Empirical Paradox, Complexity Thinking and Generating New Kinds of Knowledge [1 ed.] 1527534405, 9781527534407

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Empirical Paradox, Complexity Thinking and Generating New Kinds of Knowledge

Empirical Paradox, Complexity Thinking and Generating New Kinds of Knowledge By

Bruce J. West, Korosh Mahmoodi and Paolo Grigolini

Empirical Paradox, Complexity Thinking and Generating New Kinds of Knowledge By Bruce J. West, Korosh Mahmoodi and Paolo Grigolini This book first published 2019 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2019 by Bruce J. West, Korosh Mahmoodi and Paolo Grigolini All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-3440-5 ISBN (13): 978-1-5275-3440-7

Contents JY

Preface 1 Paradox is Fundamental 1.1 Getting Oriented . . . . . . . . 1.1.1 Physical Paradox . . . . 1.1.2 Complexity . . . . . . . 1.1.3 Aristotelian Logic . . . 1.2 Visual Paradox . . . . . . . . . 1.3 Previews of Coming Attractions

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2 Kinds of Empirical Paradox (EP) 2.1 Contradiction in theory . . . . . . 2.2 Altruism Paradox (AP) . . . . . . 2.2.1 Multilevel natural selection 2.2.2 The invisible hand . . . . . 2.3 Organization Paradox . . . . . . . 2.4 Strategic Paradox . . . . . . . . . . 2.5 Survival Paradox . . . . . . . . . . 2.6 Innovation Paradox . . . . . . . . . 2.7 Conflict Paradox . . . . . . . . . . 2.8 Control Paradox . . . . . . . . . . 2.9 Para Bellum Paradox . . . . . . . . 2.10 What have we learned? . . . . . .

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WJ 3 Thoughts on Nonsimplicity 3.1 Some background . . . . . . . . 3.2 Uncertainty and Empirical Law 3.3 Man, Machine & Management . 3.4 Nonlinearity and Contradiction 3.5 Statistics and Taylor’s Law . . 3.6 Paradox and Uncertainty . . .

CONTENTS

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4 Two Useful Models 4.1 Modeling Sociology . . . . . . . . . 4.2 Decision-making Model (DMM) . . 4.2.1 Criticality . . . . . . . . . . 4.2.2 Control of Transitions . . . 4.2.3 Committed Minorities . . . 4.2.4 Groupthink and the gadfly 4.3 Evolutionary Game Model (EGM) 4.3.1 Choice of strategies . . . . . 4.3.2 Some general observations . 4.4 Exploring simple level coupling . . 4.4.1 Joining the two models . . 4.5 Conclusions and observations . . .

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5 Self-Organized Temporal Criticality 139 5.1 The Inadequacy of Linear Logic . . . . . . . . . . . . 140 5.2 Two Brains, Two Networks . . . . . . . . . . . . . . 146 5.2.1 Intuition . . . . . . . . . . . . . . . . . . . . . 150 5.2.2 Deliberation . . . . . . . . . . . . . . . . . . . 153 5.2.3 Criticality and some extensions . . . . . . . . 154 5.3 SOTC model of two-level brain . . . . . . . . . . . . 158 5.3.1 Crucial events, swarm intelligence and resilience 160 5.3.2 Influence flows bottom-up . . . . . . . . . . 161 5.3.3 Influence flow top-down . . . . . . . . . . . . 170 5.3.4 Resilience vs vulnerability . . . . . . . . . . . 172 5.4 The Sure Thing Paradox . . . . . . . . . . . . . . . . 175

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Conclusions from SOTC model ............................................185

6 Criticality and Crucial Events 189 6.1 A little more history ..............................................................190 6.2 Properties of crucial events ......................................................194 6.3 Importance of crucial events ....................................................200 6.3.1 Making crucial events visible......................................202 6.4 Dynamic nonsimplicity matching..........................................206 6.5 Summary and closing observations..........................................210 A Master Equations 215 A.1 The Decision Making Model.................................................215 A.1.1 All to All coupling ..................................................216 A.2 SUM and SEM ......................................................................219 A.2.1 Criticality-induced network reciprocity .....................222 A.2.2 Morality stimulus on SEM at criticality………….…225 B Importance of Deception

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C Analytic Arguments and STP 233 C.1 Limits of T ............................................................................233 C.2 The special case W=1 ...........................................................234 C.2.1 Violation of the renewal condition ...........................236 Bibliography 239 Index 264

Preface As scientists it is gratifying when a theory, with which we are associated, is shown to be consistent with newly obtained experimental data. This reaction to success fades into near insignificance, however, when it is compared with the response to an experimental result that contradicts a fundamental assumption of the same theory. The reaction to failure is much stronger than the reaction to success as any experimental psychologist will tell you. When confronted with such conflict any scientist worth his salt becomes strongly motivated to determine where the theory is deficient, or where the experiment did not measure what they thought it was measuring. This is the normal evolution of scientific knowledge and it ratchets upward due to a cultivated social desire, embraced by most scientists, to understand the why of the world, as well as, the way. However, a scientist’s strongest reaction occurs when the failure is doubled, which is to say when an empirical result not only contradicts a tenet of the theory, but is also at odds with other well established experimental data. Such conflicts do not occur very often, that is, one rarely finds results contradicting both a validated theory and standardized data, by reproducible experiments, but when they do, they invariably lead to fundamental insights. The insights are a consequence of the fact that such a double contradiction constitutes an empirical paradox and its resolution constitutes not only new knowledge, but quite often a new kind of knowledge. For example, Charles Darwin was the first to identify the altruism JY

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PREFACE

paradox, which he subsequently resolved with a conjecture. His conjecture remained unproven for over a century and required the formation and maturation of the new discipline of sociobiology in order to satisfactorily resolve the controversy. A century earlier, a similar paradox was identified in an economic context by Adam Smith. We briefly discuss these and a half dozen or so other kinds of paradox that arise in organizations, in the bonds people form in social groups and some that are unique to the information age. The working hypothesis of this essay is that empirical paradox is entailed by complexity. We choose the word entail with conscious intent to emphasize the notion that empirical paradox is necessitated by complexity and a system, or network, is truly complex only when it gives rise to paradox. Herein we use this as an operational definition of complexity and by so doing finesse the necessity of having to prove a hypothesis. For those reluctant to accept a hypothesis without proof let us suggest that it is evident that every empirical paradox emerges from a complex phenomenon, for if there is a paradox the system cannot be simple and is therefore complex. This is less formal than a hypothesis, but for our purposes it comes to the same thing in the end. There are a number of examples of empirical paradox from physics, one being the transition of water to ice. The interactions of H2 O molecules in water are short range, that is, they are local interactions. However, the interactions of H2 O molecules in ice are long range, since the molecules form an interconnecting web. Any theory of phase transition must therefore contain these contradictory properties of how H2 O molecules interact in various phases. Kenneth Wilson was awarded the 1982 Noble Prize in Physics for the mathematical resolution of the empirical paradox of criticality and phase transitions in physical phenomena. As anticipated above, the mathematical theory he developed for its resolution led to a new way of understanding many-body behavior of physical processes independently of the detailed underlying dynamics. Central to this new way of knowing is scaling and universality, concepts that we use in this essay to understand all manner of empirical paradox in the physical,

PREFACE

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social and life sciences. Although we subsequently introduce a mathematical model to guide our discussion for resolving empirical paradox, we gloss over the mathematical details, since these have been presented in the scientific literature. Our intent here is to interpret the mathematics in a way that makes the implication of the model accessible to the reader relying only weakly on their degree of mathematical literacy. It is however important to verify that this model exists and that its detailed dynamics do not matter, because it is the emergent properties that are of consequence and these we discuss at some length. This essay contains the discursive content of the first complex dynamic model of empirical paradox and the clarity of the presentation will hopefully reduce the number of responses claiming: 1) that it is not even nonsense and should never have been published; 2) that it is not nonsense, but it is trivial; 3) it is brilliant, but I knew it years ago and never got around to publishing it. Nomenclature AP: altruism paradox EP: empirical paradox DMM: decision making model EGM: evolutionary game model EGT: evolutionary game theory IPL: inverse power law LFE: law of frequency of error PDG: prisoner’s dilemma game PDF: probability density function RTP: rapid transition process SEM: selfishness model SUM: success model SOC: self-organized criticality SOTC: self-organized temporal criticality TLB: two-level brain

Chapter 1

Paradox is Fundamental In this chapter we argue that the emergence of paradox is the opportunity to develop new knowledge and more often than not, a new kind of knowledge resulting from the resolution of the paradox. Typically, contradictions arise from applying existing understanding of a phenomenon about which new experimental data has become available and which contradicts the accepted interpretation of prior data. It is the contradiction that constitutes the paradox. The earlier data was interpreted in terms of simple models, but as the phenomenon becomes more complex (less simple), or its fundamental complexity became visible, using more refined experimental tools, the previous models lead to logical inconsistencies. In physics, the resolution of the paradoxical data indicating that light is both a wave and a particle, ushered in quantum mechanics and introduced a revolutionary way to understand physical reality. However, the discussion is not restricted to physical paradox, but includes visual paradox as well, allowing the reader ample opportunity to develop some intuitive feel for the fundamental role 1

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CHAPTER 1. PARADOX IS FUNDAMENTAL paradox plays in understanding the world in which we live.

