ1 such that k >σ j in G|τ1 . Since σ ∩ ω1 = ∅, there are no vertices in σ outside of G|τ1 that could potentially subvert the domination relationship between k and j . Thus, k >σ j in all of G, and so G is directional for ω = ω1 ∪ ω2 and τ = τ2 .   We now show that the construction of a directional cover and its nerve behaves nicely for induced subgraphs of G, restricting to a subset of the partition components. Lemma 4.13 Let νI := {νi | i ∈ I } be a subset of the components of the partition ν of G, for I ⊆ VN . Let GI := G|∪i∈I νi denote the induced subgraph of G on the components νI . Then the partition νI of the vertices of GI induces a directional cover UI whose nerve is N(GI , UI ) = N|I , the restriction of the original nerve N = N(G, U) to the vertices I . Proof Observe that the partition νI yields the edge set E(GI , νI ) := {(i, j ) ∈ I × I | G|νi ∪νj is directional with direction νi → νj }. This edge set is clearly a subset of the edge set E(G, ν) for the cover U of G; specifically, E(GI , νI ) = {(i, j ) ∈ E(G, ν) | i, j ∈ I }. Moreover, the set of graphs {Gij | (i, j ) ∈ E(GI , νI )} form a graph cover of GI because for any i, j ∈ I if

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there are any edges between νi and νj in G, then Gij or Gj i must have been in the cover U of G, so one of these graphs is directional, and thus (i, j ) or (j, i) is in E(GI , νI ). Thus, νI induces a directional cover UI of GI with vertex set I and edge set E(GI , νI ). Since E(GI , νI ) is precisely the edges of E(G, ν) among the vertices in I , we see that the nerve of the cover N(GI , UI ) is precisely N|I , the restriction of the nerve of G to the vertices I .   We are now prepared to prove Theorem 4.7 (reprinted below for convenience). Theorem 4.7 (DAG decomposition of the nerve) For any DAG decomposition (W, T) of the nerve N, we have that G is directional with direction ω → τ for ω = π −1 (W) and τ = π −1 (T). In particular, FP(G) ⊆ FP(G|τ ), and so for all σ ∈ FP(G), we have π(σ ) ⊆ T. Proof Let (W, T) be a DAG decomposition of the nerve N. We will show that G is directional with direction π −1 (W) → π −1 (T) by inducting on |W| in the DAG decomposition of N. The base case of |W| = 1 follows immediately from Lemma 4.11 since the first element of W must be a proper source in N. For the inductive step, assume the inductive hypothesis holds whenever |W| < m and consider a DAG decomposition (W, T) of N where |W| = m. Since N|W is a DAG, there is a topological ordering of the vertices such that the only edges in N|W are from lower numbered to higher numbered nodes; WLOG relabel the vertices of W as 1, . . . , m according to this ordering. Let W1 = W \ {m} and T1 = T ∪ {m}. It is straightforward to check that (W1 , T1 ) is also a DAG decomposition of N. Since |W1 | < m, by the inductive hypothesis, G is directional with ω1 = π −1 (W1 ) and τ1 = π −1 (T1 ) = π −1 (m) ∪ π −1 (T). We will show that G|τ1 is also directional, so that we may apply Lemma 4.12 and further refine the directional decomposition of G. Specifically, we will show that G|τ1 has direction ω2 → τ2 for ω2 = π −1 (m) and τ2 = π −1 (T1 \ {m}) = π −1 (T). By Lemma 4.13, the original directional cover of G restricts to a directional cover of G|τ1 and its nerve is N|{m} ∪T. Since m ∈ W in the original DAG decomposition of N, m is not a sink in N, so it has at least one outgoing edge. Moreover there are no edges from T back to m in a DAG decomposition. Thus m is a proper source in N|{m} ∪ T. Therefore, by Lemma 4.11, G|τ1 is directional with direction ω2 → τ2 for ω2 = π −1 (m) and τ2 = π −1 (T).

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Finally, since G is directional with ω1 = π −1 (W \ {m}), τ1 = π −1 (m) ∪ π −1 (T) and G|τ1 is directional with ω2 = π −1 (m) and τ2 = π −1 (T), we see from Lemma 4.12, that G is directional with direction ω1 ∪ ω2 → τ2 . Since π −1 (W) = ω1 ∪ ω2 and π −1 (T) = τ2 , we see G is directional with direction π −1 (W) → π −1 (T) as desired.   Next we consider when the nerve N is itself a DAG so that in the maximal DAG decomposition T is precisely the sinks of N. Theorem 4.7 guarantees that the directionality of N pulls back to G, but to prove the rest of the nerve theorem conditions, we must appeal to a result characterizing the fixed point supports of disjoint unions, proven in [8]. The disjoint union of component subgraphs is the graph consisting of those subgraphs with no edges between the components. Theorem 4.14 ([8], Theorem 11) Let G be the disjoint union of component subgraphs G1 , . . . , GN . For any nonempty σ ⊆ VG , σ ∈ FP(G)

⇔

σ ∩ VGi ∈ FP(Gi ) ∪ {∅} for all i ∈ [N ].

We can now prove Theorem 4.8 (reprinted below). Theorem 4.8 (DAG nerve) Suppose that N is a DAG, and let T = {sinks of N} and W = VN \ T. Then G is directional with direction ω → τ for ω = π −1 (W) and τ = π −1 (T). Moreover, 1. σ ∈ FP(G) ⇒ π(σ ) ∈ FP(N) = P(T) \ {∅}, where P(T) denotes the power set of T. 2. σ ∈ FP(G) ⇒ σ ∩νi ∈ FP(G|νi )∪{∅} for all i ∈ T and σ ∩νj = ∅ for all j ∈ W. Proof The fact that G is directional with direction ω → τ for ω = π −1 (W) and τ = π −1 (T) follows from Theorem 4.7 since the given choice of (W, T) is the maximal DAG decomposition of N. As a consequence of this, we have FP(G) ⊆ FP(G|π −1 (T ) ), and so we turn our attention to G|π −1 (T ) to understand FP(G). Observe that since T = {sinks of N}, there are no edges between the vertices in T, and so N|T is an independent set. Thus, there are no edges in G between the components νi for i ∈ T, and so G|π −1 (T ) is a disjoint union of the component   subgraphs G|νi . Applying Theorem 4.14, we see that σ ∈ FP G|π −1 (T ) precisely   when σ ∩ νi ∈ FP(G|νi ) ∪ {∅} for all i ∈ T. And since FP(G) ⊆ FP G|π −1 (T ) by the directionality of G, part (2) of the theorem statement follows immediately. For part (1), observe that since G is directional, σ ∈ FP(G) implies that π(σ ) ⊆ T. Since N is a DAG, by Lemma 2.15, FP(N) = P(T), and so every subset of T is an element of FP(N). Thus, σ ∈ FP(G) ⇒ π(σ ) ∈ FP(N) as desired.  

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5 Some Extensions and Applications We now turn our attention to some examples that illustrate the power of our nerve theorems. Going back to Fig. 3a,b of the Introduction, we see that this graph and its nerve satisfy the hypotheses of Theorem 4.7 and Theorem 4.8. The nerve (panel b) is a simple path that has a maximal DAG decomposition with T = {10} (10 is the unique sink node). Theorem 4.7 thus predicts that FP(G) ⊆ FP(G|ν10 ), since ν10 = π −1 ({10}). In fact, FP(G) = {ν10 }, so the network has a unique fixed point supported on ν10 . Moreover, this fixed point is stable because it corresponds to a clique (see [8, 9]). As seen in panel d, the dynamics do indeed converge to this stable fixed point. Furthermore, for solutions with initial conditions supported on the first clique (G|ν1 ), we see that the transient dynamics activate all cliques in the chain, in sequence, following the path of the nerve. In the remainder of this section, we will discuss additional examples of networks whose graphs and nerves satisfy the hypotheses of one or more of our nerve theorems: Theorem 4.7, Theorem 4.8, and Theorem 4.9. Just as in Fig. 3, we will see that the nerve not only predicts the fixed points and asymptotic dynamics, but also provides insight into the transient dynamics of the network. In the second subsection, we will see that even when we violate a key condition of directional covers, the nerve of a network covered by directional graphs can still provide accurate predictions of the dynamics.

5.1 Iterating and Combining DAG Decomposition and Cycle Nerve Theorems Recall that for any partition {ν1 , . . . , νn } of the vertices of G, there is an associated quotient map π : VG → VN = [n] defined by π(νi ) = {i} for each i ∈ [n]. We saw in Sect. 4 that if such a partition induces a directional cover, and the nerve N has a DAG decomposition (W, T), then the fixed points of the corresponding CTLN are confined to the non-DAG part T. In other words, Theorem 4.7 tells us that FP(G) ⊆ FP(G|τ ), where τ = π −1 (T). Now if we consider the restricted graph, G = G|τ , we can potentially iterate this process by finding a new directional cover, with nerve N , DAG decomposition (W , T ), and quotient map π . This would enable us to further restrict the fixed points of the original network to FP(G) ⊆ FP(G ) ⊆ FP(G ), where G = G|τ , and τ = π −1 (T ) ⊂ τ . Note that G is the original graph restricted to an even smaller subset of vertices. This kind of iteration may enable us to get more power from our nerve theorems, by further constraining FP(G).

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Figure 15 shows an example of a graph (a1) whose nerve (a2) is a DAG. Therefore, we can apply Theorem 4.8 to conclude that the fixed point supports of FP(G) must be unions of fixed points of G restricted to the green, red, or orange components. Moreover, the activity flows towards the subnetwork corresponding to the sinks in the nerve. Indeed, for a given initial condition supported on the top gray nodes, we see the solution converge to a limit cycle supported only on green nodes (a3). The green component, however, is itself a complex graph. Thus, we may consider the subgraph G of G corresponding to the non-DAG part of N. Figure 15 (right) shows G (b1) with nodes colored according to a DAG decomposition of its own nerve N (b2). (Note that G is a disjoint union of three graphs, and the nerve N is the disjoint union of nerves for each connected component of G .) Applying Theorem 4.7 allows us to restrict the fixed points of G even further, to the vertices that map to the colored nodes in b2. We see this reflected in the dynamics as well. Panel b3 shows the same limit cycle as before, only now it’s clear that the green curves in a3 correspond only to the yellow, purple, and blue neurons in b1. We can also combine the DAG nerve theorems with the cycle nerve theorem, Theorem 4.9. Figure 16a depicts the graph of a complex network whose nodes are grouped according to a partition with 12 components (8 in gray, 4 in color). In panel b we see the nerve of the induced directional cover, with the color of each node matching those in the original graph. This nerve has a DAG decomposition with the eight gray nodes in the DAG part, W, and the four colored nodes in the non-DAG part, T. Theorem 4.7 thus tells us that all fixed points of the original graph G must be contained in τ = π −1 (T), which is the set of colored nodes in panel a. Indeed, FP(G) ⊆ FP(G|τ ) for this graph. Note that we can also apply another nerve theorem to G|τ to say something stronger about FP(G). Because G|τ has a nerve that is a cycle, Theorem 4.9 tells us that any fixed point of G must intersect each of the components π −1 (i) for i in the cycle. That is, fixed points of G must contain at least one vertex from each of the four colors (red, blue, green, orange) shown in panel a. This is in fact the case, as the CTLN for G has FP(G) = {14567}. Moreover, we see that even if we choose initial conditions supported only on the gray vertices, the dynamics will converge to

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Fig. 16 Application of multiple nerve theorems. (a) A graph G for a CTLN. (b) A nerve N of G with 12 vertices. Each of the gray vertices in N corresponds to a cluster of gray nodes in (a), while the colored vertices in N correspond to the vertices with matching colors in G. (c) A solution to a CTLN with graph G and initial conditions supported on the top-most gray nodes. The activity flows down the network and converges to a limit cycle involving only the colored nodes

a part of the state space where the gray neurons are off and at least one neuron of each color is active (see Fig. 16c).

5.2 Extensions Beyond Directional Covers In Definition 4.3 of (rigid) directional covers, we required fairly stringent conditions that enabled us to prove strong results about the fixed points of a graph in terms of the fixed points of its nerve. Here we consider some examples of “weak” directional covers, where the component graphs are all directional but one of the conditions in Definition 4.3 is violated. Nevertheless, we find that the nerve provides a remarkably accurate prediction of the network dynamics. Our starting point for these examples is a pair of “nerve” graphs with a grid-like structure, shown in Fig. 17a. Each graph is a finite lattice with directed edges moving down and to the right across the grid, and an additional edge, 20 → 16, completing a cycle at the bottom. We can think of these graphs as a pair of nerves, N1 and N2 , for larger networks obtained by inserting directional graphs along the edges. In this case, a vertex i corresponds to a component νi , and an edge i → j corresponds to a directional graph G|νi ∪νj with direction νi → νj . The only difference is that the first nerve, N1 , includes the 15 → 20 edge; while the second nerve, N2 , does not (see dotted line in Fig. 17a). The nerves enable one to make concrete predictions about the network dynamics. Specifically, we can consider CTLNs where the graph G is chosen to be N1 or

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Fig. 17 A grid-like nerve shapes global sequential dynamics irrespective of the component graphs. (a) Two versions of the nerve, with and without the 15 → 20 edge. (b–c) Solutions for CTLNs where the graph is nerve 1 or nerve 2, respectively. (d) 5-clique component graphs and a corresponding directional graph for an edge in the cover. (e) 5-star component graphs and their directional graph. (f–i) Solutions for associated CTLNs for all four combinations of nerves and component graphs

N2 . Figure 17b shows a solution for a CTLN with G = N1 , where the activity is initialized with x1 (0) = 0.5 and xi (0) = 0 for i > 1 (we refer to this as initial condition 1). Note that the transient activity follows a hopscotch trajectory 1 → (2, 6) → 7 → (8, 12) → 13 → (14, 18) → 19, where the pairs (2, 6), (8, 12), and (14, 18) fire synchronously. Once the activity reaches the bottom row T, the dynamics converge to a limit cycle that follows the cycle 19 → 20 → 16 → 17 → 18 → 19 in the graph. If we choose the same initial condition for G = N2 , we obtain exactly the same result. This is because the activity never reaches node 15, and so the cut edge is not “seen.” Figure 17c shows the solution when we initialize instead at x3 (0) = 0.5, and xi (0) = 0 for i = 3 (initial condition 2). In this case, the activity follows the hopscotch trajectory 3 → (4, 8) → 9 → (10, 14) → 15 that ends in a stable fixed point at 15. This is precisely what we expect since node 15 is a sink. These basic features of the

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dynamics for a CTLN defined directly on the nerve can be taken as a prediction for the dynamics of any network G that has a directional cover with N1 or N2 as its nerve. If we consider a graph G for which N1 or N2 is the nerve of a directional cover, satisfying all the conditions of Definition 4.3, then we have additional knowledge about the fixed points of G. The first nerve, N1 , has a maximal DAG decomposition (W, T) with nodes 1–15 in W and nodes 16–20 in T. Theorem 4.7 thus predicts that all fixed points of G are supported in τ = π −1 (T). Moreover, since N|T forms a cycle, applying Theorem 4.9 we expect all fixed points to intersect each component of T. In the second nerve, N2 , 15 → 20 and thus node 15 is a sink. Any DAG decomposition for N2 must therefore include node 15 in T. Applying Theorem 4.7, we expect fixed point(s) corresponding to the cycle, as before, as well as fixed point(s) supported in ν15 = π −1 (15). We may also have fixed points whose supports are unions of those from the cycle and ν15 . These observations are all independent of the choice of components νi of G and the subgraphs G|νi ∪νj , so long as the nerve corresponds to a directional cover in accordance with Definition 4.3. What happens if a cover by directional graphs violates one of the conditions of Definition 4.3? Figure 17d shows a directional graph that can be inserted along the edges of N1 and N2 . The components are 5-cliques, and the directional graphs are the same as in Fig. 3c. Inserting these into the grid-like nerves, however, yields many nodes with splitting overlaps (unlike for the nerve in Fig. 3b). This means backwards edges within the directional graphs violate the “splitting” condition 2 of Definition 4.3, and the resulting network does not have a directional cover. Moreover, the splitting condition was essential to our nerve theorems: the fixed points of a network G obtained by inserting the Fig. 3c graph into either N1 or N2 do not satisfy the constraints given by Theorem 4.7. In fact, there are numerous fixed points supported outside G|τ for τ = π −1 (T). As another example, inserting the directional graph in Fig. 17e as G|νi ∪νj along the edges of N1 or N2 also produces a cover by directional graphs that violates the splitting condition. Here, each component is a cyclically symmetric oriented graph on five vertices, called the 5-star,6 and the directional graphs Gij = G|νi ∪νj are chosen to have forward edges from νi onto three of the nodes in νj , and backwards edges from the remaining two nodes in νj to all five nodes in νi . Again, our nerve theorems fail to predict the fixed point structure of the larger network. Nevertheless, we find that the dynamics of these networks whose graph covers violate the splitting condition are well predicted by their nerves. Figure 17f–i display the dynamics of the networks obtained from each of the four combinations: 5-clique components with nerve N1 (panel f), 5-clique components with nerve N2 (panel g), 5-star components with nerve N1 (panel h), and 5-star components with nerve N2 (panel i). In each case, the nerve dictates the global structure of the dynamics as activity flows from one component to another. Figure 17f shows the solution for a CTLN with 5-clique components and initial conditions supported in ν1 . As

6 Each

vertex k in the 5-star has two outgoing edges: k → k + 1 and k → k + 2 (indexing mod 5).