1.1

Getting Oriented How wonderful that we have met with a paradox. Now we have some hope of making progress. — Bohr [41]

This essay is a collaboration among three very different authors and even though we are all physicists, we come from different cultures, from different parts of the world and ply our trade in very different ways. But an essay should speak with a single voice and not become bogged down in statements of over-qualification, with this being the experience of one, that being the opinion of the other, with the third having a slightly more nuanced view than the other two. For that reason we adopt the first person singular for nearly every expression of personal experience and hopefully present a coherent, more robust, personality than any of us possess individually, thereby blending our unique voices into one harmony. So with that caveat, let us begin our journey into the exploration of conflict, contradiction, paradox and the generation of new kinds of knowledge. When I reached my late twenties, or early thirties, I found that the people I knew had more stamina, could run farther, climb faster, jump higher, were in all around better physical condition and were overall more physically capable than I was. The realization that I was probably less athletically capable than were my friends bothered me, so I began to actively investigate the phenomenon, but only decades later did I come up with some remarkable and totally unexpected conclusions. Not the least of which is that the world is filled with paradoxes and I had inadvertently stumbled into one called the capability paradox. The capability paradox apparently arose from the fact that not only were my friends probably more athletically capable than I was, they were probably less athletically capable than were their friends,

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with me being the exception. This circumstance seemed to imply that everyone in a large group is probably less athletically capable than almost everyone else in the same large group. Therein lies the inconsistency that constitutes the paradox. How can almost everyone in a large group be less capable than almost everyone else within the same group? This paradox is similar to the friendship paradox, or happiness paradox, both of which have been argued to be a consequence of the network structure of social media [43]. The friendship (or happiness) paradox asserts that within a social network most individuals have the experience of being less popular (happy) than their friends on average, leaving aside the fact that people are sometimes less than honest in their computer postings. The variability in the connectedness of individuals on social networks has been used to explain the counter intuitive nature of the paradox. We subsequently use statistical arguments to quantify this connectedness and hopefully develop an intuition for the kind of variability that can lead to this kind of contradiction. But before we posit any attempt to explain the cause of paradox, or at least before we develop a mathematical model to provide insight into paradox, let us examine the multiple ways paradox enters our lives and its inevitability in today’s society. Perhaps a more familiar form of paradox is a self-contradictory statement such as: This sentence is false. About which many articles have been written for both popular and scientific consumption. The logical paradox is a consequence of the breakdown of linear logical thinking in understanding what has been said. The disruption in the logical understanding of the statement arises because what is presumed true at the outset is contradicted by the end of reading the statement. Consequently, one must start again, with the opposite assumption about the statement’s truth, but obtain the same contradictory result, thereby generating an endless cycle of change and contradiction. Suppose we assume the statement is true. We then read it and conclude that the statement is, in fact, false. But if we assume the statement is false then when we read it a second time we conclude the statement is true. This is the cycle, which arises

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from the logical necessity to change the truth value of the sentence with each reading. In isolation the inability to assess the final truth of a given statement would be an exercise in logic. But the implications of such statements run much deeper. The sequential building of interdependent statements to construct a logical argument, which is the hallmark of linear logical thinking, is disrupted by the existence of such statements. If a statement of the above form is contained within a sequence of remarks, then it is not possible to go beyond the statement itself with any clarity regarding the truth of the sequence of interdependent statements. Therein lies the truth-value paradox. It is interesting, but the truth-value paradox in itself does not concern us here, although a number of its consequences are certainly of interest. There are many different classes of paradox. A particularly devastating class is called antinomy, which according to the American philosopher and logician W.V. Quine, brings on a crises of thought [195]: An antinomy produces a self-contradiction by accepted ways of reasoning. It establishes that some tacit and trusted pattern of reasoning must be made explicit and henceforward be avoided or revised. We mention this because many, otherwise educated, people view logical thinking as proceeding by an invariant set of rules, such as the syllogism, by which we reason about the world and that have been fixed since the time of Aristotle. Not only is this not true, but a belief in such a rigid system of reasoning prevents such people from developing the cognitive strategies necessary to understand the complex world in which we live. But even less extreme forms of paradox have resulted in the necessary abandoning of once accepted forms of reasoning, an example of which are Gödel’s two incompleteness theorems. You have probably heard of Gödel’s proof and may even vaguely recall that it has had a profound effect on what mathematical the-

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orems can and cannot be proven. At one time it was believed to be sufficient to prove that a line of reasoning, assuming the truth of a given theorem, which results in a paradox is a reduction to absurdity (reductio ad absurdum), and on its face is sufficient to establish the falsity of the theorem. On the one hand, in the 1920s the mathematician Hilbert championed the idea that all of mathematics could be derived from a finite set of axioms, like those of Euclid that we learned about in high school geometry. On the other hand, in the 1930s Gödel proved that one could not prove the truth of all possible statements made within a closed mathematical system, using only the axioms from within the system. Consequently, in a selfreferential closed system one cannot prove the truth or falsity of a theorem, the truth value is not decidable.1 The incompleteness theorem of Gödel implies that scientists can never be absolutely certain concerning the truth of all the statements made in a closed mathematical system and therefore by extension they can never completely rely on the complete validity of any closed mathematical model of reality [201]. Consequently, the self-referential sentence (This statement is false.) is true only if false, and false only if true. Determining the truth of this sentence might give a reasonable person a headache; but then mathematicians can be very unreasonable. Fortunately scientists have experiments from which to determine, if not the truth, at least the consistency of a theorem with the behavior of the world. But experimental observation does not obviate the need for mathematical models of the phenomenon being investigated and therein lies the rub. The point of this digression into mathematics and mathematical 1 The results of Gödel’s 1931 paper, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems," can be summarized as follows:

Any consistent axiomatic system of mathematics will contain theorems which cannot be proven. If all the theorems of an axiomatic system can be proven then the system is inconsistent, and thus has theorems which can be proven both true and false.

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logic is to establish the historical precedent, as observed by Quine, that the identification of paradox has, on a number of occasions, entailed the reconstruction of the foundations of thought. This precedent regarding the reformulation of the static rules of thinking may then subsequently facilitate the reader’s acceptance of a reformulation involving dynamic rules entailed by the kind of paradox that emerges in scientific theory. It is the latter that is the focus of this essay. What does concern us is the form of paradox that arises in science when a phenomenon is explained by a theory attributing two or more mutually incompatible characteristics to a phenomenon and yet all the attributed properties are empirically observed. This natural, or empirical, paradox is of interest to us because nature has devised ways to resolve such conflicts, since they are part of objective reality. However when we attempt to reason about phenomenon containing empirical paradox we encounter logical inconsistencies, which we herein seek to resolve. Or as King Lear might have observed: that way madness lies.