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predicted, the asymptotic behavior is of a limit cycle following the cycle 16 → 17 → 18 → 19 → 20 → 16 in the nerve. Figure 17g shows the activity with 5clique components inserted into nerve N2 , with initial conditions leading the activity to converge to a stable fixed point, as in Fig. 17c. Inserting 5-star graphs in N1 and N2 , instead of 5-cliques, also produces asymptotic behavior that settles into a repeating sequence (Fig. 17h) or a localized attractor (Fig. 17i). The structure of the inserted graphs, however, does affect the local dynamics within each component of the nerve. In Fig. 17f the transient dynamics are considerably more regular than in Fig. 17b, and the neurons within each component fire synchronously. In contrast, in Fig. 17h the transient dynamics are irregular like in Fig. 17b, and the neurons within each component do not fire synchronously. In each case, we see global aspects of the dynamics being dictated by the nerve, while local dynamics are affected by differences in the component graphs G|νi . For example, in Fig. 17g the activity converges to a stable fixed point corresponding to the 5-clique supported on ν15 ; but in Fig. 17i, the network does not converge to a fixed point because the 5-star does not support a stable fixed point. Instead, the activity settles into a limit cycle typical of 5-star CTLNs, with activity confined almost entirely to the neurons in ν15 . Taken together, these examples suggest that nerves can be predictive of network dynamics for a broader class of directional covers, including networks where our current set of nerve theorems do not apply. It is an open question how to formalize these observations into new nerve theorems that reflect the predictions on the dynamics.

6 Conclusion In this work, we investigated how the global structure of a network, as captured by the nerve of a directional cover, reflects the underlying dynamics. By replacing directional subgraphs with single edges, the nerve provides a significant dimensionality reduction of a network. Moreover, this reduced network is meaningful: in simulations, we have seen that the dynamics of a CTLN with a directional cover closely follows the dynamics of its nerve. Although the observed relationship between the dynamics of a network and its nerve is so far heuristic, we have proven a number of theorems directly connecting the fixed points of a larger network G to those of its nerve N. Specifically, we showed that whenever the nerve has a DAG decomposition (W, T), the nerve is directional with direction W → T, and this guarantees that the larger network G is also directional. In particular, the fixed points of the larger network are confined to live in the pullback π −1 (T) (Theorem 4.7). In the special case where the nerve is a DAG, we have a tighter connection between FP(G) and FP(N). Theorem 4.8 shows that every fixed point support of G projects to a fixed point of N. In other words, σ ∈ FP(G) ⇒ π(σ ) ∈ FP(N).

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Moreover, every σ ∈ FP(G) is a union of fixed point supports of the component subgraphs G|νi . Theorem 4.9 gives similar constraints on FP(G) whenever the nerve is a cycle. Due to the close relationship between fixed points and attractors in CTLNs, understanding the fixed point structure and how this is shaped by network architecture is an important step towards understanding how network connectivity shapes dynamics. The machinery developed here thus provides a useful framework for dimensionality reduction in the analysis of large networks. Moreover, it provides insight into how to engineer complex networks with desired dynamic properties from smaller building block components. Acknowledgments This research is a product of one of the working groups at the Workshop for Women in Computational Topology (WinCompTop) in Canberra, Australia (1–5 July 2019). We thank the organizers of this workshop and the funding from NSF award CCF-1841455, the Mathematical Sciences Institute at ANU, the Australian Mathematical Sciences Institute (AMSI), and Association for Women in Mathematics that supported participants’ travel. We thank Caitlyn Parmelee for fruitful discussions that helped set the foundation for this work. We would also like to thank Joan Licata for valuable conversations at the WinCompTop workshop. CC and KM acknowledge funding from NIH R01 EB022862, NIH R01 NS120581, NSF DMS-1951165, and NSF DMS-1951599.

References 1. Abeles, M.: Local Cortical Circuits: An Electrophysiological Study. Springer, Berlin (1982) 2. Aviel, Y., Pavlov, E., Abeles, M., Horn, D.: Synfire chain in a balanced network. Neurocomputing 44, 285–292 (2002) 3. Bel, A., Cobiaga, R., Reartes, W., Rotstein, H.G.: Periodic solutions in threshold-linear networks and their entrainment. SIAM J. Appl. Dyn. Syst. 20(3), 1177–1208 (2021) 4. Biswas, T., Fitzgerald, J.E.: A geometric framework to predict structure from function in neural networks (2020). Available at https://arxiv.org/abs/2010.09660 5. Borsuk, K.: On the imbedding of systems of compacta in simplicial complexes. Fundam. Math. 35(1), 217–234 (1948) 6. Curto, C., Morrison, K.: Pattern completion in symmetric threshold-linear networks. Neural Comput. 28, 2825–2852 (2016) 7. Curto, C., Degeratu, A., Itskov, V.: Encoding binary neural codes in networks of thresholdlinear neurons. Neural Comput. 25, 2858–2903 (2013) 8. Curto, C., Geneson, J., Morrison, K.: Fixed points of competitive threshold-linear networks. Neural Comput. 31(1), 94–155 (2019) 9. Curto, C., Geneson, J., Morrison, K.: Stable fixed points of combinatorial threshold-linear networks (2019). Available at https://arxiv.org/abs/1909.02947 10. Hahnloser, R.H., Sarpeshkar, R., Mahowald, M.A., Douglas, R.J., Seung, H.S.: Digital selection and analogue amplification coexist in a cortex-inspired silicon circuit. Nature 405, 947–951 (2000) 11. Hahnloser, R.H., Seung, H.S., Slotine, J.J.: Permitted and forbidden sets in symmetric threshold-linear networks. Neural Comput. 15(3), 621–638 (2003) 12. Hayon, G., Abeles, M., Lehmann, D.: A model for representing the dynamics of a system of synfire chains. J. Comput. Neurosci. 18(41–53) (2005)

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13. Leray, J.: Sur la forme des espaces topologiques et sur les points fixes des représentations. J. Math. Pures Appl. 9, 95–248 (1945) 14. Morrison, K., Curto, C.: Predicting neural network dynamics via graphical analysis. In: Robeva, R., Macaulay, M. (eds.) Algebraic and Combinatorial Computational Biology, pp. 241–277. Elsevier, Amsterdam (2018) 15. Morrison, K., Degeratu, A., Itskov, V., Curto, C.: Diversity of emergent dynamics in competitive threshold-linear networks: a preliminary report (2016). Available at https://arxiv.org/abs/ 1605.04463 16. Parmelee, C., Londono Alvarez, J., Curto, C., Morrison, K.: Sequential attractors of combinatorial threshold-linear networks. To appear in SIAM J. Appl. Dyn. Syst. (2022). Available at https://arxiv.org/abs/2107.10244 17. Parmelee, C., Moore, S., Morrison, K., Curto, C.: Core motifs predict dynamic attractors in combinatorial threshold-linear networks, PLOS ONE, 17(3) (2022): e0264456 https://doi.org/ 10.1371/journal.pone.0264456 18. Seung, H.S., Yuste, R.: Principles of Neural Science. In: Appendix E: Neural Networks, pp. 1581–1600, 5th ed. McGraw-Hill Education/Medical, New York (2012) 19. Xie, X., Hahnloser, R.H., Seung, H.S.: Selectively grouping neurons in recurrent networks of lateral inhibition. Neural Comput. 14, 2627–2646 (2002)

Combinatorial Conditions for Directed Collapsing Robin Belton, Robyn Brooks, Stefania Ebli, Lisbeth Fajstrup, Brittany Terese Fasy, Nicole Sanderson, and Elizabeth Vidaurre

Abstract While collapsibility of CW complexes dates back to the 1930s, collapsibility of directed Euclidean cubical complexes has not been well studied to date. The classical definition of collapsibility involves certain conditions on pairs of cells of the complex. The direction of the space can be taken into account by requiring that the past links of vertices remain homotopy equivalent after collapsing. We call this type of collapse a link-preserving directed collapse. In the undirected setting, pairs of cells are removed that create a deformation retract. In the directed setting, topological properties—in particular, properties of spaces of directed paths—are not always preserved. In this paper, we give computationally simple conditions for preserving the topology of past links. Furthermore, we give conditions for when link-preserving directed collapses preserve the contractability and connectedness of spaces of directed paths. Throughout, we provide illustrative examples.

R. Belton · B. T. Fasy () Montana State University, Bozeman, MT, USA e-mail: [email protected]; [email protected] R. Brooks Boston College, Chestnut Hill, MA, USA e-mail: [email protected] S. Ebli École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland e-mail: [email protected] L. Fajstrup Aalborg University, Aalborg, Denmark e-mail: [email protected] N. Sanderson Lawrence Berkeley National Lab, Berkeley, CA, USA e-mail: [email protected] E. Vidaurre Molloy College, Rockville Centre, NY, USA e-mail: [email protected] © The Author(s) and the Association for Women in Mathematics 2022 E. Gasparovic et al. (eds.), Research in Computational Topology 2, Association for Women in Mathematics Series 30, https://doi.org/10.1007/978-3-030-95519-9_7

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1 Introduction A directed Euclidean cubical complex is a subset of Rn comprising a finite union of directed unit cubes. Directed paths (i.e., paths that are nondecreasing in all coordinates) and spaces of directed paths are the objects of study in this paper. In particular, we address the question of how to simplify directed Euclidean complexes without significantly changing the spaces of directed paths. This model is motivated by several applications, where each axis of the model corresponds to a parameter of the application (e.g., time). In particular, Euclidean cubical complexes are used to model concurrency in computer programming [4– 6, 21], hybrid dynamical systems [20], and motion planning [7]. Consider the application to concurrency. In this example, each axis represents a sequence of actions a process completes in the program execution. The complex itself corresponds to “compatible” parameters (i.e., when the processes can execute simultaneously). Cubes missing from the complex correspond to parameters for which the processes cannot execute simultaneously for some reason, such as when they require the same resources with limited capacity; see Fig. 1. A directed path (dipath) in the complex represents a, possibly partial, program execution. Such executions are equivalent if the corresponding dipaths are directed homotopic. Simplifying the complexes allows for a more compact representation of the execution space, which, in turn, reduces the complexity of validating correctness of concurrent programs. A non-trivial Euclidean cubical complex contains uncountably many dipaths and more information than we need for understanding the topology of the spaces of

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0 Fig. 1 The Swiss Flag and Three Directed Paths. The gray and blue squares are the two-cubes of a Euclidean cubical complex. The bi-monotone increasing paths are directed paths starting at (0, 0) and ending at (5, 5). This complex has a cross-shaped hole in the middle. As a consequence, the solid directed paths are directed homotopic while the dashed directed path is not directed homotopic to either of the other directed paths. Each point highlighted in blue is unreachable, meaning that we cannot reach any point highlighted in blue without breaking bi-monotonicity in a path starting at (0, 0). This complex models the dining philosophers problem, a wellknown example in concurrency, where two processes require two shared resources with limited capacity [4, 12]. The two distinct paths (solid and dashed) represent which process uses both shared resources first

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dipaths. The main question we ask is, How can we simplify a directed Euclidean cubical complex while still preserving spaces of dipaths? Past links are local representations of a Euclidean cubical complex at vertices. They were introduced in [21] as a means to show that any finite homotopy type can be realized as a connected component of the space of execution paths for some P V -model. In [1], we found conditions for when the local information of past links preserve the global information on the homotopy type of spaces of dipaths. Because of these relationships between past links and dipath spaces, we define collapsing in terms of past links. We call this type of collapsing link-preserving directed collapse (LPDC). We aim to compress a Euclidean cubical complex by LPDCs before attempting to answer questions about dipath spaces. The main result of this paper is Theorem 3.9, which provides a simple criterion for such a collapsing to be allowed: A pair of cubes (τ, σ ) is an LPDC pair if and only if it is a collapsing pair in the non-directed sense and τ does not contain the minimum vertex of σ . This condition greatly simplifies the definition of LPDC and is easy to add to a collapsing algorithm for Euclidean cubical complexes in the undirected setting. Algorithms and implementations in this setting already exist such as in [15]. Furthermore, we provide conditions for when LPDCs preserve the contractability and connectedness of dipath spaces (Sect. 4) along with a discussion of some of the limitations (Sect. 5). This work provides a start at the mathematical foundations for developing polynomial time algorithms that collapse Euclidean cubical complexes and preserve dipath spaces.

2 Background This paper builds on our prior work [1], as well as work by others [5, 9, 10, 16, 21]. In this section, we recall the definitions of directed Euclidean cubical complexes, which are the objects that we study in this paper. Then, we discuss the relationship between spaces of directed paths and past links in directed Euclidean cubical complexes. For additional background on directed topology (including generalizations of the definitions below), we refer the reader to [6]. We also assume the reader is familiar with the notion of homotopy equivalence of topological spaces (denoted using % in this paper) and homotopy between paths as presented in [11].

2.1 Directed Spaces and Euclidean Cubical Complexes Let n be a positive integer. A (closed) elementary cube in Rn is a product of closed intervals of the following form: [v1 − j1 , v1 ] × [v2 − j2 , v2 ] × . . . × [vn − jn , vn ]  Rn ,

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where v = (v1 , v2 , . . . , vn ) ∈ Zn and j = (j1 , j2 , . . . , jn ) ∈ {0, 1}n . We often refer to elementary cubes simply as cubes. The dimension of the cube is the number of unit entries  in the vector j; specifically, the dimension of the cube in Eq. (1) is the sum: ni=1 ji . In particular, when j = 0 := (0, 0, . . . , 0), the elementary cube is a single point and often denoted using just v. If τ and σ are elementary cubes such that τ ⊆ σ , we say that τ is a face of σ and that σ is a coface of τ . Cubical sets were first introduced in the 1950s by Serre [17] in a more general setting; see also [2, 8, 13]. Elementary cubes stratify Rn , where two points x, y ∈ Rn are in the same stratum if and only if they are members of the same set of elementary cubes; we call this the cubical stratification of Rn . Each stratum in the stratification is either an open cube or a single point. A Euclidean cubical complex (K, K) is a subspace K  Rn that is equal to the union of a finite set of elementary cubes, together with the stratification K induced by the cubical stratification of Rn ; see Fig. 2. We topologize K using the subspace topology with the standard topology on Rn . By construction, if σ ∈ K, then all of its faces are necessarily in K as well. If σ ∈ K with no proper cofaces, then we say that σ is a maximal cube in K. We denote the set of closed cubes in (K, K) by K; the set of closed cubes in K is in one-to-one correspondence with the open cubes in K. Specifically, vertices in K correspond to vertices in K and all other elementary cubes in K correspond to their interiors in K. Throughout this paper, we denote the set of zero-cubes in K by verts(K) and note that verts(K)  Zn , since all cubes in (K, K) are elementary cubes. The product order on Rn , denoted , is the partial order such that for two points p = (p1 , p2 , . . . , pn ) and q = (q1 , q2 , . . . , qn ) in Rn , we have p  q if and only if pi ≤ qi for each coordinate i. Using this partial order, we define the interval of points in Rn between p and q as

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Fig. 2 Euclidean cubical complex in R2 with 24 zero-cubes (vertices), 28 one-cubes (edges), and six two-cubes (squares). By construction, all elementary cubes in a directed Euclidean cubical complex are axis aligned. Consider the vertex v = (3, 4). The edge e = [(2, 4), (3, 4)] (written e = [2, 3] × [4, 4] in the notation of Eq. (1)) is one of the two lower cofaces of v. Since e is not a face of any two-cube, e is a maximal cube (since it is not a face of a higher-dimensional cube)

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[p, q] := {x | p  x  q}. The point p is the minimum vertex of the interval and q is the maximum vertex of the interval, with respect to . Notationally, we write this as min([p, q]) := p and max([p, q]) := q. When q ∈ Zn and q = p + j, for some j ∈ {0, 1}n , the interval [p, q] is an elementary cube as defined in Eq. (1). If, in addition, j is not the zero vector, then we say that [v − j, v] is a lower coface of v. Using the fact that the partial order (Rn , ) induces a partial order on the points in K, we define directed paths in K as the set of nondecreasing paths in K: A path in K is a continuous map from the unit interval I = [0, 1] to K. We say that a path γ : I → K goes from γ (0) to γ (1). Letting K I denote the set of all paths in K, the set of directed paths (or dipaths for short) is − → P (K) := {γ ∈ K I | ∀i, j s.t. 0 ≤ i ≤ j ≤ 1, γ (i)  γ (j )}. − → We topologize P (K) using the compact-open topology. For p, q ∈ K, we denote − →q − → the subspace of dipaths from p to q by P p (K). We refer to (K, P (K)) as a − →q directed Euclidean cubical complex.1 The connected components of P p (K) are exactly the equivalence classes of dipaths, up to dihomotopy. If two dipaths, f and g are homotopic through a continuous family of dipaths, then f and g are called dihomotopic. Given a directed complex, certain subcomplexes are of interest: Definition 2.1 (Special Complexes) Let (K, K) be a directed Euclidean cubical complex in Rn . Let p ∈ verts(K) and let σ be an elementary cube (that need not be in K). 1. 2. 3. 4.

The complex above p is Kp := {q ∈ K | p  q}. The complex below p is Kp := {q ∈ K | q  p}. − →q The reachable complex from p is reach(K, p) := {q ∈ K | P p (K) = ∅}. The complex restricted to σ is K|σ :=

{τ ∈ K | min σ  min τ  max τ  max σ }.

5. If K = I n , then we call (K, K) the standard unit cubical complex and often denote it by (I n , I). If K = I n +x for some x ∈ Zn , then K is a full-dimensional unit cubical complex.

1 Directed Euclidean cubical complexes are an example of a more general concept known as directed space (d-spaces). To define a d-space, we have a topological space X and we define a set of dipaths P (X) ⊆ XI that contains all constant paths, and is closed under taking nondecreasing − → reparameterizations, concatenations, and subpaths. Indeed, P (K) satisfies these properties.