1.1.1

Physical Paradox

Every great and deep difficulty bears in itself its own solution. It forces us to change our thinking in order to find it. — Bohr [41] Physicists, in their attempt to understand the physical world, build models of the phenomena they observe. They follow in the tradition of Isaac Newton, who introduced quantifiable mechanical forces in the form F = ma into Natural Philosophy and thereby initiated its transformation into classical physics. His perspective tied the motion of the planets to that of apples falling from trees, all being one in the same through the force of gravity. For over a century the behavior of matter was determined by identifying the force and solving the resulting equations to predict the trajectory of a cannon ball, the speed of sound in air, the time of an eclipse, etc..

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On the scale we live our lives things seem to be deterministic and given sufficient information about the state of a material object at any one time enables its behavior to be predicted at any latter time. Rudolf Clausius in an 1850 paper summarized centuries of physical experiments in the first two laws of thermodynamics. The First Law is that energy cannot be created or destroyed, but only changed in form. This is consistent with the mechanical forces of Newton and all the experiments based on his equations of motion. The Second Law, states that heat only flows from a hot body to a cold body. This one-way flow of heat is not consistent with the time reversibility of Newton’s dynamic laws of matter and after 15 years of wrestling with this inconsistency Clausius introduced a new concept into physics, entropy. As pointed out by Haw [112], Clausius recognized that energy alone was insufficient to characterize processes involving work and heat, a second physical quantity, entropy, was required to accommodate the unidirectional flow of heat. The Second Law was then recast in the form of entropy remaining the same, or increasing, in any natural process. But even with the introduction of entropy the Laws of Thermodynamics and those of Newton remain fundamentally incompatible. It is our contention that the resolution of such fundamental incompatibility leads to new ways of knowing and the resolution of physical paradox is no different. So what is the empirical paradox (EP) and what is the new physics that its resolution entails? The predictable dynamic behavior of matter is a direct consequence of Newton’s equations of motion, and consequently its properties are determined by Newton’s laws. The equations have the property that if the direction of time is reversed the equations work just as well as they did for time flowing normally. A consequence of this property is that starting from an initial state the equations of motion predict a final state, such that if the direction of time is reversed once the final state is reached the resulting dynamics unwind the previous motion and the initial state is reformed. Both the forward and backward behavior are predicted by the equations of motion, so that the equations are reversible in time.

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The one-way flow of heat in thermodynamics indicate that the equations of heat flow are not reversible in time. This is incompatible with Newton’s force law and therefore constitutes a paradox, for how can a physical process be both time-reversible and time-irreversible simultaneously. This EP is based on two fundamental physical models of the world and we emphasize that over a century and half after this particular EP was identified it remains unresolved. But the search for its resolution and even the partial successes of that search have produced new ways of understanding the world. Ludwig Boltzmann in a mathematical tour de force gave a comprehensive foundation to the kinetic theory of gases in an 1872 paper, deriving the statistical behavior of gas particles from the deterministic equations of Newtonian mechanics. Five years later he published a sequel in which he constructed a statistical interpretation of entropy, which enabled quantification of the entropy in terms of the total number of distinct ways the energy can be shared among the gas particles, thereby resolving the irreversibility paradox. This expression for the entropy is carved on his headstone as shown in Figure 1.1. Although Boltzmann’s arguments turned out to be flawed with regard to resolving the time-reversal paradox, they did put statistics on a firm physical foundation and uncertainty became a physical aspect of the world in which we live. A remarkably entertaining and scientifically accurate historical account of the forgotten science leading from the first mysterious indicators of statistics in complex phenomena to its acknowledged ubiquity in modern science is given by Mark Haw in his book Middle World [112]. That particular story started with the seventeen century Scottish botanist Robert Brown and his discovery of the erratic motion of particles in this middle world that in size is between atoms and animals. Brown was searching for the secret of life, which he at first thought he had observed with the unquenchable motion of pollen motes in water. But on carrying out multiple experiments on a variety of inanimate as well as animate particle in the three months of June, July and August of 1827, determined the motion had to do with particle size and not life. Haw goes on to trace

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Figure 1.1: This is a bust of Ludwig Boltzmann (1844 -1906) with his equation expressing the relation, between entropy S and the number of microstates W available to the energy, carved into the stone above his head. the twists and turns of scientific discovery, involving the giants of physics, what they saw and what they did not see, up to the present day statistical treatment of uncertainty. Wave-particle duality: Physics encountered an equally unsatisfactory state of affairs at the turn of the twentieth century, when microscopic entities, quanta, were observed to have the particle property of spatial localization, as well as, the wave property of spatial extension. It was an established truth, at a time when the Industrial Revolution was shifting into high gear, that light consisted

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of waves, as verified by nearly two centuries of experimental measurements of interference, refraction and diffraction. Consequently, to say that the community of physical scientists was surprised by Einstein’s 1905 paper explaining the photoelectric effect as being due to light being made up of a string of discrete quanta, is a gross understatement. The paper was, of course, properly couched in the language of Planck’s quantum hypothesis, but it was only 17 years later that Einstein was awarded the Noble Prize in Physics for this remarkable work. Note that the phrase quantum hypothesis constitutes a heuristic assumption without a theory. It is referred to as Planck’s quantum hypothesis, because one of the towering figures in science at that time, Max Planck, had found it necessary to introduce the idea of energy occurring in discrete packets, called quanta, in order to explain another mystery in physics, that being black body radiation. This hypothesis had been published in 1900, again without an underlying physical theory. Planck was to be awarded the Noble Prize in Physics for his further development of the quantum hypothesis, just four years before Einstein received his. This earlier discussion need not concern us here. Suffice it to say that neither of these giants realized the full implications of what they had become partners in creating. So what does this have to do with empirical paradox? Recall that empirical paradox is the result of an observable phenomenon having at least two measurable properties that are logically inconsistent and the logic is the result of our theoretical understanding of the phenomenon. Our understanding of the nature of light constitutes such an EP. The beautiful separating of the colors in the sun’s light into a rainbow is explained by the wave mechanism of refraction and the diffuse edge of your shadow on the ground is explained by the wave mechanism of diffraction. These are just two of the literally hundreds of examples of the manifestation of the wave properties of light. So it is firmly experimentally established that light is a wave and therefore has extension in space. On the other hand, Einstein argued that an electron is ejected

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from a conductor by shining light on it, because light can be described by packets of energy, that is by quanta having energy given by the product of Planck’s constant and the frequency of the light. The energy of the ejected election is an integer number of such quanta. This phenomenon is what enables today’s solar panels to transform sunlight into electrical energy and the subsequent development of quantum theory is the foundation of Information Age technology. So it is firmly experimentally established that light is a particle (quantum) and is localized in space. Can an entity be both extended and localized in space simultaneously? Spooky, eh? Nature says yes it can be both through thousands of reproducible experiments. This was the EP facing the physicists of the early 20th century. They resolved the paradox by introducing an epistemological argument that came to be called wave-particle duality interpretation of quanta. This interpretation of quanta was developed, in the 1920s, by Werner Heisenberg and Niels Bohr, and was subsequently adopted by the majority of physical scientists as the Copenhagen Interpretation of quantum mechanics. In this view of quantum theory, quanta are thought to be neither particles nor waves until they are measured. The measuring process itself induces a collapse of a quanta into either the state characteristic of being a particle, or a state characteristic of being a wave. Prior to measurement a quanta is either both or neither, but the pre-measurement state is not knowable, since it cannot be accessed by experiment. This assumption of unknowability is a statement about the fundamental character of physical reality. The quantum paradox is resolved by maintaining an either/or dichotomy of that part of nature that is accessible by experiment, but it is both/and for that part of nature that is not experimentally accessible, for a physical scientist that is the best we can do. At least for the time being. The quantum paradox has been the focus of often heated argument over the last century. The wave-particle duality, or quantum paradox resolution, is now so much a part of the culture of physical science that many students of physics think it is either quaint, or

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arcane, to discuss the matter seriously, being more metaphysics than physics. But this misses the larger point having to do with paradox in general, how we resolve EPs in the natural, social and life sciences and what that resolution entails.

1.1.2

Complexity

We are all deeply conscious today that the enthusiasm of our forebears for the marvelous achievements of Newtonian mechanics led them to make generalizations in this area of predictability which, indeed, we may have generally tended to believe before 1960, but which we now recognize were false. We collectively wish to apologized for having misled the general educated public by spreading ideas about determinism of systems satisfying Newton’s laws of motion that, after 1960, were to be proven incorrect. — Lighthill [140] In the middle twentieth century a number of isolated individuals working alone, as opposed to the community of scientists as a whole, began to recognize that the tools of analysis available to them were not up to the job and began to study complexity itself as the focus of research. The concern was over understanding the patterns, order and structures that emerged from nonequilibrium systems having chaotic dynamics, as well as, those that shared matter and energy with their environment, while retaining a low entropy, and whose endogenous dynamics produce self-organized critical states. Ekeland [78] summarizes one view of complexity in his book about time, mathematics and how they have been used in the formalization of the science of complexity for nearly half a millennium. The book is a discursive popularization of the way in which the simple mathematical laws describing the universe give rise to complexity. He begins with a detailed critique of Kepler’s three laws of planetary motion emphasizing their approximate nature. The limits of quantitative methods (accurate but limited in scope) for

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determining planetary trajectories led the 19th century mathematical genius Poincaré to develop equally rigorous qualitative methods (greater range but less precision) to describe their behavior, thereby questioning the efficacy of prediction in science; commenting that all the perturbation calculations of trajectories, such as those of lunar orbits, are asymptotically divergent. Thus, the stage was set for the introduction of the unpredictable, weaving together order and chaos, leading to complexity and the ‘butterfly effect’. This last phrase was the result of an off-hand remark made by Ed Lorenz that his results in meteorology implied that the flapping of a butterfly’s wings in Brazil could stir up weather patterns that might ultimately result in a tornado in Texas. We now step from the complexity of reversible microsystems to irreversible, dissipative macrosystems with the introduction of thermodynamic potentials to determine dynamics. We also introduce catastrophe theory to identify and categorize qualitative changes in a system’s behavior, such as phase changes in physical systems and tipping points in social systems. A very different view of complexity is presented by Morin [170] in a collection of essays on the various metaphysical, as well as, practical implications of complexity. He introduces the notion of blind intelligence, resulting from the simplification of complex phenomena to make them orderly and predictable. In his approach to understanding complex phenomena Morin emphasizes the necessity to transcend the limitations of linear logic to resolve paradox: ”The modern pathology of mind is in the hyper-simplification that makes us blind to the complexity of reality.” He goes on to attack the notion of either/or choices and discusses the formation of alternatives, again emphasizing that our logic-based mathematical models are illsuited to handle true complexity, which are marred by uncertainty. Thus, he asserts that even more than self-organization, complexity involves self-creation and the paradox that such understanding entails, including the observer being part of what is being observed. He, more or less, ends with the notion of complex thinking; a process that is capable of unifying incompatible ideas by pragmatically af-

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firming that nothing is isolated and everything is interrelated. The overwhelming significance of complexity in the modern world was first systematically articulated by the mathematician Norbert Wiener in a work that introduced into science a new paradigm, cybernetics. In his book of the same name [265], Wiener sets the stage with a brief introduction to contemporary scientific luminaries that had contributed to the formation of this nascent science, capturing a new view of science and society after World War II. Cybernetics is concerned with how a quantity was being exchanged between humans and machines and how this quantity facilitated communication and could be used for control. This newly identified quantity was information and its measure is entropy, as hit upon by N. Wiener, R.A. Fisher and C. Shannon at essentially the same time, albeit for different purposes. Wiener discusses some concepts that have challenged the imagination of thinkers for millennia and others that they were only then becoming aware of; the irreversibility of time entailing uncertainty; the necessity for linear feedback loops for stability in physiological systems, homeostasis; ergodic theory and information, with the implied use of probability density functions (PDFs) to construct the entropy measure of information; the analogy between computing machines and the brain, at a time when the state of the art was the vacuum tube computer Eniac. Wiener was not bashful about speculating concerning the potential utility of his new paradigm of science in areas for which he had no special training, but he had spent his life discussing these ideas with the finest minds in the world. He was convinced, and argued convincingly, that because of an area’s scientific complexity, such as psychopathology, that the computing machine and the human brain have much in common and each could significantly benefit by study from a perspective focusing on their complexity. In retrospective I would say that this work [265] ushered in the modern world with the first version of complexity science (Version 0.1) and in so doing was prescient in anticipating our present day dependence on information. Given the multiple definitions of complexity and the variety of phenomena that have been described as being complex, but bear no

1.1. GETTING ORIENTED

15

mechanism in common, we are going to adopt a strategy successfully employed by the physical/mathematics community half a century ago. In the middle of the last century it became clear that dynamic systems come in two forms, those that are linear and those that are not, and the richness of dynamic structures of the latter bear no resemblance to those of the former. Consequently, the community tacitly agreed to adopt the nomenclature of calling the latter system nonlinear and thereby to define its members by what they are not. This led some to criticize this as describing a "zoo of non-elephants", and in that they were accurate. However, it has turned out that this modest change in nomenclature has bee very useful. We therefore replace the term complex with the term nonsimple in the remainder of this essay, but in discussions that reference previously done work were the term complex was used we also use that term, in the hope of avoiding confusion.

1.1.3

Aristotelian Logic

No, no, you’re not thinking; you’re just being logical. — Bohr [41] We argue that complex (nonsimple) phenomena, by virtue of being nonsimple, entail paradox and in so doing, violates the two thousand year Western tradition of Aristotelian logic; the tradition being that a statement A and its negation A (not A) cannot be simultaneously true. Said the other way around, a simple system cannot contain contradictions, by definition, and is therefore free of paradox. Even the simplest example, such as the logical paradox given previously, has two poles that contradict one another. The fact that the truth of such a statement cannot be determined rules out the notion that such a statement is logically simple, which is part of the surprise, because its construction appears so uncomplicated. In a nonsimple organization, consider the paradox of having both A and A within a common context. This could be viewed as the Yin and Yang of Taoism; the thesis and antithesis existing together

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as depicted in Figure 1.2. The combination of A and A existing together manifest a duality in opposition to one another, but they are also synergistic and interrelated within the larger system, as in the wave-particle duality of quantum phenomena. The resolution of the paradox that is implicit in Figure 1.2 requires abandoning the either/or - thinking of Aristotelian logic that implies A or A, but not both together, to embrace the both/and - thinking of A and A together, achieved by balancing the tension of contradiction dynamically. This management of the tension of paradox, whether within individuals, groups or organizations produces flexibility and resilience, while fostering more dynamic decision making [211].

If our contention regarding the essential nature of paradox in nonsimple phenomena is true we would expect to see EP within every scientific discipline as the discipline matures over time. This is, in fact, what is observed. Empirical paradox has been observed in every scientific discipline, examples of which we discuss in some detail are the altruism paradox (AP) in macroevolutionary biology [68] and sociobiology [70, 268], organizational paradox in management [211], strategy paradox in economics [209], and we have already discussed wave-particle duality in physics. The result has been that the resolution of EP, which can be a subtle concept, whose nuanced definition shifts according to the discipline in which it is used, is discipline-specific and consequently often cannot be used to resolve EP within other disciplines. This implies that the mathematical models, designed around a specific mechanism, are often not generalizable to phenomena outside that discipline. We believe that this limitation has been overcome using network theory, as we subsequently explain. But for the time being we focus on the myriad ways paradox disrupts our simple mental pictures of people, society and the world in general and consequently how we are forced to think about them.

1.1. GETTING ORIENTED

17

Figure 1.2: top: A nine year old’s view of the world (Gabriel West, with permission). bottom: In Taoism the fusion of the two cosmic forces, the light and dark representing respectively Yin and Yang, each containing a ’seed’ of the other. This becomes more than a metaphor for paradox when dynamics is properly taken into account. But perhaps with something less than cosmic forces.