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2.2 Past Links of Directed Cubical Complexes An abstract simplicial complex is a finite collection S of sets that is closed under the subset relation, i.e., if A ∈ S and B is a set such that ∅ = B ⊆ A, then B ∈ S. The sets in S are called simplices. If the simplex A has k + 1 elements, then we say that the dimension of A is dim(A) := k, and we say A is a k-simplex. For example, the zero-simplices are the singleton sets and are often referred to as vertices. Since every element of a set A ∈ S gives rise to a singleton set in the finite set S, A must be finite. In a topological space embedded in Rn , the link of a point v is constructed by intersecting an arbitrarily small (n − 1)-sphere around v with the space itself. In Rn , the link of a point is an (n − 1)-sphere. Moreover, if v ∈ Zn , the link inherits the stratification as a subcomplex of Rn , and can be represented as a simplicial complex whose i-simplices are in one-to-one correspondence with the (i + 1)dimensional cofaces of v. The past link of v is the restriction of the link using the set of lower cofaces of v instead of all cofaces. Thus, we can represent each simplex in the past link as a vector in {0, 1}n \ {0}, where the vector j ∈ {0, 1}n \ {0} represents the cube [v − j, v] in the simplex-cube correspondence. As a simplicial complex, the past link of v in Rn has n vertices {xi }1≤i≤n , and j represents the simplex {xi |1 ≤ i ≤ n, ji = 1} of dimension ||j||1 − 1; for example, (1, 0, 0) represents a vertex and (1, 0, 1) represents an edge. We are now ready to define the past link of a vertex in a Euclidean cubical complex: Definition 2.2 (Past Link) Let (K, K) be a directed Euclidean cubical complex in Rn . Let v ∈ Zn . The past link of v is the following simplicial complex: n lk− K (v) := {j ∈ {0, 1} \ {0} | [v − j, v] ⊆ K}.

As a set, the past link represents all elementary cubes in K for which v is the maximum vertex. As a simplicial complex, it describes (locally) the different types of dipaths to or through v in K; see Fig. 3. We conclude this section with a lemma summarizing properties of the past link, most of which follow directly from definitions: Lemma 2.3 (Properties of Past Links) Let (K, K) be a directed Euclidean cubical complex in Rn . Then, the following statements hold for all v ∈ Zn :  − 1. lk− p∈Rn lkKp (v). K (v) =

− 2. If (K , K ) is a subcomplex of (K, K), then lk− K (v) ⊆ lkK (v). − 3. lk− K (v) = lkKv (v). 4. If there exists w ∈ Zn such that K = [w − 1, w], then lk− K (w) is the complete simplicial complex on n vertices. 5. lk− K (v) is a subcomplex of the complete simplicial complex on n vertices.

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(b) Past Link of x

Fig. 3 Past link in the Open Top Box. (a) The maximum vertex of this complex is v = (v1 , v2 , v3 ). The past link lk− K (v) is the simplicial complex comprising three vertices and two edges (shown in blue/cyan). These simplices are in one-to-one correspondence with the set of lower cofaces of v (highlighted in pink). For example, the edges of lk− K (v), which are labeled e1 and e2 , are in one-to-one correspondence with the elementary two-cubes that are lower cofaces of v (σ1 = [(v1 − 1, v2 , v3 − 1), v] and σ2 = [(v1 , v2 − 1, v3 − 1), v], respectively). In the vector notation for simplices of lk− K (v), we write e1 = (1, 0, 1) and e2 = (0, 1, 1). (b) The past link of a vertex x that is neither the minimum nor the maximum vertex in the complex

Proof Statement 1: If K = ∅, then all past links are empty and the equality trivially holds. If K = ∅, then verts(K) is a finite non empty set. Thus, there exists q ∈ Rn such that for all w ∈ verts(K), q  w. Let j ∈ lk− K (v). Then, [v−j, v] ⊆ K and so v−j ∈ verts(K). Hence, q  v−j, which means − that j ∈ lk− p ∈ Rn lkKp (v). The reverse inclusion follows from the Kq (v) ⊆ fact that each of these statements holds if and only if. Statement 2: Observe that if j ∈ lk− K (v), then, by definition of the past link, [v − j, v] ⊆ K . Since K ⊆ K, we have [v − j, v] ⊆ K ⊆ K. Therefore, we can conclude that j ∈ lk− K (v). Statement 3: By Statement 2 (which we just proved), we have the following − inclusion lk− Kv (v) ⊆ lkK (v). To prove the inclusion in the other direction, let j ∈ lk− K (v). Since v − j  v, then [v − j, v] ⊆ Kv . Therefore, we conclude − that lk− K (v) ⊆ lkKv (v). Statement 4: Since K = [w − 1, w], we know that K is full-dimensional, and so for all j ∈ {0, 1}n , [w − j, w] ⊆ K. Thus, by definition of past link, we have that n the past link of w is: lk− K (w) := {0, 1} \ {0}, which is the complete simplicial complex on n vertices. Statement 5: Let L = K ∩ [v − 1, v]. By definition of past link, we − know lk− L (v) = lkK (v). By Statement 2, since L is a subcomplex of [v − 1, v], − − we know lkL (v) ⊆ lk− [v−1,v] (v). By Statement 4, lk[v−1,v] (v) is the complete − simplicial complex on n vertices. Therefore, lkK (w) is the complete simplicial complex on n vertices.  

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2.3 Relationship Between Past Links and Path Spaces The topology of the past links is intrinsically related to spaces of dipaths. Specifically, in [1] we prove that the contractability and/or connectedness of past links of vertices in directed Euclidean cubical complexes with a minimum vertex2 implies that all spaces of dipaths with w as initial point are also contractible and/or connected. Theorem 2.4 (Contractability [1, Theorem 1]) Let (K, K) be a directed Euclidean cubical complex in Rn that has a minimum vertex w. If, for all − →k vertices v ∈ verts(K), the past link lk− K (v) is contractible, then the space P w (K) is contractible for all k ∈ verts(K). An analogous theorem for connectedness also holds. Theorem 2.5 (Connectedness [1, Theorem 2]) Let (K, K) be a directed Euclidean cubical complex in Rn that has a minimum vertex w. Suppose that, for all v ∈ verts(K), the past link lk− K (v) is connected. Then, for all k ∈ verts(K), − →k the space P w (K) is connected. Furthermore, we proved a partial converse to Theorem 2.5. Specifically, the converse holds only if K is a reachable directed Euclidean cubical complex as defined in Statement 3 of Definition 2.1. This is expected: properties of parts of the directed Euclidean complex which are not reachable from w, do not influence the dipath spaces from w. Theorem 2.6 (Realizing Obstructions [1, Theorem 3]) Let (K, K) be a directed Euclidean cubical complex in Rn . Let w ∈ verts(K), and let L = reach(K, w). − →v Let v ∈ verts(L). If the past link lk− L (v) is disconnected, then the space P w (K) is disconnected.

3 Directed Collapsing Pairs Although simplicial collapses preserve the homotopy type of the underlying space [14, Proposition 6.14] and hence of all path spaces, this type of collapsing in directed Euclidean cubical complexes may not preserve topological properties of spaces of dipaths. In this section, we study a specific type of collapsing called a link-preserving directed collapse. We define link-preserving directed collapses in Sect. 3.1 and give properties of link-preserving directed collapses in Sect. 3.2.

2 In [1], the minimum (initial) vertex was often assumed to be 0 for ease of exposition. We restate the lemmas and theorems here using more general notation, where K has a minimum vertex w.

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3.1 Link-Preserving Directed Collapses Since we are interested in preserving the dipath spaces through collapses, the results from Sect. 2.3 motivate us to study a type of directed collapse (DC) via past links, introduced in [1]. However, we call it a link-preserving directed collapse (LPDC) (as opposed to a directed collapse) since we show in the last sections of this paper that when the spaces of dipaths starting from the minimum vertex are not connected, the following definition of collapse does not preserve the number of components. Definition 3.1 (Link Preserving Directed Collapse) Let (K, K) be a directed Euclidean cubical complex in Rn . Let σ ∈ K be a maximal cube, and let τ be a proper face of σ such that no other maximal cube contains τ (in this case, we say that τ is a free face of σ ). Then, we define the (τ, σ )-collapse of K as the subcomplex obtained by removing everything in between τ and σ : K = K \ {γ ∈ K | τ ⊆ γ ⊆ σ },

(2)

and let K denote the stratification of the set K induced by the cubical stratification of Rn (thus, K  K). We call the directed Euclidean cubical complex (K , K ) a link-preserving directed collapse (LPDC) of (K, K) if, for all v ∈ verts(K ), the past link lk− K (v) is − − homotopy equivalent to lk− K (v) (denoted lkK (v) % lkK (v)). The pair (τ, σ ) is then called an LPDC pair. Remark 3.2 (Simplicial Collapses) The study of simplicial collapses is known as simple homotopy theory [3, 19], and traces back to the work of Whitehead in the 1930s [18]. The idea is very similar: If C is an abstract simplicial complex and α ∈ C such that α is a proper face of exactly one maximal simplex β, then the following complex is the α-collapse of C in the simplicial setting: C = C \ {γ ∈ C | α ⊆ γ ⊆ β}. Note that we use only the free face (α) when defining a simplicial collapse, as doing so helps to distinguish between discussing a simplicial collapse and a directed Euclidean cubical collapse. In addition, we always explicitly state “in the simplicial setting” when talking about a simplicial collapse. Applying a sequence of LPDCs to a directed Euclidean cubical complex can reduce the number of cubes, and hence can more clearly illustrate the number of dihomotopy classes of dipaths within the directed Euclidean cubical complex. For an example, see Fig. 4. However, it is not necessarily true that LPDCs preserve dipath spaces. We discuss the relationship between dipath spaces and LPDCs in Sect. 4.

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Fig. 4 Collapsing the Swiss Flag. A sequence of vertex collapses is presented from the top left to bottom right. At each stage of the sequence, the faces and vertices shaded in blue and purple represent the vertex collapsing pairs with the blue Euclidean cube being σ and the purple vertex being τ . The result of the sequence of LPDCs is shown in (f) is a one-dimensional directed Euclidean cubical complex and one two-cube. Observe that this directed Euclidean cubical − →(5,5) complex clearly illustrates the two dihomotopy classes of P 0 (K). (a) Swiss flag. (b) First stage. (c) Second stage. (d) Third stage. (e) Fourth stage. (f) After collapses

3.2 Properties of LPDCs We give a combinatorial condition for a collapsing pair (τ, σ ) to be an LPDC pair; namely, the condition is that τ does not contain the vertex min(σ ). From the definition of an LPDC, we see that finding an LPDC pair requires computing the past link of all vertices in verts(K ). In [1], we discussed how we can reduce the check down to only the vertices in σ since no other vertices have their past links affected. In this paper, we prove we need to only check one condition to determine if we have an LPDC pair. The one simple condition dramatically reduces the number of computations we need to perform in order to verify we have an LPDC. This result given in Theorem 3.9 depends on the following lemmas about the properties of past links on vertices. Lemma 3.3 (Properties of Past Links in a Vertex Collapse) Let (K, K) be a directed Euclidean cubical complex in Rn . Let σ ∈ K and τ, v ∈ verts(σ ) such that τ  v. If τ is a free face of σ and K is the (τ, σ )-collapse, then the following two statements hold:

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n 1. lk− K|σ (v) = {j ∈ {0, 1} \ {0} | min(σ )  v − j}. − n 2. lk− K |σ (v) = lkK|σ (v) \ {j ∈ {0, 1} \ {0} | v − j  τ }.

Proof To ease notation, we define the following two sets: J := {j ∈ {0, 1}n \ {0} | min(σ )  v − j} I := {j ∈ {0, 1}n \ {0} | v − j  τ }. First, we prove Statement 1 (that lk− K|σ (v) = J ). We start with the forward − inclusion. Let j ∈ lkK|σ (v). By the definition of past links (see Definition 2.2), we know that [v − j, v] ⊆ K|σ . By the definition of K|σ (see Definition 2.1), we know that min(σ )  min([v − j, v]) = v − j. This implies j ∈ J . Therefore, lk− K|σ (v) ⊆ J . For the backward inclusion, let j ∈ J . Then, since v ∈ verts(σ ) and σ is an elementary cube by assumption, and min(σ )  v − j by definition of J , we have v − j ∈ verts(σ ). Since σ ∈ K, all faces must be in K; hence, [v − j, v] ⊆ K|σ . − Therefore, j ∈ lk− K|σ (v), and so lkK|σ (v) ⊇ J . Since we have both inclusions, then Statement 1 holds. Now, we prove Statement 2 (that lk− K |σ (v) = J \ I ). Again, we prove the inclusions in both directions. For the forward inclusion, let j ∈ lk− K |σ (v). − − By Statement 2 of Lemma 2.3, we have lkK |σ (v) ⊆ lkK|σ (v), and so, we obtain j ∈ lk− / I . Assume, for a K|σ (v) = J . Next, we must show that j ∈ contradiction, that j ∈ I . Then, by definition of I , v − j  τ . Since τ  v, we obtain the partial order v − j  τ  v. This implies that [τ, v] ⊆ [v − j, v]. Since [v − j, v] is an elementary cube in K |σ , then its face [τ, v] must also be an elementary cube in K |σ . Setting γ = [τ, v] and observing τ = τ ⊆ γ ⊆ σ , we observe that γ is not an elementary cube in K by Eq. (2). This gives us a contradiction and so j ∈ / I. (v) ⊆ J \ I . Therefore, lk− K |σ Finally, we prove the backward inclusion of Statement 2. Let j ∈ J \ I . Then, by Statement 1, j ∈ lk− K|σ (v) and either τ ≺ v−j or τ is not comparable to v−j under . Thus, by Eq. (2), [v − j, v] is an elementary cube of K |σ . Thus, by Definition 2.2, − we have that j ∈ lk−  K |σ (v). Hence, J \ I ⊆ lkK |σ (v), and so Statement 2 holds.  Using Lemma 3.3, we see why τ cannot be the vertex min(σ ) when performing an LPDC. If τ = min(σ ), then n lk− K |σ (v) = {j ∈ {0, 1} \ {0} | min(σ )  v − j}

\ {j ∈ {0, 1}n \ {0} | v − j  min(σ )} = {j ∈ {0, 1}n \ {0} | min(σ )  v − j and v − j " min(σ )} = {j ∈ {0, 1}n \ {0} | min(σ ) ≺ v − j} = {j ∈ {0, 1}n \ {0} | j ≺ v − min(σ )}.

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Fig. 5 Removing the minimum vertex of a cube. Consider the directed Euclidean cubical complex in (a), which as a subset of R3 is a single closed three-cube; call this three-cube σ . Letting τ = min(σ ), we observe that the past link of v = max(σ ) is contractible before the (τ, σ )-collapse and is homeomorphic to S1 after the collapse. Thus, the past links before and after the collapse are not homotopy equivalent, and so this collapse is not an LPDC. (a) Initial complex. (b) After collapse n If v is the maximum vertex of σ , then we obtain lk− K |σ (v) = {0, 1} \ {0, v − min(σ )}. This computation gives us the following corollary, which we illustrate in Fig. 5 when K is a single closed three-cube.

Corollary 3.4 (Caution for a (min(σ ), σ )-Collapse) Let (K, K) be a directed Euclidean cubical complex in Rn . Let σ ∈ K, τ = min(σ ), and v ∈ verts(σ ). If τ is a free face and K is the (τ, σ )-collapse, then the past link of v in K |σ is: {j ∈ {0, 1}n \ {0} | j ≺ v − min(σ )} In particular, if v = max(σ ) and k = dim(σ ), then the past link is the complete complex on k elements before the collapse, and, after the collapse, it is homeomorphic to Sk−2 . Thus, (τ, σ ) is not an LPDC pair. The following lemma shows under which condition a directed Euclidean cubical collapse induces a simplicial collapse in the past link. Lemma 3.5 (Vertex Collapses that Induce Simplicial Collapse of Past Links) Let (K, K) be a directed Euclidean cubical complex in Rn . Let σ ∈ K and τ, v ∈ verts(σ ) such that τ  v and τ = min(σ ). If τ is a free face of σ and K is − the (τ, σ )-collapse, then lk− K (v) is the (v − τ )-collapse of lkK (v) in the simplicial setting. Proof Consider Kv . Since τ, v ∈ verts(σ ) and σ is maximal in K, we know [min(σ ), v] and [τ, v] are elementary cubes in Kv . Since τ is a free face of σ , we further know that [min(σ ), v] is the only maximal proper coface of [τ, v] in Kv . By definition of past link (Definition 2.2), we then have that v − min(σ ) and v − τ are simplices in lk− Kv (v), and v − min(σ ) is the only maximal proper − − coface of v − τ in lkKv (v). Hence, v − τ is free in lk− Kv (v). Moreover, lkK (v) is v

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Fig. 6 Past link of an “uncomparable” vertex before and after a collapse. Consider the directed Euclidean cubical complex shown, comprising a single three-cube σ and all of its faces. Let τ = [x, y]. Since v and max(τ ) = y are not comparable, by Lemma 3.6, the past link of v is the same before and after the collapse. Indeed, we see that this is the case for this example. The past link of v is the complete complex on two vertices, both before and after. (a) Initial complex. (b) After collapse

the (v−τ )-collapse of lk− Kv (v). One can see this by using Statement 2 of Lemma 3.3 (v) can be characterized as the (v − τ )-collapse of lk− by which lk− Kv (v). K v

− By Statement 3 of Lemma 2.3, we know that lk− K (v) = lkKv (v) and −   that lk− K (v) = lkK (v), which concludes this proof. v

Next, we prove two lemmas concerning relationships of the past link of a vertex in the original directed Euclidean cubical complex and in the collapsed directed Euclidean cubical complex. These relationships depend on where v is located with respect to τ . In the first lemma, we consider the case where min(τ )  v, and we present a sufficient condition for past links in K and the (τ, σ )-collapse to be equal. See Fig. 6 for an example that illustrates the result of this lemma. Lemma 3.6 (Condition for Past Links in K and K to be Equal) Let (K, K) be a directed Euclidean cubical complex in Rn . Let τ, σ ∈ K such that τ is a face of σ . If τ is a free face of σ and K is the (τ, σ )-collapse, then, for all v ∈ verts(K) such − that max(τ )  v, we have lk− K (v) = lkK (v). − Proof By Statement 2 of Lemma 2.3, we have lk− K (v) ⊆ lkK (v). Thus, we only − − − need to show lkK (v) ⊆ lkK (v). Suppose j ∈ lkK (v). By the definition of the past link (see Definition 2.2), we know that [v − j, v] is an elementary cube in K. By assumption, max(τ )  v. Thus, by Eq. (2), [v − j, v] is not removed from K and thus is an elementary cube in K . Thus, j ∈ lk−   K (v).