18

1.2

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Visual Paradox A visual paradox is when you look at something and you see something that can’t exist or doesn’t exist despite the fact you are looking at it. — Curfs [65]

Most of us believe that we see the world objectively, not distorted by our mental constructs or the way we think the world ought to be. But like most unexamined beliefs this turns out not to be true. The least complicated example of the dependence of what we see on what we believe, turns out to be an optical illusion, also known as an ambiguous figure, or for our purposes here, what we call a visual paradox. An artist who was a master of visual paradox was Escher [82], whose favorite image seemed to be that of a person walking down (up) a flight of stairs only to arrive at the floor above (below) him. After some analysis of the image the deceptive use of how objects occupy space that entails the visual contradiction is found, but the cognitive understanding does not completely quiet the discomfort. His etchings are often so filled with visual contradictions that like in Figure 1.3 only a section of one such print is displayed and this contains approximately half a dozen contradictions. I cannot be more precise as to the number, because what constitutes a contradiction depends on which way is up. I remember how surprised and intrigued I was the first time I saw a visual paradox in a book I was reading. An image that matches my memory is given by the classical face-vase ambiguous image found in almost every college general psychology book. On first viewing the image one either ’sees’ a vase or two faces peering at one another, but whichever registers on the brain first is not important, because that percept is not stable. That percept will not persist over time. Instead the percept, say it is the vase, is replaced by the other, the opposing faces, which is also not stable in time. This forms an unstable, or a multi-stable, perception in which the exchanging of percepts in the brain flickers like the lights in a disco club of the 1970s.

1.2. VISUAL PARADOX

19

Figure 1.3: Here is a cut-out from an Escher print that defies our notions of up and down., with a person emering from a staircase within the wall, and two people going down opposite sides of the same staircase. Try rotating the sketch 90 degrees in either direction. It will appear just as right, or just as wrong, from either perspective.

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Scientists have been studying such multi-stable perception phenomena for nearly two centuries and parallel strategies for their understanding have evolved: the sensory, or bottom-up, and the cognitive, or top-down explanations. It is not the intent of this essay to layout the detailed science underlying each of the exemplar EP. However, we do provide a bit more detail in the discussion of ambiguous figures, because it has a wealth of empirical data supporting an insight that with only minor changes becomes universal in the context of the mathematical model of the brain developed in Chapter 5. That insight is [136]: The bottom-up approach assumes that perceptual reversals result from cycles of passive adaptation, recovery, and mutual inhibition of competing neural units or channels in early visual areas....The top-down approach, in apparent contrast, assumes perceptual reversals as the result from active high-level/ cognitive processes like attention, expectation, decision-making, and learning. Much like the logical paradox given earlier, there is a conflict between the two percepts of the single physical image. In the logic case it was the truth of the statement that did not persist from one reading to the next. Here it is not the truth that flickers, but the percept that flashes back and forth and that appears to have a quite different source. Or does it? When I first look at Figure 1.4 my mind focuses on one percept or the other, either/or, but not both. This observation is not unlike the results obtained from dichotic stimulation with different monocular percepts, or as also put by Kleinschmidt et al. [135]: An observer can only be aware of one of the two incompatible percepts at any given moment, but over time experiences striking spontaneous reversals (’flips’) between the two percepts.

1.2. VISUAL PARADOX

21

Figure 1.4: Some people first see two faces looking at one another, while some clearly see a vase. This is the face-vase paradox and whichever is seen first, eventually the other will be seen, and then one’s perception will flicker between the two. The neural processing of the different percepts is done in different parts of the brain and has been recorded using functional magnetic resonance imaging (fMRI) for bistable percepts having a constant physical stimulus [135]. The subjects in the experiments were instructed to register their experience of the visual scene by means of key-presses denoting a perceptual transition, or the retention of one percept, or the other. This enabled the scientists to associate a visual experience with an fMRI image of brain activity: part of the brain lights up for the experience of one percept, an entirely different part of the brain lights up for the experience of the other percept;

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the transition from one percept to the other coincides with observed bilateral activity. Figure 1.5 suggests a way to understand this visual paradox mathematically by modeling it using a double-well potential. Imagine a ball rolling down an inclined plane, as depicted in Figure 1.5. If the ball is released on the curve below the top of the barrier it will slosh around on that side of the barrier, losing a little energy with each pass, until it comes to rest at the potential minima on the left. If however, it is released at a level higher than the barrier, it will roll down the hill and then up and over the barrier, rolling up the curve on the right hand side and coming to a stop. It will then reverse direction and repeat the process until friction bleeds it of sufficient energy that it can no longer cross the barrier. It will come to rest on whichever side that sufficient loss of energy occurs.

Figure 1.5: Depicted is a generic quartic, or double-well potential. It has two minima separated by a barrier. A ball is shown with an energy that exceeds that of the barrier, so the ball will slosh back and forth forever in the absence of friction,

1.2. VISUAL PARADOX

23

Of course the double-well potential we envision in the case of visual paradox has nothing to do with balls and gravity. This potential function is a mechanical analog of how the attention in the brain divides over time between two percepts, those being the different perceptual states of the one stable stimulus, the ambiguous figure. Imagine that the perceptual potential describes the level of attention required to see one percept, or the other, each percept being located at a potential minima in direct analogy with physically separate regions of the brain that light up in the fMRI. Once the brain occupies a minima, we experience one percept, and it requires effort exceeding the barrier height to change. According to the top-down school of thought cognition increases the level of attention necessary to overcome the barrier separating one minima (percept) from the other. Whereas, according to the bottom-up school of thought there are spontaneous fluctuations in the attention level that can drive the brain from one minimum to the other. Therefore to complete the analogy with the visual paradox we could assume one of two things: either the height of the potential barrier fluctuates in time, or the level of attention fluctuates in time. In either case, at intermittent times, the level of attention exceeds the barrier height and as a result our focus can shift from one percept to the other. The barrier separating the potential minima is initially high, so the brain is focused on one perceptual state or the other. However, with practice the barrier height can be changed, as can the level of attention (analog of particle’s energy). What is important is the level of attention relative to the barrier height, both of which can be changed, such that the rate of switching between percepts can be increased or decreased. Eventually it requires little or no effort to switch back and forth between them. In fact, given enough practice, it ought to become possible to remove the potential barrier entirely. When this removal is achieved the paradox is resolved and the brain ought to be able to register both percepts at the same time. Of course, one can also accomplish this gestalt by increasing the level of attention above the existing barrier, but this enhanced level must be maintained in order for the simultaneous recognition of the percepts

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to persist. A third option is to have a combined level of attention and fluctuations that matches the barrier height, a kind of stochastic resonance. Such stochastic resonance has been observed in the performance of multiple other cognitive tasks, including the transmission of information when it is in the form of a harmonic signal [187] and the enhancement of neuron sychronization by external noise. The more elaborate the physical image the greater the level of attention required to resolve the visual paradox. An example of the latter can be achieved by examining the image depicted in Figure 1.6. Here the brain registers either a young girl in a bonnet, a wench, her ear partially covered by flowing hair, with her chin pointing to the left and a ribbon around her neck, or alternatively a witch with a large nose capped with a wart. The wench’s ear becomes the witch’s eye and the wench’s ribbon morphs into a tentative smile on the witch’s lips. When I look at this figure I stay with one percept, or the other, longer than in the simpler case of the face-vase, but more importantly it took me much longer to hold the two percepts of the wench-witch visual paradox, with the same degree of confidence that I did the simpler face-vase ambiguous figure. The reason for introducing the perception potential is that it provides a relatively painless way to experientially quantify the varying levels of difficulty in: (1) discriminating between the complimentary percepts and (2) experiencing the two percepts simultaneously. In resolving the visual paradox, by being able to experience both percepts simultaneously, we have operationally made the transition from the either/or way of resolving paradox to the both/and way. Is there any difference between the visual paradox and wave-particle duality? Aside from the fact that the physics paradox involves deliberation and the visual paradox involves a much faster form of thinking. We subsequently argue that thinking has two components, one that is fast and is usually associated with intuition and the other is slow and is usually associated with cognition. The fast component explains the rapid transition between percepts once it is initiated, while the slow component explains the dwell time with either percept

1.2. VISUAL PARADOX

25

Figure 1.6: You either see a wench from the rear with her chin pointing to the left, or an witch with the wench’s chin morphed into the witch’s rather large nose.

and the reluctance to initiate a transition. A bimodal potential is used in Chapter 5 to determine the dynamics of paradox, including the visual. We use both the fast and slow thinking to show how the brain can be made to flicker back and forth between the percepts, or to retain both simultaneously. We develop the mathematics for this perceptual potential based on the dynamics of nonsimple networks, which can model the level of attention your brain exerts to resolve the visual paradox. The dynamic nonsimple network is shown to be sufficiently flexible to model

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EP in a multitude of other disciplines, as well.