In the following lemma, we consider the case where max(τ )  v, and we present a sufficient condition for past links in the (τ, σ )-collapse and the (min(τ ), σ )collapse to be equal. See Fig. 7 for an example that illustrates this result. Lemma 3.7 (Comparing Past Links in a General Collapse with Past Links in a Vertex Collapse) Let (K, K) be a directed Euclidean cubical complex in Rn such

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Fig. 7 Two collapses with same past links. For example, in the directed Euclidean cubical complex K shown in (a), let σ be the three-cube, and let τ = [x, y]. We look at the past link of the vertex v. In the original directed Euclidean cubical complex, the past link of v is the complete complex on three vertices. By Lemma 3.7, the past link of v is the same in both the (τ, σ )-collapse and the (x, σ )-collapse since max(τ ) = y  v. By Lemma 3.8, we also know that the past links of v in K and the (x, σ )-collapse are homotopy equivalent. Indeed, we see that this is the case. (a) Original complex. (b) The (τ, σ )-collapse. (c) The (x, σ )-collapse

that there exists cubes τ, σ ∈ K with min(τ ) a free face of σ . Let K be the (τ, σ ) be the (min(τ ), σ )-collapse. If v ∈ verts(K ) and max(τ )  v, collapse and let K  and lk− (v) = lk− (v). then v ∈ verts(K)  K K  If τ is a zero-cube (and hence in verts(K)), Proof We first show v ∈ verts(K).  which means that v ∈ verts(K).  On the other hand, if τ is not a zerothen K = K, cube, then we have min(τ ) ≺ max(τ )  v. In particular, min(τ ) = v. And so, by  as a (min(τ ), σ )-collapse and since v ∈ K, we conclude that v ∈ K.  definition of K − Next, we show lk− (v) = lk (v). By Statement 2 of Lemma 2.3, we  K K − − have lk− (v) ⊆ lk (v). Thus, what remains to be proven is lk−   (v). K K (v) ⊆ lkK K − Let j ∈ lkK (v). By definition of the past link (Definition 2.2), we know that [v − j, v] ⊆ K . Consider two cases: v − j  min(τ ) and v − j  min(τ ). Case 1

Case 2

(v − j  min(τ )): Since v − j  min(τ )  max(τ )  v, we know that τ ⊆ [v − j, v]. Thus, by Eq. (2), we have [v − j, v]  K , which is a contradiction. So, Case 1 cannot happen. (v − j  min(τ )): If v − j  min(τ ), then, by the definition of  a (min(τ ), σ )-collapse in Definition 3.1, we know that [v − j, v] ⊆ K − and thus j ∈ lkK(v).

− Hence, lk−  (v). Since we have both subset inclusions, we conK (v) ⊆ lkK − − clude lkK (v) = lkK(v).  

In general, the minimal vertex of τ is not free in K and hence, there is no vertex collapse. In the main theorem, the previous lemma is applied to a subcomplex of K; specifically, it is applied to the restriction to the unit cube corresponding to σ , where all vertices, including min(τ ) are then free. The results carry over to K. The next result states that vertex collapses result in homotopy equivalent past links as long as we are not collapsing the minimum vertex of the directed Euclidean cubical complex.

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Lemma 3.8 (Past Links in a Vertex Collapse) Let (K, K) be a directed Euclidean cubical complex in Rn . Let σ ∈ K and let τ ∈ verts(σ ) such that τ = min(σ ). Let v ∈ verts(K) with v = τ . If τ is a free face of σ and K is the (τ, σ )-collapse, − then lk− K (v) % lkK (v). Proof We consider three cases: Case 1 Case 2 Case 3

(v ∈ / verts(σ )): By definition of past link (Definition 2.2), if v ∈ / verts(σ ), − then the past links lk− (v) and lk (v) are equal. K K (τ  v): By Lemma 3.6, if τ = max(τ )  v, again we have equality of − the past links lk− K (v) and lkK (v). (v ∈ verts(σ ) and τ  v): By Lemma 3.5, we know that lk− K (v) is (v) in the simplicial setting. Since simplicial the v − τ -collapse of lk− K collapses preserve the homotopy type (see e.g., [14, Proposition 6.14]), − we conclude lk−   K (v) % lkK (v).

We give an example of Lemma 3.8 in Fig. 7 by showing how the LPDC induces a simplicial collapse on past links. Lastly, we are ready to prove the main result. Theorem 3.9 (Main Theorem) Let (K, K) be a directed Euclidean cubical complex in Rn such that there exist cubes τ, σ ∈ K with τ a free face of σ . Then, (τ, σ ) is an LPDC pair if and only if min(σ ) ∈ / verts(τ ). Proof Let v = max(σ ) and k = dim(σ ). Let (K , K ) be the (τ, σ )-collapse of K. Let (L, L) be the cubical complex such that L = K|σ . Since σ ∈ K, we know L = σ (i.e., L is a unit cube). Since L is a single unit cube and σ is a maximal elementary cube, all proper faces of σ , including τ and min(τ ), are free faces in L.  be the (min(τ ), σ ) L) Thus, let (L , L ) be the (τ, σ )-collapse of L, and let (L, collapse of L. We first prove the forward direction by contrapositive (if min(σ ) ∈ verts(τ ), then (τ, σ ) is not an LPDC pair). Assume min(σ ) ∈ verts(τ ). By Corollary 3.4, k−1 and lk− (v) is homeomorphic to Sk−2 . we obtain lk−  L (v) is homeomorphic to B L Since min(σ ) ∈ verts(τ ), we know that min(σ ) = min(τ ). Since τ is a face of σ , we know max(τ )  max(σ ) = v. Since min(σ ) = min(τ ) ∈ verts(τ ) and since τ is a proper face of σ , we know that v = max(τ ). Thus, v ∈ verts(L ). Applying − Lemma 3.7, we obtain lk−  (v). Putting this all together, we have: L (v) = lkL − k−1 lk− % Sk−2 % lk−  (v) = lkL (v), L (v) % B L − and so lk− L (v) % lkL (v). Since no faces of σ are in K \ L, the past link of v remains the same outside of L − in both K and K . Thus, lk− K (v) % lkK (v) and so we conclude that (τ, σ ) is not an LPDC pair, as was to be shown. Next, we show the backwards direction. Suppose min(σ ) ∈ verts(τ ). Let v ∈ verts(K ), and consider two cases: max(τ )  v and max(τ )  v.

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R. Belton et al. − (max(τ )  v): By Lemma 3.6, we have lk− K (v) = lkK (v). − − Hence, lkK (v) % lkK (v). Since v was arbitrarily chosen, we conclude that (τ, σ ) is an LPDC pair. − (max(τ )  v): By Lemma 3.7, we have that lk−  (v). L (v) = lkL Since min(σ ) ∈ verts(τ ), we know that min(τ ) = min(σ ). Applying − Lemma 3.8, we obtain lk−  (v). Again, since no faces of σ are L (v) % lkL removed from K and K to obtain L and L , the past link of v remains the − same outside of L in both K and K . Thus, lk− K (v) % lkK (v). Since v was arbitrarily chosen, we conclude that (τ, σ ) is an LPDC pair.  

4 Preservation of Spaces of Dipaths In [1], we proved several results on the relationships between past links and spaces of dipaths. One result, Theorem 2.4, states that for a directed Euclidean cubical complex with a minimum vertex, if all past links are contractible, then all spaces of dipaths starting at that minimum vertex are also contractible. If we start with a directed Euclidean cubical complex with a minimum vertex that has all contractible past links, then all spaces of dipaths from the minimum vertex are contractible by this theorem. We explain how those relationships extend to the LPDC setting in this section. Applying an LPDC preserves the homotopy type of past links by definition. Hence, applying the theorem again, we see that any LPDC also has contractible dipath spaces from the minimum vertex. Notice that the minimum vertex is not removed in an LPDC, since it is a vertex and minimal in all cubes containing it (including the maximal cube). We give an example of this in Example 4.1. Example 4.1 (3 × 3 Filled Grid) Let K be the 3 × 3 filled grid. For all v ∈ verts(K), lk− K (v) is contractible. By Theorem 2.4, this implies that all spaces of dipaths starting at 0 are contractible. Applying an LPDC such as the edge [(1, 3), (2, 3)] results in contractible past links in K and so all spaces of dipaths in K are also contractible. See Fig. 8. We can generalize this example to any k d filled grid where k, d ∈ N. An analogous result holds for connectedness (Theorem 2.5). If we start with a directed Euclidean cubical complex such that all past links are connected, then all dipath spaces are connected. Any LPDC results in a directed Euclidean cubical complex that also has connected dipath spaces. See Example 4.2. Example 4.2 (Outer Cubes of the 5 × 5 × 5 Grid) Let K = [0, 5]3 \ [1, 4]3 , which, as an undirected complex, is homeomorphic to a thickened two-sphere. For all v ∈ verts(K), lk− K (v) is connected. By Theorem 2.5, this implies that for − → all v ∈ verts(K), the space of dipaths P v0 (K) is connected. Applying an LPDC such

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Fig. 8 (a) The 3×3 filled grid has contractible past links and dipath spaces. The pair comprising of the purple edge [(1, 3), (2, 3)] and the blue square [(1, 2), (2, 3)] is an LPDC pair. (b) The result of performing the LPDC. All past links are contractible and so all dipath spaces are also contractible

as with the vertex (5, 0, 0) in the cube [(4, 0, 0), (5, 1, 1)] results in connected past − → links in K and so all spaces of dipaths P v0 (K ) are connected. We can generalize this example to any k d grid where d ≥ 3 and the inner cubes of dimension d are removed. Both Theorem 2.4 and Theorem 2.5 have assumptions on the topology of past links and results on the topology of spaces of dipaths from the minimum vertex. We may ask if the converse statements are true. Does knowing the topology of spaces of dipaths from the minimum vertex tell us anything about the topology of past links? The converse to Theorem 2.4 holds. To prove this, we first need a lemma whose proof appears in [21]. Lemma 4.3 (Homotopy Equivalence [21, Prop. 5.3]) Let (K, K) be a directed − →q−j Euclidean cubical complex in Rn . Let p, q ∈ Zn . If P p (K) is contractible for − →q − all j ∈ lk− K (q), then P p (K) % lkKp (q). Thus, we obtain: Theorem 4.4 (Contractability) Let (K, K) be a directed Euclidean cubical complex in Rn that has a minimum vertex w. The following two statements are equivalent: − → 1. For all v ∈ verts(K), the space of dipaths P vw (K) is contractible. 2. For all v ∈ verts(K), the past link lk− K (v) is contractible. Proof By Theorem 2.4, we obtain Statement 2 implies Statement 1. Next, we show that Statement 1 implies Statement 2. Let v ∈ verts(K). For all j ∈ lk− K (v), the cube [v−j, v] is a subset of K, which means that v−j ∈ verts(K). − →v−j Thus, by assumption, all dipath spaces P w (K) are contractible. By Lemma 4.3, − → − we know that P vw (K) % lk− Kw (v) = lkK (v). Again, since v ∈ verts(K), the dipath − →   space P vw (K) is contractible. Therefore, lk− K (v) is contractible.

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As a consequence of this theorem, we know that if we start with a directed Euclidean cubical complex with contractible dipath spaces starting at the minimum vertex, then any LPDC also result in a directed Euclidean cubical complex with all contractible dipath spaces starting at the minimum vertex, and vice versa. Corollary 4.5 (Preserving Directed Path Space Contractability) Let (K, K) be a directed Euclidean cubical complex in Rn that has a minimum vertex w. Let τ, σ ∈ K such that τ is a face of σ . If τ is a free face of σ , let (K , K ) be the (τ, σ )− → collapse. If K is an LPDC of K, then the spaces of dipaths P vw (K) are contractible − → for all v ∈ verts(K) if and only if the spaces of dipaths P kw (K ) are contractible for all k ∈ verts(K ). Proof We start with the forwards direction by assuming that the spaces of − → dipaths P vw (K) are contractible for all v ∈ verts(K). Theorem 4.4 tells us that − all past links lk− K (v) are contractible for all v ∈ verts(K). This implies that lkK (k) is contractible for all k ∈ verts(K ) because K is an LPDC of K. Applying − → Theorem 4.4 again, we see that all spaces of dipaths P kw (K ) are contractible for all k ∈ verts(K ). Next we prove the backwards direction by assuming that the spaces of − → dipaths P kw (K ) are contractible for all k ∈ verts(K ). Let v ∈ verts(K). Either v ∈ verts(K ) or v ∈ / verts(K ). Case 1 Case 2

(v ∈ verts(K )): By Theorem 4.4, we know that lk− K (v) is contractible. (v) is also contractible. Since K is an LPDC of K, then lk− K (v ∈ / verts(K )): If v ∈ / verts(K), then τ is a vertex and v = τ . Observe that lk− σ (τ ) is contractible since σ is an elementary cube and τ does not − contain min(σ ). Furthermore, notice that lk− K (τ ) = lkσ (τ ) because τ is a − free face of σ . Hence, lkK (τ ) is contractible.

Therefore lk− K (v) is contractible for all v ∈ verts(K). Applying Theorem 4.4, we − → get that P vw (K) is contractible for all v ∈ verts(K).   Using Theorem 2.5 and the partial converse to the connectedness theorem [1, Theorem 3], we get that any LPDC of a directed Euclidean cubical complex with connected dipath spaces and reachable vertices results in a directed Euclidean cubical complex with connected dipath spaces. Corollary 4.6 (Condition for LPDCs to Preserve Connectedness of All Directed Path Spaces) Let (K, K) be a directed Euclidean cubical complex in Rn that has a minimum vertex w. Let (L, L) = reach(K, w). Let (τ, σ ) be an LPDC pair in L, − → and let L be the (τ, σ )-collapse. The spaces of dipaths in P kw (L) are connected − → for all v ∈ verts(L) if and only if the spaces of dipaths P vw (L ) are connected for all v ∈ verts(L ). We note that reachability is a necessary condition. Below we give an example of a directed Euclidean cubical complex K that has all connected dipath spaces but an LPDC yields a directed Euclidean cubical complex with a disconnected path space.

185

>

> > > >

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> > >

>

>

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>

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Fig. 9 The bowling ball before and after the collapse described in Example 4.7. − → Observe P 0(5,5,5) (K) has one connected component. Additionally, σ = [(4, 1, 1), (5, 2, 2)] (highlighted in blue) and τ = [(5, 1, 1), (5, 2, 2)] (highlighted in purple) is an LPDC pair. After − →(5,5,5) collapsing (τ, σ ), P 0 (K) changes from having one connected component to three connected components. The three connected components are represented by the three dipaths. (a) Original complex. (b) After collapse

Example 4.7 (Bowling Ball) Let K be the complex resulting from taking the boundary of the 5 × 5 × 5 grid union the closed cubes [(4, 1, 1), (5, 2, 2)] and [(4, 3, 3), (5, 4, 4)], then removing the open cubes [(4, 3, 3), (5, 4, 4)] and [(5, 3, 3), (5, 4, 4)]. See Fig. 9a. Notice that some vertices of K are unreachable, for example, vertex (4, 1, 1). Furthermore, all past links of vertices in K are connected and so all dipath spaces starting at 0 are also connected. After performing an LPDC with τ = [(5, 1, 1), (5, 2, 2)] and σ = [(4, 1, 1), (5, 2, 2)], the dipath space between 0 and (5, 5, 5) changes from having one connected component to three connected components, as shown in the figure. This example shows that the reachability condition in Corollary 4.6 is necessary for preserving connnectedness in LPDCs. LPDCs can also preserve dihomotopy classes of dipaths starting at the minimum vertex of many directed Euclidean cubical complexes that have disconnected past links. Recall the Swiss flag as discussed in Fig. 4. The Swiss flag has disconnected past links at (3, 4) and (4, 3), yet there exists a sequence of LPDCs that results in a directed Euclidean cubical complex that highlights the two dihomotopy classes of dipaths between 0 and (5, 5). Example 4.8 gives another similar situation. Example 4.8 (Window) Let K be the 5 × 5 grid with the following two-cube interiors removed: [(1, 1), (2, 2)], [(3, 1), (4, 2)], [(1, 3), (2, 4)], [(3, 3), (4, 4)]. See Fig. 10a. K has disconnected past links at the vertices (2, 2), (4, 2), (2, 4), (4, 4) − → so K does not satisfy Corollary 4.5 or Corollary 4.6. Observe that P (K)0(5,5) has six connected components. We can perform a sequence of LPDCs that preserves the dihomotopy classes of dipaths between 0 and (5, 5) at each step. First, we apply vertex LPDCs to remove the two-cubes along the border. Then we can apply four edge LPDCs and one vertex LPDC to get a graph of vertices and edges. This graph − → (5,5) more clearly illustrates the six dihomotopy classes of dipaths in P (K)0 .

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(5,5)

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Fig. 10 Link-preserving DCs of the window. A sequence of LPDCs is presented from (a)–(e). The directed Euclidean cubical complex in (b) comes from performing several vertex LPDCs to remove the two-cubes along the border of K. In (b)–(d), the LPDC pairs (τ, σ ) are highlighted in purple and blue respectively. The result of the sequence of LPDCs is a graph of vertices and edges that more clearly illustrates the dihomotopy classes of dipaths in the dipath space

In summary, LPDCs preserve the connectedness and/or contractability of dipath spaces starting at the minimum vertex as long as K has all reachable vertices and all dipath spaces starting at the minimum vertex in K connected and/or contractible to begin with. If K does not have these properties, the first step could be to remove all unreachable vertices and cubes before collapsing. In the next section, it becomes clear that this will not suffice, if the dipath spaces are not all connected or contractible.

5 Discussion LPDCs preserve spaces of dipaths in many examples (see Sect. 4), in particular, if they are all trivial in the sense of either all connected or all contractible and the directed Euclidean cubical complex is reachable from the minimum vertex. However, LPDCs do not always preserve spaces of dipaths. We discuss some of those instances here. One limitation of LPDCs is that the number of components may increase after an LPDC as we saw in Example 4.7 or, as we see in Example 5.1, they may decrease.

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Fig. 11 The LPDC for the window that changes dipath space. The LPDC of the edge [(2, 4), (3, 4)] changes the dipath space between 0 to (5, 5) from having six connected components to five connected components. (a) Initial complex. (b) After collapse

Example 5.1 (A Sequence of LPDCs of the Window that Decreases the Number of Connected Components of the Dipath Space) Consider K as given in Example 4.8. After applying vertex LPDCs that remove the two-cubes on the border of K, − → we can apply an LPDC to the edge [(2, 4), (3, 4)]. Now P (K )0(5,5) has five − → (5,5) connected components; whereas, the dipath space P (K)0 has six connected components. See Fig. 11. This example shows that there are both “good” and “bad” ways to apply a sequence of LPDCs to a directed Euclidean cubical complex. As illustrated in Example 4.8, there exists a sequence of LPDCs that preserves the six − → (5,5) connected components in P (K)0 . However, if we perform a sequence of LPDCs that removes the edge [(2, 4), (3, 4)] as in this example, then we get a directed Euclidean cubical complex that does not preserve the dihomotopy classes of dipaths − → (5,5) in P (K)0 . Example 5.1 illustrates the need to investigate other properties if we want to preserve dipath spaces when performing an LPDC. In Example 4.7, the problem was the existence of unreachable vertices. In Example 5.1, the vertex (2, 4) is a deadlock after the LPDC: only trivial dipaths initiate from there; whereas, before collapse, that was not the case. This seems to suggest that the introduction of new deadlocks should not be allowed; in practice, this would require an extra—but computationally easy—check on vertices of σ . In the non-directed setting, if K is obtained from K by collapsing a collapsing pair (τ, σ ), then not only is the inclusion of K in K a homotopy equivalence. K is a deformation retract of K. The following example removes any hope of such a result in the directed setting: Example 5.2 (LDPC of the Four-Cube with No Directed Retraction to the Collapsed Complex) Let (I 4 , I 4 ) be the standard unit four-cube. Let τ be the vertex (1, 1, 0, 0), and σ be the cube [0, 1]. Since τ is free and not the minimum vertex of σ , the pair (τ, σ ) is an LPDC pair. Thus, let (K , K ) be the collapsed complex. Next, we show that there is no directed retration, i.e., no directed map from I 4 to K that is the identity on K .

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Suppose, for a contradiction, that f : I 4 → K is such a directed retraction. Let p1 = (0, 1, 0, 0), p2 = (1, 0, 0, 0), q1 = (1, 1, 1, 0), and q2 = (1, 1, 0, 1). By the product order on R4 , we have p1 , p2  τ and τ  q1 , q2 . Since the points p1 , p2 , q1 , and q2 are vertices of I 4 and are not equal to τ , we also know that p1 , p2 , q1 , and q2 are points in K . Since f is a directed retraction, we have that p1 = f (p1 )  f (τ ) and that p2 = f (p2 )  f (τ ). Similarly, we obtain that f (τ )  f (q1 ) = q1 and that f (τ )  f (q2 ) = q2 . Let x1 , x2 , x3 , x4 ∈ I such that f (τ ) = (x1 , x2 , x3 , x4 ). Then, p1  f (τ ) ⇒ x2 ≥ 1 and hence x2 = 1, p2  f (τ ) ⇒ x1 ≥ 1 and hence x1 = 1, f (τ )  q1 ⇒ x4 ≤ 0 and hence x4 = 0, f (τ )  q2 ⇒ x3 ≤ 0 and hence x3 = 0. Thus, f (τ ) = (1, 1, 0, 0) = τ , which is not in K and hence a contradiction. In fact, this argument extends to (I k , I k ) for k ≥ 4. As further evidence that such a (τ, σ )-collapse does not preserve the directed topology, consider the spaces of dipaths in (I 4 , I 4 ) and (K , K ). We would need dipaths in the original space to map to dipaths in the collapsed space. However, notice that the dipath from p1 to q1 through τ cannot be mapped to a dipath in (K , K ). We observe that vertex LPDCs appear to not introduce the problems of unreachability and deadlocks. These observations lead us to suspect that studying unreachability, deadlocks, and vertex LPDCs can help us better understand when LPDCs preserve and do not preserve dipath spaces between the minimum and a given vertex. We leave this as future work. In summary, we provide an easy criterion for determining when we have an LPDC pair, as well as discuss various settings for when LPDCs preserve spaces of dipaths. Fully understanding when LPDCs preserve spaces of dipaths between two given vertices is a step towards developing algorithms that compress directed Euclidean cubical complexes and preserve directed topology. Acknowledgments This research is a product of one of the working groups at the Women in Topology (WIT) workshop at MSRI in November 2017. This workshop was organized in partnership with MSRI and the Clay Mathematics Institute, and was partially supported by an AWM ADVANCE grant (NSF-HRD 1500481). This material is based upon work supported by the US National Science Foundation under grant No. DGE 1649608 (Belton) and DMS 1664858 (Fasy), as well as the Swiss National Science Foundation under grant No. 200021-172636 (Ebli). We thank the Computational Topology and Geometry (CompTaG) group at Montana State University for giving helpful feedback on drafts of this work.

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References 1. Belton, R., Brooks, R., Ebli, S., Fajstrup, L., Fasy, B.T., Ray, C., Sanderson, N., Vidaurre, E.: Towards directed collapsibility. In: Acu, B., Danielli, D., Lewicka, M., Pati, A.N., Ramanathapuram Vancheeswara, S., Teboh-Ewungkem, M.I. (eds.) Advances in Mathematical Sciences, Association for Women in Mathematics Series. Springer, New York (2020) 2. Brown, R., Higgins, P.J.: On the algebra of cubes. J. Pure Appl. Algebra 21(3), 233–60 (1981) 3. Cohen, M.M.: A Course in Simple-Homotopy Theory, vol. 10. Springer Science & Business Media, Berlin (2012) 4. Dijkstra, E.W.: Two starvation-free solutions of a general exclusion problem (1977). Manuscript EWD625, from the archives of UT Austin 5. Fajstrup, L., Goubault, E., Raussen, M.: Algebraic topology and concurrency. Theoret. Comput. Sci. 241–271 (2006) 6. Fajstrup, L., Goubault, E., Haucourt, E., Mimram, S., Raussen, M.: Directed Algebraic Topology and Concurrency, 1st edn. Springer, Berlin (2016) 7. Ghrist, R.: Configuration spaces, braids, and robotics. In: Braids: Introductory Lectures on Braids, Configurations and Their Applications, pp. 263–304. World Scientific, Singapore (2010) 8. Goubault, E.: The Geometry of Concurrency. PhD thesis, École Normale Supérieure, Paris (1995) 9. Grandis, M.: Directed homotopy theory, I: The fundamental category. Cahiers de Topologie et Géométrie Différentielle Catégoriques 44(4), 281–316 (2003) 10. Grandis, M.: Directed Algebraic Topology: Models of Non-Reversible Worlds, vol. 13. New Mathematical Monographs. Cambridge University Press, Cambridge (2009) 11. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002) 12. Hoare, C.A.R.: Communicating sequential processes. Commun. ACM 21(8), 666–677 (1978) 13. Jardine, J.F.: Cubical homotopy theory: A beginning (2002). Technical report 14. Kozlov, D.: Combinatorial Algebraic Topology, vol. 21. Algorithms and Computation in Mathematics. Springer, Berlin (2008) 15. Lachaud, J.-O.: Cubical complex (2018). Software, https://projet.liris.cnrs.fr/dgtal/doc/nightly/ moduleCubicalComplex.html. Part of the Toplogy package of the DGtal library 16. Raussen, M., Ziemia´nski, K.: Homology of spaces of directed paths on Euclidean cubical complexes. J. Homotopy Related Struct. 9(1), 67–84 (2014) 17. Serre, J.-P.: Homologie singulière des espaces fibrés. Ann. Math. 54(3), 425–505 (1951) 18. Whitehead, J.H.C.: Simplicial spaces, nuclei and m-groups. Proc. Lond. Math. Soc. s2-45(1), 243–327 (1938) 19. Whitehead, J.H.C.: Simple homotopy types. Am. J. Math. 72(1), 1–57 (1950) 20. Wisniewski, R.: Towards modelling of hybrid systems. In: Proceedings of the 45th IEEE Conference on Decision and Control, pp. 911–916. IEEE, Piscataway (2006) 21. Ziemia´nski, K.: On execution spaces of PV-programs. Theoret. Comput. Sci. 619, 87–98 (2016)

Lions and Contamination, Triangular Grids, and Cheeger Constants Henry Adams, Leah Gibson, and Jack Pfaffinger

Abstract Suppose each vertex of a graph is originally occupied by contamination, except for those vertices occupied by lions. As the lions wander on the graph, they clear the contamination from each vertex they visit. However, the contamination simultaneously spreads to any adjacent vertex not occupied by a lion. How many lions are required in order to clear the graph of contamination? We give a lower bound on the number of lions needed in terms of the Cheeger constant of the graph. Furthermore, the lion and contamination problem has been studied in detail on square grid graphs by Brass et al. and Berger et al., and we extend this analysis to the setting of triangular grid graphs.

1 Introduction In the “lions and contamination” pursuit-evasion problem [6, 9, 16], lions are tasked with clearing a square grid graph consisting of vertices and edges. At the start of the problem, all vertices occupied by lions are considered cleared of contamination, and the rest of the vertices are contaminated. In each time step, the lions move along the edges of the grid, and each new vertex they occupy becomes cleared. However, the contamination can also travel along the edges of the grid not blocked by a lion and re-contaminate previously cleared vertices. How many lions are needed to clear the grid, independent of the starting position of the lions? Certainly n lions can clear an n × n grid graph by sweeping from one side to the other. One might conjecture that n lions are required to clear an n × n grid graph. However, in general it is not yet known whether n − 1 lions suffice or not. As a lower bound, the paper [9] proves that at least , n2 - + 1 lions are required to clear an n × n grid graph. The details of the discretization certainly matter, in the following

H. Adams · L. Gibson () · J. Pfaffinger Colorado State University, Fort Collins, CO, USA e-mail: [email protected]; [email protected]; [email protected] © The Author(s) and the Association for Women in Mathematics 2022 E. Gasparovic et al. (eds.), Research in Computational Topology 2, Association for Women in Mathematics Series 30, https://doi.org/10.1007/978-3-030-95519-9_8

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sense. For a n × n × n grid graph, one might expect that n2 lions are required, but [6] shows that when n = 3, only 8 = n2 − 1 lions suffice to clear a 3 × 3 × 3 grid. We consider the case of planar triangular grid graphs, under various models of lion motion. Given a strip discretized by a triangular grid graph, n lions can clear a strip of height n when they are allowed to move one-at-a-time (this is allowed in the standard model of lion motion). However, we conjecture that n lions do not suffice when all lions must move simultaneously. In the setting of simultaneous motion (which we refer to as “caffeinated lions,” as the lions never take a break), we show that , 3n 2 - caffeinated lions suffice to clear a strip of height n. Furthermore, via a comparison with [6, 9], we show that , n2 - lions are insufficient to clear a triangulated rhombus, in which each side of the rhombus has length n. Lastly, for an equilateral triangle discretized into smaller triangles, with n vertices per side, we conjecture n that √ lions are not sufficient to clear the graph. 2 2 Furthermore, in the setting of an arbitrary graph G, we give a lower bound on the number of lions needed to clear the graph in terms of the Cheeger constant of the graph. The Cheeger constant, roughly speaking, is a measure of how hard it is to disconnect the graph into two pieces of approximately equal size by cutting edges [12]. The use of the Cheeger constant in graph theory is inspired by its successful applications in Riemannian geometry [10]. Our bound on the number of lions in terms of a graph’s Cheeger constant is quite general (it holds for any graph), and therefore we do not expect it to be sharp for any particular graph. We give background definitions and notation in Sect. 3, study triangular grid graphs in Sect. 4, and explain the connection to the Cheeger constant in Sect. 5. We ask many open questions in Sect. 6 that we hope will inspire future work.

2 Related Work The lion and contamination problem considered in this paper is only one of many pursuit-evasion problems that occur in graphs. See [30] for an early treatment of an evasion problem on graphs, and see [17] for a bibliography of related problems and papers. The graph-clear problem explored by [21] is a model for surveillance tasks and can be used to help determine the number of robots needed to patrol a large enclosed area. In this problem, the contamination that must be cleared lives on the edges of the graph, whereas in the lion and contamination problem we consider the contamination lives on the vertices. Pursuit-evasion problems have applications to air traffic control [5], robot motion planning [24, 25], defense [20], collision avoidance [18], and tracking [19]. Perhaps the most developed graph-based pursuit-evasion challenge is the problem of Cops and Robbers. This is a particular form of graph evasion problem which involves a set of cops moving along the vertices of a graph that attempt to capture a single robber also moving on the vertices of a graph. In this version the cops and robbers alternate moving, rather than both moving at the same time. The robber is

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caught if a cop is able to move onto the vertex the robber is occupying. For a vertex v, define N[v] to be the set containing v and all vertices adjacent to v. A vertex v is called a corner if there exists a vertex u such that N[v] ⊆ N[u]. A single cop is able to catch the robber on a finite graph if and only if some sequence of deleting corners results in a single vertex remaining. This condition is known as the graph being dismantlable. For more information see [8]. There are many variants of pursuit-evasion problems that take place in Euclidean space instead of in graphs; see [13] for a survey. The most famous such problem may be the lion and man problem proposed by Rado in 1925 [26], in which a single man is chased by a single lion of the same speed. The lion and man problem has been studied both with continuous time and with discrete time [4, 32]. In many pursuitevasion problems, there are numerous sensors or intruders, for example as in [11, 27, 28]. Kalman filters can be used for efficient target tracking [22, 29, 33], and neural networks can obtain dynamic coverage while learning previously unknown domains [31]. A related class of evasion problems take place in mobile sensor networks, as studied from the topological perspective in [1, 14, 15]. Here both space and time are continuous, and ball-shaped sensors wander in a bounded domain. The perspective taken is that of minimal sensing, in that the sensors do not know their locations but instead only measure connectivity information. Furthermore, the motion of the sensors is arbitrary—perhaps the sensors are floating in the ocean or blown in the air by wind. There is no speed bound for either the sensors or the intruders. One would like to detect when the sensors have necessarily captured all possible intruders, but without using location information. A slightly different perspective is taken in [3], namely, what random models of sensor motions lead to faster mobile coverage? The relation to the lion and contamination problem in this paper can be partially viewed as a passage from continuous to discrete. If one approximates a continuous plane instead by a square or triangular grid graph, then the number of lions needed to clear that graph provides a discrete approximate analogue of the number of sensors that might be needed in a continuous model on the plane. In contrast with the aforementioned models, in the lion and contamination pursuit evasion problem that we will consider (defined rigorously in the following section), both space and time are discrete, and the pursuers and evaders move simultaneously with bounded speed.

3 Notation and Definitions We begin by providing notation and definitions for various models of lion motion, for how the contamination spreads, and for various types of triangular grid graphs. For additional background information, we refer the reader to [6, 9].

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3.1 Graphs In this paper we consider only finite simple graphs. A finite simple graph G = (V , E) contains a finite set of vertices V . The set of (undirected) edges E is a collection of 2-subsets from V . We may also use the notation V (G) or E(G) to denote the vertex set of G and edge set of G, respectively. If {u, v} is an edge in E, then we say that the vertices u and v are adjacent, and we often denote this edge as uv. Visually, we represent adjacency by drawing a line segment between the vertices u and v. An important concept for this paper will be the boundary of a set of vertices in a graph. Definition 1 Let G be a graph with vertex set V . We define the boundary of a vertex subset S ⊆ V , denoted ∂S, to be the collection of all vertices in S that share an edge with some vertex of V \ S. That is, ∂S = {v ∈ S | uv ∈ E(G) for some u ∈ / S}. Definition 2 A walk π in a graph G is an ordered list (π(1), π(2), . . . , ) of vertices where each π(t) ∈ V (G), and π(t) is adjacent to π(t + 1) for all t. We note that a walk is allowed to retrace its steps—i.e., the vertices visited need not be distinct.