1.3

Previews of Coming Attractions

As shown in this chapter a visually ambiguous figure is the resolution of a paradox, since a single ambiguous figure is a both/and realization of the two percepts that produces the ambiguity within our brain. It is the cognitive interpretation that gives rise to paradox, as can be explained in a number of ways. We should not have been, but were, quite surprised when we ran across the bimodal potential interpretation of the wench-witch figure in the book The Dynamics of Ambiguity by Caglioti [51]. In his book Caglioti went even further and used quantum mechanics to exploit the symmetry of the two percepts. We also have occasion to use quantum reasoning in the resolution of paradox, which we do in Chapter 5. Caglioti lamented the fact, a quarter century ago, it was not possible to go further because of the lack of experiments with which to determine the parameters in his hypothesized potential. Recall, as we discussed, the experiments determining the areas of brain activity associated with the two percepts were not published until six years after his book came out [135], but they made no mention of his insights. This is an indication of the all too frequent failure on the part of investigators to read about what had been previously done in a research area of interest, but had been published outside the ostensible disciplinary area of the problem, e.g., research on a neurophysiology problem published in a physics journal. This is particularly true with phenomena such as the altruism paradox (AP) may appear in quite different disciplines, such as biology, economics and sociology. But to be fair this occurred before the Internet made the search for overlapping research interests relatively painless. To clearly establish the ubiquity of EP the discussion in Chapter 2 is devoted to reviewing the nature of paradox by way of example, as it has been revealed in numerous disciplines over the last two centuries. Some forms of paradox have long pedigrees, resulting

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in an impressive literature in which scientists, authors and other scholars have taken turns at cracking this or that particular chestnut, with varying degrees of success. Other kinds of paradox have only recently been identified, as has already been mentioned. A second reason for presenting these examples of EP is to suggest the intimate connection between paradox and nonsimplicity and to make the case that the former is entailed by the latter. A preliminary explanation of paradox is given in Chapter 3, where we briefly review how science has come to understand nonsimplicity through the use of statistics. The first successful attempt to understand the divergence from the simple to the complicated was provided by the central limit theorem and normal statistics, characterized by a mean and variance. But we show that the conditions for the central limit theorem of the Machine Age are violated by the truly nonsimple phenomena of dramatic importance in the Information Age. The latter are quite often described by Pareto’s inverse power-law (IPL) statistics, which we discuss. The Industrial Revolution of the nineteenth and early twentieth centuries argued against the existence of paradox, just as it was not accepting of uncontrolled variability. On the other hand, the Information Revolution of the late twentieth and early twenty-first centuries accepted all manner of variability and subsequently even embraced paradox, albeit reluctantly. Society consists of interconnections by family, friends, acquaintances, or work relations, making it unavoidable that an individual’s behavior, or decisions, depend, at least in part, on the choices made by other people. The social network in which we are embedded influences the opinions we hold, the products we buy, as well as, the activities we pursue. Therefore exploring the basic principles that give rise to social processes in which individual behavior aggregates into collective outcomes has provided significant insight into the networked decision-making process. In Chapter 4 we introduce a network-based model of decision making that is used as the basis for an eventual understanding of EP in Chapter 5. The detailed mathematics underlying the model

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is relegated to an Appendix, so as to not interrupt the flow of the narration. A number of calculations that reveal the interpretive context of the model is given in the text. The decision making model (DMM) assumes that the intuitive component of decision making is based on an individual imitating the opinions of others and parallels a physics-based model of phase transitions. The DMM shares a number of the mathematical properties of the physics models, such as scaling, criticality and universality, but has severed its connections to its physical heritage and relies solely on sociologically based assumptions. This model makes decisions fast and without rationality. A second modeling strategy discussed in Chapter 4 addresses rationality in decision making and this is game theory, or more properly, it is evolutionary game theory (EGT). This modern form of game theory incorporates dynamics into the decision making process allowing individuals to modify their behavior over time in response to how others in the network behave. The review of EGT in this chapter is brief, given the large number of excellent books and the extensive literature on the subject of game theory that is readily available to the reader. However, we do compare and contrast DMM with an evolutionary game model (EGM) in this chapter. Chapter 5 pulls the two modeling strategies together to form a new kind of two-level model. In this composite model the two networks, one modeling DMM and the other modeling EGM, are coupled together in a unique manner that enables the internal dynamics of a network to spontaneously converge on a critical state that simultaneously satisfies the self-interest of each member of society, while also achieving maximum benefit for society as a whole. The dynamics of what we call the self-organized temporal criticality (SOTC) model [151, 152, 262] is self-adjusting and does not require the external fine tuning of a parameter to reach criticality. In the final chapter we explore the interconnections between criticality, crucial events and temporal complexity. The purpose being to clarify the underlying reasons for the interpretations that may have been obscured by the general discussions of the earlier chap-

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ters. The phenomenon of complexity (nonsimplicity) matching is used to emphasize the conditions necessary to efficiently exchange information from one nonsimple network with another. The information transfer mechanism can be traced back half a century to Ross Ashby [17] and his book on cybernetics. Unlike this earlier work we argue that the nonsimplicity of interest for the resolution of EP can be expressed in terms of crucial events, which are renewal, with IPL statistics of inter-event time intervals, generated by criticality in general and the SOTC model in particular. Nonsimple processes, ranging from physical to biological and through sociological, host crucial events and the fact that they drive the information transport between nonsimple networks is reviewed.

Chapter 2

Kinds of Empirical Paradox (EP) To clearly establish the ubiquity of EP the discussion in this chapter reviews the nature of paradox by way of example, as it has emerged in numerous disciplines over the last two centuries. Some forms of paradox have long pedigrees, resulting in an impressive literature in which scientists, authors and other scholars have taken turns at cracking this or that particular chestnut, with varying degrees of success. As society has evolved new kinds of paradox have been identified, as has already been mentioned. A second reason for presenting these examples of EP is to suggest the intimate connection between paradox and nonsimplicity and to make the case that the former is entailed by the latter. 31

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2.1

Contradiction in theory A man should look for what is, and not for what he thinks should be. — Einstein [77]

The downside of any theory is that over time it slowly replaces the phenomenon it was developed to explain, at least in the mind of the practitioner, including that of the scientist who developed the theory. This may, in part, be a fundamental limitation on how we humans think. It is much easier to formulate arguments using the well-defined constructs of a mathematical theory and to marvel at its elegant simplicity than it is to determine the consistency of such arguments directly from data. Of course, if the formal theory is a faithful representation of the behavior of the heuristic theory then the two ways of arguing should coincide. But when they do dovetail we are filled with awe and that proverbial tingling up one’s spine is a testament to how rare such clarity is actually achieved in practice. It is no wonder that when a theory has undergone a number of such challenges and emerged victorious, we are reluctant to accept the outcome of new experiments that cannot be explained by the same existing theory. We are surprised, intrigued and also attracted by the failure of theory to explain new experimental results. The first reaction of a theoretician to a divergence between theory and experiment is to question the experiment and the first reaction of the experimentalist is to question the theory. There is nothing wrong with these visceral reactions, as long as when the theoretician finds the experiment to be valid, the light of critical analysis is then turned against theory, and when the experimentalist finds the theory to be valid it spurs the reexamination of the experimental protocols. We are even more reluctant to embrace suggested conceptual changes that contradict long-held beliefs that not only have prior experimental support, but have been accepted by a large number of our friends and colleagues in that area of study. These nascent concepts are dictated by the additional experimental results and

2.2. ALTRUISM PARADOX (AP)

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must be made compatible with existing theory. This is the situation that many scientists all too often encounter when their research concerns nonsimple phenomena. They are forced to use the concepts developed for the understanding of simpler processes, but which contradict one another when they become the constituent parts of an explanation of more subtle phenomena. The maiden theory must resolve the EP, while continuing to explain the old results. Rather than proceeding with an abstract discussion in which every word has to be properly qualified, for the sake of expediency, let us proceed by way of example. In this chapter we choose to examine paradox from a wide range of disciplines in an attempt to determine what they may have in common. This review hopefully lays sufficient groundwork for the theory of EP presented in subsequent chapters.