3.2 Lion Motion Each lion occupies a vertex of the graph. In this evasion problem, time is discrete. At each turn, a lion can either stay where it is, or move across an edge to an adjacent vertex. Multiple lions are allowed to occupy the same vertex. The main restrictions on lion motion that we consider are caffeinated lions and polite lions. In the caffeinated model, all lions must move at every time step, and in the polite model, at most one lion can move at once. Definition 3 (Caffeinated Lions) In the caffeinated model of lion motion, every lion must move at each turn. In other words, between turns no lion is allowed to remain in place at its current location. Definition 4 (Polite Lions) In the polite model of lion motion, at most one lion is allowed to move at each turn. All other lions must remain at their current vertex. When we simply say “lions” without specifying, we mean lions that are neither caffeinated nor polite—any lion can move or stay put at any turn.

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3.3 Contamination Motion Every vertex that is not originally occupied by a lion is contaminated. When a lion moves to a new vertex, that new vertex becomes cleared of contamination. A cleared vertex v at time step t becomes recontaminated at the next time step t + 1 if • it is not occupied by a lion at time step t + 1, and • if it is adjacent to a contaminated vertex u at time t, and there is no lion that crosses the edge from v to u between times t and t + 1 (which would block the contamination from moving across this edge). Note that in the same time step that a lion leaves a vertex, it can become recontaminated. We let C(t) denote the set of cleared vertices at time t. We say that the lions have cleared or swept the graph G if at any time t, all of the vertices in the graph are cleared of contamination. Definition 5 (Sweep) A sweep of a graph is the movement of the lions that results in the graph being completely cleared of contamination at time t, i.e. C(t) = V (G). We are interested in finding the fewest number of lions required to clear a graph. We note that if G is a connected graph, and if k lions can sweep G from a certain starting position, then k lions can sweep G from any starting position. However, this is not necessarily the case if the lions are caffeinated—unless G contains an odd cycle (see Sect. 4.2.2).

3.4 Triangular Grid Graphs We begin by bounding the number of lions needed to clear triangular grid graphs, which we define now. Let Tn be the nth triangular number for n ∈ N. That is, Tn = n(n+1) 2 . Definition 6 (Triangular Grid) Let Pn be the planar graph which forms an equilateral triangle of side-length n − 1 (with n vertices on each side), subdivided into a grid of equilateral triangles as drawn in Fig. 1. Note that Pn has

n(n+1) 2

= Tn vertices and

3n(n−1) 2

edges.1

 The number of vertices are the nth triangular number defined by ni=0 i = n(n+1) 2 , since Ti is constructed by taking Ti−1 and adding i vertices. Denote the number of edges in Tn as En . These numbers are determined by the recurrence relation En = En−1 + 3(n − 1). To see this, suppose that Tn−1 is drawn and place n vertices below it. Each of the n − 1 vertices in row n − 1 will have two edges drawn between itself and the vertices in the last row of n vertices. The last row of n vertices will be connected together by another n − 1 edges. So the equivalence relation is established, and it satisfies En = 3n(n−1) for n ≥ 1. 2 1 Proof:

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Fig. 1 The triangular grid graph P5

Fig. 2 Example of R2,3 and R3,3 (from left to right)

Definition 7 (Triangular Lattice) Let Rn,l be the planar graph which forms a parallelogram of height n vertices and length l vertices, subdivided into a grid of equilateral triangles as drawn in Fig. 2.

4 Lions and Contamination on Triangular Grid Graphs We study the number of lions needed to clear triangular grid graphs, for various shapes of graph domain, and under various different models of lion motion.

4.1 Sufficiency of n Lions on a Triangulated Strip Theorem 1 Let n and l be positive integers. Then n lions suffice to clear the grid Rn,l of contamination. Proof There is a specific sweeping formation that we use for the n lions to clear the grid of contamination, but since there is no restriction on lion movement and the lions live on a connected graph, all lions can move into this clearing position without issue. In the graph Rn,l we let the bottom row be labeled row 1, the next row up is row 2, and this continues to row n. We can apply a sweeping method using n lions which move until they are positioned on the leftmost vertices of the grid, i.e. the leftmost diagonal column. We complete the sweep in the following manner as illustrated in Fig. 3: Let the lion on row 1 travel one step to the right. The vertex it previously occupied is cleared and protected from future contamination in the next step. Next,

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Fig. 3 Sweeping formation on R3,3 . The red vertices are contaminated, the green vertices are cleared, and the locations of lions are marked with ‘L’.

Fig. 4 Under caffeinated lion motion, n caffeinated lions cannot sweep Rn,l by moving from left to right, as contamination can travel along the diagonal edges and recontaminate previously cleared vertices. The red vertices are contaminated, the green vertices are cleared, and the locations of caffeinated lions are marked with ‘X’

let the lion in row 2 travel one step to the right. The vertex it previously occupied is cleared and protected from future contamination in the next step. Now, we let the lion in row 3 move one step to the right. Similarly, the previously occupied vertex is cleared and protected. We repeat this for n lions. Once each lion has moved one step to the right, the previous column is cleared and we repeat the process again starting with the lion on row 1. We continue this sweep until the entire length of the graph Rn,l is cleared.   We note that this clearing sequence is obtainable with polite lions, but not with caffeinated lions, as demonstrated in Fig. 4.

4.2 Sufficiency of  3n 2  Caffeinated Lions on a Triangulated Strip We now restrict lion movement so that the lions are caffeinated, i.e. each lion must move at every step. We show that , 3n 2 - caffeinated lions suffice to clear the triangular lattice Rn,l . Indeed, we provide a set formation where the lions may start in order 3n to clear the grid using only , 3n 2 - caffeinated lions. In order to prove that , 2 caffeinated lions are sufficient independent of their starting positions, we must also show that the caffeinated lions can move to this initial formation from any starting position.

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Fig. 5 The sweeping procedure for caffeinated lions using a wall of triangles

4.2.1

Sweeping Formation for  3n 2  Caffeinated Lions

Consider the graph Rn,l . We let the , 3n 2 - lions line vertically such that the lions create a “wall of triangles” in which each row alternates having one or two lions, starting with one lion in the top row. These lions form a vertical wall that stretches from the bottom to top of the grid; see to Fig. 5(left) for this formation. We will show in Lemma 1 that given any initial starting position, the caffeinated lions can travel into this initial sweeping formation. Once in this formation, the caffeinated lions can sweep first to the left and then to the right to clear the contamination. When the lions reach a corner of the grid, the lions sharing horizontal edges with the still contaminated vertices will continue their horizontal sweep. The remaining lions rotate in a three cycle (or swap positions with a lion on an adjacent vertex) to remain caffeinated while the other lions sweep. Refer to Fig. 5 for an example.

4.2.2

Can Lions Get to Some Predetermined Starting Position?

We will use the following lemma to show that caffeinated moving lions may move to arbitrarily specified positions in the Rn,l graph, no matter where their initial starting positions are. This allows the caffeinated lions to move to the initial sweeping formation shown in Sect. 4.2.1, regardless of their starting positions. Lemma 1 Consider any two vertices u, v in the finite connected graph Rn,l . Let M be the length of the shortest path between u and v. Then for any m ≥ M, there is a walk from u to v of length exactly m.

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Fig. 6 Figure accompanying the proof of Lemma 1

Fig. 7 The graph S6

Proof We are given the graph Rn,l ; see Fig. 6. We label two arbitrary distinct vertices u = v on this grid, where u denotes the starting point of a lion, and v is the desired ending location of the lion. We will proceed by induction on m. The case m = M is clear since M is defined as the length of the shortest walk from u to v in this finite connected graph. For the inductive step, suppose there is a walk of length m from u to v; we claim there exists a walk of length m + 1 from u to v. Consider the walk of length m. Let the lion travel along this walk until it has taken m − 1 steps. At this moment, the lion will be on a vertex adjacent to v; call this vertex q. Since every two adjacent vertices in the graph Rn,l are part of a common 3-cycle, there is some vertex s adjacent to both q and v. In the next step, let the lion move from vertex q to s. Once on vertex s, the lion has traveled a walk of length p, and then moves one more step to v in a walk of length m + 1. By induction, it is possible to travel from u to v in a walk of length exactly m steps for any m ≥ M.   Remark 1 A weaker analogue of Lemma 1, where M may be larger than the length of the shortest path but is still finite, more generally holds true for any connected graph G containing an odd cycle. The property in Lemma 1 does not hold on the chessboard graph, i.e. a n × n square grid graph (see Fig. 7). This is because given any two vertices u and v in this graph, there is a parity (even or odd) such that any walk between u and v necessarily has length of that specified parity. The same is true for any 2-colorable (i.e. bipartite) graph. Corollary 1 Consider the graph Rn,l . Given any k caffeinated lion starting positions u1 , . . . , uk and any k specified ending positions v1 , . . . , vk , there exists some

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M ∈ N so that we can have all lions move to arrive at the specified ending positions at (exactly) time step M. Proof Let 1 ≤ i ≤ k be arbitrary. By Lemma 1, we know that there exists an integer Mi ∈ N such that for any vertices ui , vi ∈ Rn,l , and any m ≥ Mi , there exists a walk of length exactly m between ui and vi . Now, let M = max{Mi | 1 ≤ i ≤ k}. For any m ≥ M, and for all 1 ≤ i ≤ k, there exists a walk of length exactly m from ui to vi . Hence the k caffeinated lions can simultaneously move from their initial locations to their desired positions in exactly m simultaneous steps.   Corollary 2 , 3n 2 - caffeinated lions suffice to clear the grid Rn,l , no matter their starting locations. Proof It follows from Corollary 1 that the caffeinated lions can go into the sweeping formation from any starting position. We have already shown in Sect. 4.2.1 that the caffeinated lions can clear the graph from here.  

4.3 Insufficiency of  n2  Lions on a Triangulated Square We will use some of the methods and proofs from [6] to show that , n2 - (noncaffeinated) lions cannot clear Rn := Rn,n . In this proof, we stretch the “rhombus” graph Rn to instead be drawn as a square triangulated by right triangles. It should be noted that this grid holds all of the same properties as before (the isomorphism type of the graph is unchanged). We define Sn to be the n×n square grid graph with n vertices per side as discussed in [6]; see Fig. 7. When each square is subdivided via a diagonal edge, drawn from the top left to the bottom right, then we obtain a graph isomorphic to Rn as shown in Fig. 8. The following two lemmas from [6] hold in an arbitrary graph. Lemma 2 (Lemma 1 of [6]) Let k be the number of lions on a graph. The number of cleared vertices cannot increase by more than k in one time step. Lemma 3 (Lemma 2 of [6]) Let k be the number of lions. If there are at least 2k boundary vertices in the set C(t) of cleared vertices, then the number of cleared Fig. 8 The graph R6 drawn as a square instead of a rhombus

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vertices cannot increase in the following step: |∂C(t)| ≥ 2k implies |C(t + 1)| ≤ |C(t)|. For the specific case of square grid graphs Sn , [6] defines a “fall-down transformation”. This transformation T takes any subset of the vertices of Sn , and maps it to a (potentially different) subset of the same size. The first step in the fall-down transformation moves the subset of vertices down each column as far as possible. The number of vertices in any column remains unchanged by this step. The second step in the fall-down transformation is to then move each vertex as far as possible to the left-hand side of its row, maintaining the number of vertices in any row. In the case of a square grid Sn , [6] proves that the fall-down transformation does not increase the number of boundary vertices in a set: Lemma 4 (Lemma 4 of [6]) In the graph Sn , the fall-down transformation T is monotone, meaning that the number of boundary vertices in a subset S of vertices from Sn does not increase upon applying the fall-down transformation. Since the vertex set of Sn is the same as the vertex set of Rn , we immediately get a fall-down transformation for T acting on Rn that maps a subset of vertices in Rn to a subset of vertices in Rn . We will show that this new fall-down transformation has the same monotonicity property, which is not a priori clear as the boundary of a set of vertices in Sn might be smaller than the boundary of that same set of vertices in Rn . Note that in general, if v is a boundary vertex of a set S in Sn , then v is a boundary vertex of S in Rn . This implies that a set of vertices in Sn has no more boundary vertices than if that set of vertices were in Rn . Another comment is that when defining the fall down transformation T on Sn , the choice of down vs. up or of left vs. right does not matter. But these choices do matter when defining a fall-down transformation on Rn , due to the diagonal edges as drawn in Fig. 8. We want to emphasize we have chosen to map down and to the left; the following lemma in part depends on this choice. Refer to Fig. 9 to see that Lemma 5 fails if we instead allow T to move down and to the right. Fig. 9 Given the initial starting position of a set S in blue on Sn and Rn , notice that applying a modified fall-down transformation down and to the right results in 3 boundary vertices in Sn and 4 boundary vertices in Rn , shown in green

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Fig. 10 Rn

Fig. 11 Sn

Lemma 5 Let S be a set of vertices in Rn (or equivalently, in Sn ). The set of boundary vertices of T (S) in Rn is the same as the set of boundary vertices of T (S) in Sn . Proof We will show that the set of boundary vertices of T (S) in Rn is the same as the set of boundary vertices of T (S) in Sn . First suppose that a vertex v of T (S) is a boundary vertex in Rn . Referring to the diagram in Fig. 10, this implies that one of d, c, b or a is not in T (S). However, we can reduce this to the case that one of c or b is not in T (S), since d ∈ / T (S) ⇒ c ∈ / T (S), and since a ∈ / T (S) ⇒ b ∈ / T (S). If c ∈ / T (S), then v is a boundary vertex of T (S) in both Rn and Sn . The same is true if b ∈ / T (S). This proves that if v is a boundary vertex of T (S) in Rn , then it is a boundary vertex of T (S) in Sn . On the other hand, referring to Fig. 11, if v is a boundary vertex of T (S) in Sn , then one of c or b is not in T (S), which implies that v is a boundary vertex of T (S) in Rn as well. Therefore the set of boundary vertices of T (S) in Rn is the same as the set of boundary vertices of T (S) in Sn . We know that T is monotone on Sn , i.e. that the number of boundary vertices does not increase as a result of applying T . Lemma 5 therefore immediately implies that T is also monotone on Rn , i.e. that the number of boundary vertices of a set S in Rn is no more than the number of boundary vertices of T (S) in Rn . Corollary 3 In the graph Rn , the fall-down transformation T is monotone, meaning that the number of boundary vertices in a subset S of vertices from Rn does not increase upon applying the fall-down transformation. This now brings us to the last lemma. Lemma 6 (Lemma 5 of [6]) Any vertex set S on Rn satisfying n2 2

+

n 2

has at least n boundary vertices.

n2 2

−

n 2

< |S|
|C(t)|. But by Lemma 6, there are at least n ≥ 2k boundary vertices of C(t) at time t, and so Lemma 3 tells us that |C(t + 1)| ≤ |C(t)|, which is a contradiction. Thus k ≤ , n2 - lions do not suffice to clear Rn .  

4.4 Conjectured Insufficiency of

n √ 2 2

Lions on a Triangle

In this section we consider the triangular grid graph Pn with n vertices on each n of its three sides. We conjecture that , √ - lions are not capable of clearing the 2 2 triangular graph Pn —though we give only an incomplete possible attempted proof strategy/outline. By contrast, the square graphs Sn and Rn with n vertices per each of their four sides require at least n/2 lions to clear. We ask in Question 7 in Sect. 6 how many lions are necessary to clear the triangular graph Pn or the square graphs Sn and Rn . We use the notation α ≈ β to mean that there is some small constant κ such that |α − β| ≤ κ. Conjecture 1 It is not possible to clear Pn with fewer than ≈ ,

n √ 2 2

lions.

The bound in this first conjecture is derived from later conjectures, which would provide a kind of “isoperimetric inequality” for sets C in Pn . We will show how Conjecture 1 would be a consequence of the isoperimetric inequalities in the following Conjectures 2 and 3. We must first establish some new notation. There are a number of ways that vertices in a set C can be packed into the graph Pn to have relatively few boundary vertices. For the following definitions, we orient Pn as drawn in Figs. 12 and 13. A row packing occurs when the vertices of C fall so Fig. 12 A row packing on P6 with 13 vertices

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Fig. 13 An ice cream cone packing on P6 with 13 vertices

that each row of Pn is completely filled before the row above it contains any vertices in C, and within each row the vertices are filled left to right. The orientation for this packing is shown in Fig. 12. We define an ice cream cone packing to be a packing where the diagonal rows beginning in the lower left corner are filled one at a time, with the lowest vertex in each diagonal being filled first, and the next diagonal row cannot contain vertices in C unless the diagonal row below it is already filled. The orientation for this packing is shown in Fig. 13. The following two conjectures, if true, would give a type of isoperimetric inequality in the graph Pn . The next conjecture describes what we believe are the optimal ways to place vertices in Pn while minimizing the number of boundary vertices. Conjecture 2 If C is a set of vertices in Pn , then |∂C| is at least as large as the minimum of • the number of boundary vertices in a row packing with |C| vertices, and • the number of boundary vertices in an ice cream cone packing with |C| vertices. Conjecture 3 If C is a set of vertices in Pn with |C| ≈ T,√Tn - , then there are at least ≈ , √n - boundary vertices in C. 2

Possible Proof Sketch of Conjecture 3, using Conjecture 2 As we add vertices to an ice cream cone packing, the number of boundary vertices is nondecreasing (while the last diagonal row is still empty). For j ≤ n − 1, the number of boundary vertices in an ice cream cone packing of ≈ Tj vertices is ≈ j . By contrast, in a row packing, after the bottom row is full, the number of boundary vertices is nonincreasing as we add more vertices. For j ≤ n − 1, the number of boundary vertices in a row packing of ≈ Tn − Tj vertices is ≈ j . Due to these monotonicity properties, Conjecture 2 would imply that when the number of vertices we place is Tj ≈ Tn − Tj , then we will have at least ≈ j boundary vertices. We solve for j in the equation Tj ≈ Tn − Tj to obtain 2Tj √≈ Tn , i.e. j √ (j + 1) ≈ Tn , √ −1± 1+4Tn 2 n ≈ 4T = Tn . giving j + j − Tn ≈ 0. We solve to find j ≈ 2 2 Hence if C is a set of vertices in Pn with |C| ≈ T,√Tn - , then there are at least √ j ≈ Tn ≥ , √n - boundary vertices in C, where in the last inequality we have used 2

2

the bound Tn = n(n+1) ≥ n2 . 2 We can now show how Conjecture 1 would follow from the above two (unproven) conjectures.

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Possible Proof of Conjecture 1 from Conjectures 2 and 3 n Suppose for a contradiction that k is at most ≈ , √ -, and that k lions can clear Pn . 2 2 Thus at the final time step, there must be Tn cleared vertices. Lemma 2 tells us that |C(t + 1)| − |C(t)| ≤ k for all times t. There must have been a time t such that the number of vertices in C(t) was approximately T,√Tn - and |C(t + 1)| > |C(t)|. At that same time, the number of boundary vertices |∂C(t)| would have to be at least ≈ , √n - by Conjecture 3. By Lemma 3, since |∂C(t)| is at least 2k, the number 2 of boundary vertices cannot increase in the next time step, so in fact |C(t + 1)| ≤ n |C(t)|. This is a contradiction, and thus fewer than ≈ , √ - lions cannot clear Pn . 2 2

5 Connection to Cheeger Constant In graph theory, the term Cheeger constant refers a numerical measure of how much of a bottleneck a graph has. The term arises from a related quantity, also known as a Cheeger constant, that is used in differential geometry: the Cheeger constant of a Riemannian manifold depends on the minimal area of a hypersurface that is required to divide the manifold into two pieces [10]. For both graphs and manifolds, the Cheeger constant can be used to provide lower bounds on the eigenvalues of the Laplacian (of the graph or of the manifold). In the graph theory literature, there are several different quantities known as the Cheeger constant, and they are each defined slightly differently. For some more background and applications of Cheeger constants to graphs, see [12, Chapter 2]. In this section, we show a relationship between the Cheeger constant of a graph and the number of lions needed to clear this graph of contamination. Definition 8 For a graph G, let the Cheeger constant be " g := min

|∂S| min{|S|, |S|}

# : S ⊆ V (G), S = ∅, S = V (G) ,

where ∂S = {v ∈ S | uv ∈ E(G), u ∈ / S} (i.e. the set of boundary vertices), and S is the set V (G) \ S. This is the same constant that is studied in [7], where it is called the vertex isoperimetric number hin . Note that ∂S is defined differently than the boundary used in [12] (for us, ∂S is a subset of S instead of S). Other definitions of the Cheeger constant of a graph may count the number of boundary edges instead of boundary vertices. For a connected graph, we have 0 < g ≤ 1. For the upper bound, consider a connected graph G with more than one vertex. If |S| = 1, then min{|S|, |S|} = 1 and since G is connected, |∂S| = 1. Thus any connected graph has a Cheeger constant at most 1 since g takes the minimum of all vertex subsets. For the lower bound, note that if S is neither empty nor all of V (G), then |∂S| ≥ 1 when G is connected.

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Fig. 14 From left to right, the Cheeger constants are g = 15 , g = 13 , g = 25 . These Cheeger constants can be realized by taking the blue vertices to be the set S

If G is disconnected then g = 0 since |∂S| = 0 if we take S to be a connected component. Roughly speaking, a larger g tells us that overall the graph has more connections; a smaller g says that the graph is easier to break into pieces. Consider the graphs in Fig. 14 for a few examples of Cheeger constants. We now give lower bounds on the number of lions needed to clear a graph in terms of the Cheeger constant. Recall a graph with polite lions is one in which at each turn, at most one lion can move. We first consider polite lions, before moving to the case of arbitrary lions. Theorem 3 Let G be a connected graph with vertex set V and with Cheeger constant g. If k ≤ 12 , |V2 | -g, then G cannot be cleared by k polite lions. Proof Suppose for a contradiction that k ≤ 12 , |V2 | -g polite lions can clear G. Since the number of cleared vertices can increase by at most one in each step with polite lions, in the process of clearing there must be a time t satisfying |C(t)| = , |V2 | -. Now, by the definition of the Cheeger constant g and the observation that min{|C(t)|, |C(t)|} ≤ , |V2 | -, we have that , |V2 | -g ≤ |∂C(t)|. Since 2k ≤ |∂C(t)|, this implies by Lemma 3 that |C(t + 1)| ≤ |C(t)|. Therefore the number of cleared vertices is always at most , |V2 | -, contradicting the clearing of G with k lions.   Theorem 4 Let G be a connected graph with Cheeger constant g. If k ≤ G cannot be cleared by k lions. Proof Suppose for a contradiction that k ≤

g|V | 4+g

g|V | 4+g ,

then

lions can clear G. We have that

| |V | k k(2 + ≤ g|V 2 , and hence 2k ≤ g( 2 − 2 ). This implies that for any x satisfying |V | k 0 ≤ x ≤ 2 , we have 2k ≤ g( 2 − x). By definition of the Cheeger constant, g ≤ |V|∂S| , where S is any subset of V of size |V2 | ± x. Combining these two facts | 2 −x implies that for any x satisfying 0 ≤ x ≤ k2 , we have 2k ≤ g( |V2 | − x) ≤ |∂S|, where S ⊆ V is any set of size |S| = |V2 | ± x. By Lemma 2 the size of the cleared g 2)

set can increase by at most k in any step. Therefore there is some time t when the number of cleared vertices is within ± k2 of |V2 | and when |C(t + 1)| > |C(t)|. But since 2k ≤ |∂C(t)| at this time step t, Lemma 3 then implies that the number of cleared vertices cannot increase in the next step, giving a contradiction. Therefore k lions do not suffice to clear the graph G.  

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The advantage of using the Cheeger constant is that it gives a lower bound on the number of lions needed to clear an arbitrary graph. That is not to say that the bound obtained by this method is near the optimal number of lions. Rather, Theorems 3 and 4 gives a weak bound for any graph, including graphs that do not have obvious symmetry that can be used to discover a better bound. For example, suppose the Cheeger constant is g = 1, which happens if G is a complete graph. If g = 1, then Theorem 4 implies that more than |V |/5 lions are needed to clear the graph G, and Theorem 3 implies that at least |V |/4 polite lions are needed. If g = 12 , then Theorem 4 says that more than |V |/9 lions are needed, and Theorem 3 implies that at least |V |/8 polite lions are needed.

6 Conclusion and Open Questions The lion and contamination problem is a pursuit-evasion problem classically defined on square grid graphs. We have explored extensions of this problem by considering restricted models of lion motion—such as caffeinated and polite lions—and by studying the case of triangular grid graphs. We found an upper bound on the number of lions that that would suffice to clear a triangulated parallelogram graph under the typical lion motion and using caffeinated lions. We also found a lower bound for the number of lions to clear a triangulated square graph using techniques from [6]. We explored a possible lower bound on the number of lions needed to clear a discretized triangle graph, and provided a sketch of a possible proof. Lastly, we used the Cheeger constant to give a lower bound on the number of lions needed to clear contamination on an arbitrary connected graph using two types of lion motion: polite lions and arbitrary lions. While we do not expect the Cheeger constant bound to typically be tight, particularly for graphs with extra structure or symmetry, it is quite general and applies to any connected graph. We end with a collection of open questions, that we hope will inspire future work on the lions and contamination problem with other families of graphs.

Question 1 Let G be a connected graph and let H be a connected subgraph. If k lions can clear G, then can k lions clear H ? Prove or find a counterexample. This question is not a priori obvious even if V (H ) = V (G). Nor is this question obvious if H is an induced subgraph of G, which means that V (H ) ⊆ V (G) and that two vertices in H are connected by an edge in H exactly when they are connected by an edge in G.

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Remark 2 Note that the square grids are subgraphs of Rn (with diagonals removed), so if the above question is answered in the affirmative, then our results might be able to be used to prove some results from [6]—see Theorem 2.

Question 2 We define a sweep of a graph to be monotonic if every vertex that is cleared never again becomes recontaminated. If a graph G admits a sweep by k lions, then does it necessarily admit a monotonic sweep by k lions? And if not, then what is the smallest possible counterexample, both in terms of k and in terms of the number of vertices in the graph? For these questions, the lions are allowed to choose their starting positions. See [23] for a different type of sensor motion (also in which contamination lives on the edges) in which the existence of a sweep implies the existence of a monotonic sweep.

Question 3 Consider the graphs C(n, k) which have n evenly-spaced vertices around a circle, and all edges of length at most k steps around the circle. For k = 0, this is n distinct points; for k = 1 this is a circle graph; for k = 2 a bunch of triangles form. How many lions are needed to clear these graphs of contamination? For k = 1 it is clear that 2 lions suffice. For k > 1 one can see that 2k lions suffice (note 4 lions are likely needed for k = 2 and n large). What are the best lower bounds we can get on the number of lions needed?

Question 4 What can we say about strongly regular graphs, of type (n, k, λ, μ)? What bounds on the number of lions needed can we give in terms of n, k, λ, μ?

Question 5 Given a way to discretize a Euclidean domain into a graph, what can be said about the number of lions needed to clear the graph, perhaps as the number of vertices in discretization goes to infinity? For domains in the plane one might expect the number of lions to scale in relationship with a length, and for domains in Rn one might expect the number of lions to scale in relationship to a (n − 1)-dimensional volume. In what settings are results along these lines true? See Corollary 9.1 of [2] for related ideas. What can be said about the relationship between the number of lions needed for different types of discretizations, say triangular versus square versus hexagonal triangulations of a planar domain, or different regular lattices in a three-dimensional domain, perhaps as the number of vertices goes to infinity?

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Question 6 There are several related notions of Cheeger constants on graphs, all of which are different discretizations of the Cheeger constant of a manifold. In Sect. 5 we consider a connection between the Cheeger constant that is sometimes referred to as the vertex isoperimetric number hin [7] and the number of polite lions needed. In this Cheeger constant, the size of a boundary is counted using vertices. A more commonly studied discretization, however, is the Cheeger constant hG of a graph [12] where the size of a boundary is counted using edges. Suppose we only considered monotonic sweeps— sweeps in which the number of cleared vertices is not allowed to ever decrease, as in Question 2—by caffeinated lions (lions which must move at every step). Is the Cheeger constant hG relevant for bounding the number of lions (from below) needed for monotonic sweeps by caffeinated lions?

Question 7 How many lions are necessary to clear the triangular graph Pn or the square graphs Sn and Rn ? Are n lions needed for all three of these graphs?

References 1. Adams, H., Carlsson, G.: Evasion paths in mobile sensor networks. Int. J. Robot. Res. 34(1), 90–104 (2015) 2. Adams, B., Adams, H., Roberts, C.: Sweeping costs of planar domains. In: Research in Computational Topology, pp. 71–92. Springer, Berlin (2018) 3. Adams, H., Ghosh, D., Mask, C., Ott, W., Williams, K.: Efficient evader detection in mobile sensor networks. Preprint (2021). arXiv:2101.09813 4. Alonso, L., Goldstein, A.S., Reingold, E.M.: “Lion and man”: Upper and lower bounds. ORSA J. Comput. 4(4), 447–452 (1992) 5. Ba¸sar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory. SIAM, Philadelphia (1998) 6. Berger, F., Gilbers, A., Grüne, A., Klein, R.: How many lions are needed to clear a grid? Algorithms 2(3), 1069–1086 (2009) 7. Bobkov, S., Houdré, C., Tetali, P.: λ∞ , vertex isoperimetry and concentration. Combinatorica 20(2), 153–172 (2000) 8. Bonato, A., Nowakowski, R.J.: The Game of Cops and Robbers on Graphs. American Mathematical Society, Providence (2011) 9. Brass, P., Kim, K.D., Na, H.-S., Shin, C.-S.: Escaping off-line searchers and a discrete isoperimetric theorem. In: International Symposium on Algorithms and Computation, pp. 65– 74. Springer, Berlin (2007) 10. Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Proceedings of the Princeton Conference in Honor of Professor S. Bochner, pp. 195–199 (1969) 11. Chin, J.-C., Dong, Y., Hon, W.-K., Ma, C.Y.-T., Yau, D.K.Y.: Detection of intelligent mobile target in a mobile sensor network. IEEE/ACM Trans. Netw. 18(1), 41–52 (2009)

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12. Chung, F.R.K.: Spectral graph theory. In: Regional Conference Series in Mathematics. Published for the Conference Board of the mathematical sciences by the American Mathematical Society (1997) 13. Chung, T.H., Hollinger, G.A., Isler, V.: Search and pursuit-evasion in mobile robotics. Auton. Robot. 31(4), 299–316 (2011) 14. de Silva, V., Ghrist, R.: Coordinate-free coverage in sensor networks with controlled boundaries via homology. Int. J. Robot. Res. 25(12), 1205–1222 (2006) 15. de Silva, V., Ghrist, R.: Coverage in sensor networks via persistent homology. Algebr. Geom. Topol. 7(1), 339–358 (2007) ˙ nski, P.: Offline variants of the “lion and man” problem: 16. Dumitrescu, A., Suzuki, I., Zyli´ Some problems and techniques for measuring crowdedness and for safe path planning. Theor. Comput. Sci. 399(3), 220–235 (2008) 17. Fomin, F.V., Thilikos, D.M.: An annotated bibliography on guaranteed graph searching. Theor. Comput. Sci. 399(3), 236–245 (2008) 18. Fox, D., Burgard, W., Thrun, S.: The dynamic window approach to collision avoidance. IEEE Robot. Autom. Mag. 4(1), 23–33 (1997) 19. Hájek, O.: Pursuit Games: An Introduction to the Theory and Applications of Differential Games of Pursuit and Evasion. Courier Corporation, North Chelmsford (2008) 20. Isaacs, R.: Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. Courier Corporation, North Chelmsford (1999) 21. Kolling, A., Carpin, S.: The GRAPH-CLEAR problem: definition, theoretical properties and its connections to multirobot aided surveillance. In: 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems (2007) 22. La, H.M., Sheng, W.: Flocking Control of a Mobile Sensor Network to Track and Observe a Moving Target, pp. 3129–3134. IEEE, Piscataway (2009) 23. LaPaugh, A.S.: Recontamination does not help to search a graph. J. ACM 40(2), 224–245 (1993) 24. Latombe, J.-C.: Robot Motion Planning, vol. 124. Springer Science & Business Media, Berlin (2012) 25. LaValle, S.M., Hutchinson, S.A.: Optimal motion planning for multiple robots having independent goals. IEEE Trans. Robot. Autom. 14(6), 912–925 (1998) 26. Littlewood, J.E.: Littlewood’s Miscellany. Cambridge University Press, Cambridge (1986) 27. Liu, B., Brass, P., Dousse, O., Nain, P., Towsley, D.: Mobility improves coverage of sensor networks. In: Proceedings of the 6th ACM International Symposium on Mobile Ad Hoc Networking and Computing, pp. 300–308 (2005) 28. Liu, B., Dousse, O., Nain, P., Towsley, D.: Dynamic coverage of mobile sensor networks. IEEE Trans. Parallel Distrib. Syst. 24(2), 301–311 (2013) 29. Olfati-Saber, R.: Flocking for multi-agent dynamic systems: Algorithms and theory. IEEE Trans. Autom. Control 51(3), 401–420 (2006) 30. Parsons, T.D.: Pursuit-evasion in a graph. In: Theory and Applications of Graphs, pp. 426–441. Springer, Berlin (1978) 31. Qu, Y., Xu, S., Song, C., Ma, Q., Chu, Y., Zou, Y.: Finite-time dynamic coverage for mobile sensor networks in unknown environments using neural networks. J. Franklin Inst. 351(10), 4838–4849 (2014) 32. Sgall, J.: Solution of David Gale’s lion and man problem. Theor. Comput. Sci. 259(1–2), 663– 670 (2001) 33. Su, H., Chen, X., Chen, M.Z.Q. Wang, L.: Distributed estimation and control for mobile sensor networks with coupling delays. ISA Trans. 64, 141–150 (2016)

A Topological Approach for Motion Track Discrimination Tegan H. Emerson, Sarah Tymochko, George Stantchev, Jason A. Edelberg, Michael Wilson, and Colin C. Olson

Abstract Detecting small targets at range is difficult because there is not enough spatial information present in an image sub-region containing the target to use correlation-based methods to differentiate it from dynamic confusers present in the scene. Moreover, this lack of spatial information also disqualifies the use of most state-of-the-art deep learning image-based classifiers. Here, we use characteristics of target tracks extracted from video sequences as data from which to derive distinguishing topological features that help robustly differentiate targets of interest from confusers. In particular, we calculate persistent homology from time-delayed embeddings of dynamic statistics calculated from motion tracks extracted from a wide field-of-view video stream. In short, we use topological methods to extract features related to target motion dynamics that are useful for classification and disambiguation and show that small targets can be detected at range with high probability.