2.2

Altruism Paradox (AP) The world is a dangerous place to live; not because of the people who are evil, but because of the people who don’t do anything about it. — Einstein [77]

In every war and natural disaster there are men and women who become heroes; people who risk their lives for strangers, seeking neither reward, nor acclaim. Ordinary people stand in awe of such sacrifice and wonder, if called upon, would they be able to rise to the challenge. The truth is that most of us do not know how we, or others, will act under those conditions; we do not understand what enables one person to step into a dangerous situation, while others hesitate and avoid putting themselves at risk. How and why we respond to what we are called upon to do in times of crisis remains a mystery, but in spite of that we value the act and acknowledge its significance for the individual and for society. Take for example the only woman ever to be awarded the Medal of Honor, Dr. Mary Edwards Walker (1832-1919), the first female

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CHAPTER 2. KINDS OF EMPIRICAL PARADOX (EP)

U.S. Army surgeon. This is the highest award in recognition of bravery that the United States can bestow on a member of its military; or as stipulated in the citation: ...the highest U.S. military decoration...for gallantry and bravery in combat at the risk of life above and beyond the call of duty. Mary was born in the small upstate New York town of Oswego on November 26, 1832. She was brought up on a farm, attended the first free schoolhouse in Oswego, which was started by her parents, and later completed her non-professional education at Falley Seminary in Fulton, New York. She subsequently paid her way through Syracuse Medical College, from which she graduated with honors in 1855, at the age of 23, the only woman in her class. Six years later, at the start of the Civil War, after a marriage to a classmate from medical school ended in divorce, due to his philandering, she tried to join the Union Army as a surgeon, but was rejected because of her gender. She declined the offer to become a nurse and instead volunteered to be an unpaid field surgeon near the Union front lines. Some two years later Dr. Walker was appointed assistant surgeon of the 52nd Ohio Infantry. As a medical doctor she frequently ventured alone into enemy territory to provide medical care for civilians and to assist Confederate Army field surgeons. She was alone because the Union soldiers thought it too dangerous and refused to accompany her on these expeditions. It is probably fair to say that she was one of the only physicians to assist both sides in the Civil War with complete disregard for her personal safety. She was captured by Confederate troops on April 10, 1864, and arrested as a spy, shortly after helping a Confederate Army surgeon perform an amputation. She was a prisoner of war until she was exchanged for a Confederate surgeon from Tennessee on August 12, 1864. After the war, Dr. Walker was recommended for the Medal of Honor by Generals William Tecumseh Sherman and George Henry

2.2. ALTRUISM PARADOX (AP)

35

Figure 2.1: Mary E. Walker MD was the first female US Army surgeon. She received the Medal of Honor, shown here, for her heroic service during the American Civil War. Thomas. On November 11, 1865, President Andrew Johnson signed a bill to award her the medal. She is shown wearing the Medal of Honor in Figure 2.1. The medal citation reads as follows: Whereas it appears from official reports that Dr. Mary E. Walker, a graduate of medicine, "has rendered valuable service to the Government, and her efforts have been earnest and untiring in a variety of ways," and that she was assigned to duty and served as an assistant surgeon in charge of female prisoners at Louisville, Ky.,

36

CHAPTER 2. KINDS OF EMPIRICAL PARADOX (EP) upon the recommendation of Major Generals Sherman and Thomas, and faithfully served as contract surgeon in the service of the United States, and has devoted herself with much patriotic zeal to the sick and wounded soldiers, both in the field and hospitals, to the detriment of her own health, and has also endured hardships as a prisoner of war four months in a Southern prison while acting as contract surgeon; and Whereas by reason of her not being a commissioned officer in the military service, a brevet or honorary rank cannot, under existing laws, be conferred upon her; and Whereas in the opinion of the President an honorable recognition of her services and sufferings should be made. It is ordered, That a testimonial thereof shall be hereby made and given to the said Dr. Mary E. Walker, and that the usual medal of honor for meritorious services be given her.

Then there is the other side of the human coin, which some would say is the more common and that is selfishness. The mystery of selflessness is particularly compelling because the behavior of the hero is completely inconsistent with what we are taught by most of science is the fundamental nature of being human. While it is true that many, if not most, religions teach a different version of human nature, one more benign, science typically concedes that people have a dark side that all too often dominates their behavior. Thomas Hobbs, a 14th century English clergyman, summarized it as life outside the boundaries of society being: "solitary, poor, nasty, brutish and short". Although reluctantly, science has been forced to agree that this summary was not inaccurate. But what of heroism, art, music and the other matters of the spirit? What is their origin? The puzzle has to do with actions not fitting the accompanying thoughts; the lack of understanding of the incompatibility in the way we act to the way we think. In our minds we explore all the viable

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alternatives before entering into a dangerous situation, but few of us, rehearse sacrificing our life for another individual, or for a cause. The latter is a different and an even more subtle enigma, since it is a consciously chosen act consistent with one’s thinking, rather than being in conflict with it. It is the contradiction that we address in this essay and which we attempt to understand, subsequently using a mathematical model to focus our thinking.

2.2.1

Multilevel natural selection

There are many versions of the altruism paradox (AP), which is a consequence of this particular paradox being identified in a number of different disciplinary contexts. This multiplicity argues against there being any one discipline-specific mechanism able to account for its occurrence in them all. For clarity let us consider a specific AP; one that has been tantalizing scientists for a couple of centuries. It was first recognized by Charles Darwin that some individuals in a number of species act in a manner that although helpful to others, may jeopardize their own survival and yet this property is often characteristic of that species. He identified such altruism as contradicting his theory of evolution [68]: It is extremely doubtful whether the children of such [altruistic] individuals would be reared in greater number than the children of selfish and treacherous members of the same tribe...Therefore it hardly seems probable...that the standard of their excellence could be increased through natural selection, that is, by the survival of the fittest. It is not surprising that the proposed theory to resolve this AP crossed the boundary between biology and sociology to form the new discipline of sociobiology. The theory of sociobiology was developed in the last century, at least in part, for the purpose of explaining how and why Darwin’s theory of biological evolution is compatible with

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sociology. Or more succinctly, how social behavior can be explained in terms of Darwinian evolution. In their rethinking of the theoretical foundations of sociobiology Wilson and Wilson [268] did a remarkable job of explaining the reasons for the disarray in which the modern theory of sociobiology, at that time found itself. Their review is of interest to us here because the status of the theory is the result of a half century of attempts to resolve the AP and is crystallized in a quotation with which Wilson and Wilson open their seminal paper [68]: It must not be forgotten that although a high standard of morality gives but a slight or no advantage to each individual man and his children over the other men of the same tribe...an increase in the number of well-endowed men and an advancement in the standard of morality will certainly give an immense advantage to one tribe over another. The crux of the problem lies in the distinction between what is best for the group to function as an adaptive unit resulting from interactions within a social group, compared with interactions between social groups. This failure of the group to achieve maximum fitness through the self-sacrifice of individuals within the group forms the AP. Darwin proposed a resolution to this problem by speculating that natural selection is not restricted to the lowest element of the social group, the individual, but can occur at all levels of a biological hierarchy, as paraphrased by Wilson and Wilson [268]: Selfish individuals might out-compete altruists within groups but internally altruistic groups out-compete selfish groups. This is the essential logic of what has become known as multilevel selection theory. Difficulties arose in sociobiology with the rejection of Darwin’s multilevel selection hypothesis in the 1960s and the subsequent ambiguities, contradictions, and confusion that framed the subsequent

2.2. ALTRUISM PARADOX (AP)

39

theoretical discussion of the next half century in terms of the individual. We do not believe we can improve on the detailed critique of the alternatives to multilevel selection theory presented by Wilson and Wilson, even though there are a few points with which we may not completely agree. However, going into that level of detail here is not necessary, since the reconvergence of scientific consensus on the use of multilevel selection theory to resolve the AP in sociobiology has been accomplished. We concur with the multilevel selection hypothesis of Darwin, but not with the biological requirement of the need for strictly altruistic individuals. Historically, the alternative theories to multilevel selection were based on the failure to make the altruistic actions of individuals compatible with the more universally accepted action of individual acting in their own self-interest. In particular, we disagree with the closing encapsulation of Darwin’s original insight [268]: "Selfishness beats altruism within groups. Altruistic groups beat selfish groups. Everything else is commentary." Subsequently we present a mathematical argument that does not rely on the either/or choice for the behavior of the individual.