1 Introduction The ability to identify and label objects in imagery and video has long been of interest to communities across the civilian and military sectors. As higher resolution imaging systems have evolved, there has been corresponding growth in the development and use of automated algorithms to perform these identification tasks. Recently, deep learning methods have demonstrated dramatic improvements in both identification and classification tasks for data arising from applications

T. H. Emerson · S. Tymochko () Pacific Northwest National Laboratory, Seattle, WA, USA e-mail: [email protected]; [email protected] G. Stantchev · J. A. Edelberg · M. Wilson · C. C. Olson US Naval Research Laboratory, Washington, DC, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected] © The Author(s) and the Association for Women in Mathematics 2022 E. Gasparovic et al. (eds.), Research in Computational Topology 2, Association for Women in Mathematics Series 30, https://doi.org/10.1007/978-3-030-95519-9_9

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ranging from self-driving cars, natural language processing, and image classification [2, 8, 12]. Due to their success, deep learning algorithms have become the default technique when presented with high-volume, high-resolution image analysis tasks. However, not all imaging challenges are appropriate for Deep Learning techniques; in particular, in cases where labelling data is difficult or where high-resolution sensors (e.g., in the infrared domain) are expensive or unavailable. We consider here the problem of detecting small targets at range. This is a difficult task to automate because of two competing requirements: (1) wide field-ofview (WFOV) cameras are required for sufficient coverage of a scene but (2) high resolutions are needed in order to accurately detect and classify objects moving within the scene. The latter requirement is especially critical because the scene may contain moving objects whose tracks appear similar to the target’s at the given resolution, thus registering as false alarms if classification decisions are made solely based on change detection. Such objects are referred to as dynamic confusers. In addition, the WFOV requirement, creates a scenario where there is generally not enough spatial information present in an image sub-region containing a possible detected target to use correlation-based methods to differentiate it from confusers. As a result, rather than rely on correlation-based methods (e.g., classification via convolutional neural networks) we instead use characteristics of target tracks extracted from video sequences as data from which to derive distinguishing topological features that clearly identify target class. In essence, we use topological methods to extract features related to target movement dynamics that are useful for classification and disambiguation. We present here a novel approach for differentiating between targets and confusers based on an analysis of motion tracks derived from object and change detection algorithms. Our approach performs well as a stand-alone technique on the test set considered but could also be considered as a preprocessing step prior to a deep learning implementation. The approach leverages mathematical techniques for time series analysis (i.e., time-delayed embeddings) to ensure that informative dynamic invariants are preserved such that a topological analysis of the resulting point cloud data yields features that are useful for classification. Section 2 provides an introduction to the existing tools and techniques we utilize (i.e., time-delay embedding, topological data analysis, and motion track generation). Next, in Sect. 3 we provide a description of the data set, discuss the experimental design and how we address the pre-processing data challenges. In Sect. 4 we outline our framework for analyzing motion tracks. Experimental results are presented in Sect. 5, followed by discussion of the results and the implications for future work in Sect. 6.

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2 Background 2.1 Time-Delay Embedding Time-Delay Embedding (TDE) is a technique for time-domain signal analysis based on the seminal work of Takens [13] and Whitney [16]. The underlying assumption of TDE is that, given a dynamical system describing a physical process, some measured aspect of the system is a time-varying statistic from which certain uniquely identifying dynamic invariants may be extracted given a “proper” embedding of the observed statistic. Such an embedding is constructed in turn by using algorithms such as false nearest neighbors [9] and mutual information [6] to select, respectively, an appropriate embedding dimension, D, and time-delay, τ , to apply to an observed N −(D−1)τ time series, z = {zn }N . n=1 , such that Z = {(zn , zn+τ , · · · , zn+(D−1)τ )}n=1 Thus, the output of a TDE is a cloud of D-dimensional points constructed from delayed copies of the observed 1-dimensional time series as shown in Fig. 1. In the context of the target vs. confuser problem, this translates to the assumption that the motion patterns of each object can, in theory, be modeled by a dynamical system and that the motion of an object as projected onto the sensor is the observed dynamic statistic. Even though some dynamic information has been lost due to this projection, Takens’s theorem guarantees that certain dynamic invariants describing the topology of the underyling manifold on which the dynamics evolve will still be preserved and observable via the delay embedding process. Given this preservation of topological invariants, we leverage tools from topological data analysis to help generate uniquely identifying features from the projected motion tracks.

[ 1,

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]

,D

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Fig. 1 Example of how a point cloud is created from a 1-dimensional statistic. We define the notation generally and visually demonstrate the result for a given 1-dimensional statistic, a delay of τ = 2, and an embedding dimension of D = 3

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2.2 Topological Data Analysis Topological Data Analysis (TDA) describes a broad collection of techniques which aim to extract invariants of and/or characterize data based on the mathematical notion of “shape” or “connectedness”. The use of TDA has grown in recent years due to the flexibility of the toolset and increased computational efficiency. In this section we provide a brief high-level description of two TDA tools: Persistent Homology (PH) and Persistence Images (PIs). We omit the use of technical formulations and refer the interested reader to [3] and [4] for formal definitions and to [10] for algorithms and implementation. 2.2.1

Persistent Homology

Homology in general can be intuitively understood as an algebraic framework for consistently counting distinct classes of “holes” and connected components of various dimensions present in a topological space. Homology can be used to analyze data, that is, a finite collection of points in a D-dimensional space, by associating it with a specific combinatorial topolological space whose homological structure (the algebraic relationship among the equivalence classes of holes at various dimension) can be computed [5]. In particular, the number of equivalence classes of k-dimensional holes at a given scale, is referred to as the kth Betti number [7]. This association is scale-dependent and consequently so would be the resulting homological structure. Persistent Homology (PH) is a common tool in TDA that aims at extracting and encoding scale-dependent topological information from data by tracking how the homology structure of certain topological spaces associated with the data varies as a function of the scale parameter [4]. PH can be computed in a variety of ways depending on the data under consideration. For our approach we will be computing the PH of a point cloud in a real-valued Euclidean space. Within this context, scale is equivalent to distance, which for a fixed value can be used to construct a topological space called the Vietoris-Rips (VR) complex [5]. The VR complex is formed by adding edges, faces, tetrahedral cells, and so forth based on the pairwise intersections of balls of radius equal to the chosen distance parameter. The Betti numbers of the associated VR complex can then be computed using linear algebra. Varying the distance parameter and evaluating the associated homology structure allows the tracking of scales across which each equivalence class of k-dimensional holes persists. The collection of intervals, the end points of which represent respectively the birth/disappearance of a homological feature, is called a Barcode Diagram. For each dimensional homology we can summarize the information in a variety of ways. In early TDA approaches each homology dimension was typically summarized in a Persistence Diagram (PD)–a multiset of points in the plane where the x−value of a point corresponds to the scale at which the hole appeared and the y−value indicates the scale at which the hole disappeared/collapsed (see for instance [5]). The space of PDs can be endowed with a metric structure allowing one to measure the similarity between

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point clouds. However, the metrics on PDs are typically expensive to compute and tend to limit the tools available to differentiate between configurations of point clouds. Therefore, significant effort has been spent in recent years on developing alternative representations of the information captured in a PD such that broader machine learning techniques can be applied. One such representation, a Persistence Image (PI), is described next. 2.2.2

Persistence Images

Persistence images are generated by centering 2-dimensional probability distributions on each point in a PD to form a surface, overlaying a grid, and then integrating the area under the surface over each discretized region of the grid. See [1] for the steps involved in forming a PI. Typically the 2-dimensional distribution is chosen to be Gaussian requiring a choice of covariances. The other associated parameter is the size of the grid. It has been shown [1, 17] that the performance of PIs is robust to the choice of these parameters. For our implementation we chose a grid size/resolution of 25 and the default parameter settings for the PI code found at [14].

3 Track Generation and Conditioning The data we consider are acquired using a stationary wide field-of-view system which captures images of the same scene over time. The scene contains some objects of interest (called “targets”) but there also could be false alarms such as birds flying by or bugs near the lens (called “confusers”). Images are collected using a commercial camera. Using custom change detection algorithms we are able to track objects as they move within or across the field-of-view as shown in Fig. 2. The resolution of the imaging system is sufficient to detect movement but, as can be seen from the zoomed-in image clips in Fig. 2, is not high enough to resolve the moving objects. It is possible to do targeted high-resolution imaging on objects of interest, but without further processing the number of nontrivial confusers is prohibitive. The experiment and results we present in the subsequent section have been considered as standalone, however, it is plausible to employ the presented methodology as a first step to reduce the number of false alarms to such an extent that targeted imaging could then be employed to extract images on which a deep learning framework could be applied.

3.1 Motion Track Generation Track data are derived from visible, monochromatic, imagery that contains dynamic targets of interest as well as dynamic confusers. Imagery is processed frame-byframe using a custom target detection processor. In particular, raw detections are

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Fig. 2 Example of analyzed data. The image chips contained in each green box correspond to a zoomed-in view, at a single frame, of the object generating the corresponding motion track

generated via a combination of temporal and spatial processing, the results of which are fused to allow the continued detection of slow-moving targets while simultaneously providing a live adaptive non-uniformity correction of the imaging system. These frame-by-frame detections are then correlated in space and time resulting in individual track files corresponding to each moving object in the image series. The target tracks are then determined via comparison with ground truth data. Finally, an operator reviews the source imagery and track information to separate the remaining non-target tracks into different confuser classes.

3.2 Motion Sub-Track Generation Figure 2 further highlights another challenge for analysis of motion tracks: tracks of differing lengths. Not all objects appear in the image at the same time and nor do they stay in the field-of-view for the same number of frames. To address this issue we generate sub-tracks of the same length for all tracks which allows for an appropriate comparison between all tracks. Here, the analyzed data set consists of four confuser motion tracks and one target track. The five evaluated tracks each have a ground-truth label. For each track, tj , we have a set of (x, y)-coordinate pairs corresponding to the j

j

N

j where i counts the location in the image indexed by time, that is, tj = {(xi , yi )}i=1 number of frames and Nj is the total number of frames for which the j th object is detected. Let N ∗ < Nj be a desired sub-track length. For each track we generate the ∗ j j j kth sub-track of the j th object as sk = {(x(k−1)+i , y(k−1)+i )}N i=1 . Each sub-track is a subset of its parent track and all sub-tracks are the same length N ∗ such that they

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Fig. 3 Examples of normalized sub-tracks taken from each of the five test objects

may be directly compared both within and across classes. In short, the kth sub-track is an N ∗ -length window extracted from the parent track and the (k+1)th sub-track is generated by shifting the extraction window by one. Larger time shifts are possible but are not investigated here. Because an object needs to retain its dynamic representation regardless of the j initial location within the field of view, we normalize each sub-track such that s˜k ∈ [0, 1]×[0, 1]. We perform this normalization by first subtracting the minimum value from within a sub-track from all elements composing the sub-track and then divide that result by the maximum value of the result to force all elements to map to the range [0, 1]. This normalization is performed independently for both the x- and y-components comprising the sub-track. Figure 3 shows a single, normalized subtrack from each of the five original tracks. Notice that while three of the confuser sub-tracks are visually distinct from the target, the third confuser is more similar to the target than to the other confusers. Finally, we generate a vector of statistics from a sub-track by projecting 2-dimensional track coordinates into a 1-dimensional vector representation by j j calculating sk = s˜k ·v where v is a 2×1 vector with random entries drawn uniformly from (0, 1].1 That random vector is formed once, remains fixed, and is applied to all sub-tracks from all objects in the scene. All told, given Nj and a choice of N ∗ , we j produce a set, Sj , comprised of Kj = (Nj − N ∗ ) vectors sk for each moving object in the scene. The generation of normalized sub-tracks and their associated statistics completes the preprocessing stage and yields the data on which we perform our experiments.

1 Note that almost any projection (away from a set of measure zero) produces a valid statistic on the underlying dynamics and satisfies the requirements of Takens’ theorem. Using different random projections did not significantly vary the results.

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4 Topological Approach for Motion Track Classification In many situations the exact variables that are needed to build explicit, accurate dynamic models for different systems or processes are unknown or the number required is prohibitive. For some applications the technology needed to capture relevant measurements may not even exist. As a result there is a need for other means of analyzing signals/data that can be acquired. Takens’s Theorem, discussed in Sect. 2, provides a rigorous mathematical framework for doing exactly this, which we can leverage for disambiguating target dynamics from those of confusers. To do so, we assume that the dynamic systems governing the motion mechanics of confusers differ from those governing the targets of interest. Moreover, we assume that the motion pattern/shape projected through the field of view of the sensor is a representative statistic of system dynamics. Finally, we assume that the dynamical trajectories of the underlying systems evolve on different manifolds whose topology (e.g. shape, dimension, equilibria, etc.) may be coarsely captured by homological differences. Given these assumptions, and partially inspired by prior successes using topological features to differentiate between different motion types [15] and dynamical systems [1], we describe below a novel approach for extracting topological features from motion tracks.

4.1 Experimental Design j

We assume that we begin with normalized sub-tracks of equal length, {sk }, generated as specified in Sect. 3.2. These are the inputs to the time delay embedding. The number of sub-tracks of each class used for these experiments are shown in Table 1. For each sub-track we perform a time-delayed embedding. For visualization sake we have embedded the data into the plane (i.e., we set the embedding dimension as D = 2) using a time-delay of τ = 1. These parameters are also used in the persistent homology computation. The Persistent Homology of each 2-D point cloud is computed using the Python Ripser package [14]. For this first round of exploratory work we have chosen to look at the 0-dimensional homology only. Once we have computed the homology we generate the corresponding persistence images (see Fig. 4). For 0-dimensional homology the persistence image is generally a vector and not an image. This is due to the fact that with the Vietoris-Rips filtration, all 0-dimensional homological features are born at a birth value of 0. Thus, rather than using a 20 × 20 persistence image, we can simplify it to a length 20 vector which Table 1 Number of sub-tracks of different lengths coming from targets and confusers

Sub-track length 50 75 100

Target 491 466 441

Confuser 1111 1011 911

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Fig. 4 Track classification loop: the persistent homology of the cloud of embedded track points is computed and then summarized using a persistence image. Here, we have shown the Vietoris-Rips complex at a single fixed scale arising from a confuser and a target subtrack. Note: this figure is for the sake of visualization, in practice there will not be negative coordinate values in the time delay embeddings

essentially captures the same information as the first column of the full persistence image. We will refer to these as “persistence vectors” for simplicity. These labeled j persistence vectors, pk , are the inputs to our classification algorithm. Each of the J objects in the scene yields a set, Pj , of Kj persistence vectors which ultimately yields a superset P = {Pj }Jj=1 of persistence vectors. For this initial work we have implemented a k-nearest neighbors classifier using Python’s SciKit-Learn [11]. We use the default parameters for the classifier, where k = 5.

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We split each subset comprising the superset equally to yield training and test sets on which to test classification performance. We then explore the effect of track length on classification performance by testing subtracks consisting of 50, 75, and 100 measurements. As a comparison, we also apply k−nearest neighbors to the normalized sub-track statistics of the same length. We do acknowledge this is not a standard benchmark for classifying time series data. However, it will allow us to compare how much additional information is gained using the time delay embedding and topological methods. In the future we plan to compare our method to more standard time series classification methods. Figure 4 shows an outline of the full track classification loop.

5 Results The results of the experiments described in Sect. 3 are presented in Table 2. “Track Length” refers to the number of measurements used to generate a subtrack. “Feature Method” indicates whether the k−nearest neighbors algorithm was j applied to the 1-D statistic, sk , computed from the normalized sub-track or the H0 j persistence vector, pk , arising from embedding the statistic. Within the “Confusion Matrix” column we indicate the true class label by sub-column heading and the class predicted by the k−nearest neighbors algorithm as sub-rows. In the following, as well as in Sect. 6, we associate the target and confuser classes with the notion of “positive” and “negative” classes, respectively.

Table 2 Classification results using a k-Nearest Neighbors Classifier. Columns labels indicate true class row labels indicate the predicted class. The k-Nearest Neighbors classifier was trained using 50% of the data and the reported results are for the remaining 50% of the data Track length 100

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Feature method Statistic of normalized subtrack Persistence vector of embedded statistic Statistic of normalized subtrack Persistence vector of embedded statistic Statistic of normalized subtrack Persistence vector of embedded statistic

Confusion matrix ↓Predicted/True → Confuser Target Confuser Target Confuser Target Confuser Target Confuser Target Confuser Target

Confuser 0.91 0.00 1.00 0.00 0.81 0.02 1.00 0.00 0.76 0.02 1.00 0.01

Bold values specify the confusion matrix for the best performing feature method

Target 0.09 1.00 0.00 1.00 0.19 0.98 0.00 1.00 0.24 0.98 0.00 0.99

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For each of the three track lengths we see that the persistence vector of the embedded statistic is better able to differentiate between the two classes. At all three lengths the persistence vectors are able to correctly classify all of the confusers and it is not until the shortest length that we see the first misclassification of a target. Even at this shortest length the persistence vectors only have a 1% false negative rate. In the context of our motivating application a false negative is more acceptable than a false positive and both feature methods achieve at most a 2% false negative rate. However, we see that the 1-D statistic produces high false positive rates that increase as the track length is decreased. Excessive false positives degrade our ability to extract targeted higher-resolution images for better classification at the rates required to aid with time- and distance-dependent response decisions.

6 Conclusion We have implemented a novel framework for distinguishing between small targets and confusers based on analysis of motion tracks derived from wide-field-of-view images. Our approach leverages the rich theory of both time-delayed embeddings and topological data analysis. We tested our approach on a data set comprised of five motion tracks where one track is the positive (target) class and the other four belong to the negative (confuser) class. We find that the topological features are a strong discriminator between the classes and are robust to sub-track length. We do note that as there is only one target class, it is expected that the subtracks have similar signatures. In the future we hope to test our method on a larger dataset with more target tracks to test how robust this method is to variations caused by different targets. Additionally, we will perform further benchmarking against alternative approaches such as the Fourier transform and deep learning methods. We expect our approach will remain competitive based on the success shown in this paper. We will also utilize more sophisticated methods of selecting time-delay embedding parameters not implemented in this initial, proof-of-concept work. We also plan to expand the set of homological dimensions considered and apply deep learning image classifiers to the resulting persistence images. Acknowledgments The authors would like to acknowledge support from the Office of Naval Research. The authors would like to thank the editors and anonymous reviewers whose comments helped improve the quality of this paper.

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