2.2.2

The invisible hand

Adam Smith posed essentially the same paradox in a economic/social setting in An Inquiry into the Nature and Causes of the Wealth of Nations by maintaining that people act in their own self-interest and in so doing, society as a whole benefits [209]: It is not from the benevolence of the butcher, the brewer, or the baker that we expect our dinner, but from their regard to their own interest. On the biology side: How can natural selection be explained genetically, using microbiology, since sometimes an individual’s behavior would be an expression of the selfish gene [70], whereas at other times it would be the expression of an altruistic gene?

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On the social side: How could it be that people act individually in their own self-interest, which is to say selfishly, and yet the interpretation of the collective result is that they have behaved altruistically, since society as a whole benefits? In the two and a half centuries since Smith published his magnum opus, scholars have disputed all aspects of the AP, including its underlying assumptions on what constitutes the social nature of being human. Analogous arguments have persisted in the AP controversy surrounding macroevolution for over a century. The two alternatives, selfishness and selflessness, stand in such sharp contrast to one another that their incompatibility appears irrefutable and yet their intellectual incompatibility is resolved at the operational level of ordinary life, each and every day. How is that possible?

2.3

Organization Paradox We cannot solve our problems with the same thinking we used when we created them. — Einstein [77]

Organizational paradox was reviewed by Smith and Lewis [210], who studied 360 articles on the topic in their paper, pointing out that this was only a sample of the number of papers published in this area, which increased at a rate of 10% per year for the twenty years preceding their paper. This remarkable growth rate is a measure of the fact that an ever-increasing number of scholars see value in adopting a paradox perspective in order to understand the consequences of nonsimplicity in a business environment. But as they remark, their review highlights the lack of conceptual and theoretical coherence in that literature and the need to explain four categories of paradox relating to organizations: learning paradox that involves the replacement of outdated modes of doing things and replacing them through innovation, taking note of the consequent tensions between the past, present and future; belonging paradox that refers

2.3. ORGANIZATION PARADOX

41

to the tensions that arise between the individual and the collective, that is, tensions of identity; organizing paradox that is entailed by competing designs and processes to realize a given outcome, including tensions between competition and collaboration; performing paradox that is the result of the tensions between the differing, and often conflicting, demands of varied internal and external stakeholders. A more discipline-oriented division of these and similar paradoxes might make the separation along the lines of psychology and sociology, but that level of detail would take us too far afield. Of course these four types of paradox are not independent, but rather operate interdependently. To a large extent this interdependence is a consequence of nonsimplicity and results in two diametrically opposed views of organization leadership. The more familiar view is that stability is required in order for organizations to prosper, while leaders seek to improve products over time. The leadership strives to reach accommodation between conflicting challenges while at the same time achieving a stable resolution of conflict. Smith et al. [211] favor the alternative view of replacing the either/or -strategy of stability with the both/and- leadership strategy: We disagree profoundly with this image of leadership, because it is rooted in a mischaracterization of the business environment. The fundamental challenges we focus on...are fundamental paradoxes that persist over time...as the business environment and the actors change, stability breaks down...culminating in a crisis that prompts a leader to impose a different order...one in which the goal of leadership is to maintain a dynamic equilibrium in the organization. A system in dynamic equilibrium is unlike a system in static equilibrium. Take for example homeostasis in physiology, in which a perturbation creates an imbalance in the system, resulting in a response to relax back to equilibrium. As its name implies, static equilibrium is the system state when all the forces are in balance

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CHAPTER 2. KINDS OF EMPIRICAL PARADOX (EP)

and the only activity, say in a physical system, is due to thermal fluctuations and the dissipation produced by internal friction. On the other hand, dynamic equilibrium is the balancing of opposing forces in time, with the system endogenously adapting and adjusting to maintain its balance in a steady state, far from true static equilibrium. The latter is often mistakenly identified as homeostasis [211] when it should more accurately be termed homeodynamics. For the time being we note that despite the remarkable insight demonstrated in their review [211], the dynamic equilibrium model has not been mathematically tested, that is, it has not been rendered into a set of self-consistent mathematical equations, whose solutions can be tested against experimental/observational data. Therefore, the logic of the implications drawn have not been subjected to rigorous mathematical analysis. However, it does provide a number of criteria which any such mathematical model must satisfy in order to be acceptable to this community. This will be addressed in the context of the dynamic decision making model (DMM) in due course. Smith and Lewis [210] develop a qualitative model for the foundation of a theory with which to resolve paradoxes that invariably arise in the business environments of large organizations. Contradictions always exist in the real world, nonsimple, organized social groups, particularly in businesses, some of which form fundamental paradoxes that cannot be resolved by adopting to support one side of the paradox over the other. Neither side of the contradiction is seen to be without value and a successfully organized group is one that devises and implements enabling strategies to cope with paradox. Lewis [138] defines paradox as: ..contradictory yet interrelated elements that exist simultaneously and persist over time. This definition highlights two components of paradox: (1) underlying tensions — that is, elements that seem logical individually but inconsistent and even absurd when juxtaposed — and (2) responses that embrace tensions simultaneously.

2.4. STRATEGIC PARADOX

43

There always exists a tension between the short-term needs of the individual and the long-term needs of the organized group to which they belong, addressing the challenges of the future, the delicate balance between stability and change. Subsequently, we take a networked-based approach to modeling in order to capture this emergence of the difference in behavior of the group from that of the individual. The individual in our model must recognize that inconsistency is often desirable and be capable of holding multiple, conflicting views, and be able to transition from either/or -thinking to both/and -thinking in leadership [211].

2.4

Strategic Paradox In matters of truth and justice, there is no difference between large and small problems, for issues concerning the treatment of people are all the same. — Einstein [77]

Smith et al. [211] proposed a new way for leaders to address contradictory, but simultaneous challenges, starting from a rejection of stability as being a necessary condition for an organization to prosper. As previously mentioned, contradictions always exist in nonsimple organizations, some of which form a fundamental paradox, which cannot be resolved by adopting to support one side of the conflict, or the other. Both sides of the contradiction have value and the successful leader is one that can devise strategies that enable the organization to cope with paradox without becoming doctrinaire. The resolution of paradox demands that diametrically opposed alternatives be examined and it turns out that such alternatives not only contradict one another, but depend on one another, as well. Of central importance to economic organization is the strategic paradox, which is a consequence of the need to commit to a strategy despite the uncertainty of the outcome, knowing full well that the strategies that lead to economic success are the same strategies that can lead to economic disaster [211]:

44

CHAPTER 2. KINDS OF EMPIRICAL PARADOX (EP) Strategic paradoxes are essentially dilemmas that cannot be resolved. Tensions continually arise between today’s needs and tomorrow’s (innovation paradoxes), between global integration and local interests (globalization paradoxes), and between social missions and financial pressures (obligation paradoxes).

The goal of the leader is to establish a dynamic stability within the organization that embraces paradox, one that persists over time. There always exists the tension between the short term needs of the individual and the long term needs of the organization to prepare for addressing the challenges of the future, the delicate balance between stability and change. The seductive attraction of consistency in decision making, mistaking it for stability, must be avoided. A leader must recognize that inconsistency is often desirable and that an organization, as well as a person, must be capable of holding multiple, conflicting views, all of which are true. The individual, or organization, must transition from either/or -thinking to both/and thinking, which, in order to satisfy competing demands in the long term, requires a leader to frequently shift their short term positions [211]. Or as expressed by the poet [81]: A foolish consistency is the hobgoblin of little minds, adored by little statesmen and philosophers and divines. With consistency a great soul has simply nothing to do. He may as well concern himself with his shadow on the wall. Speak what you think now in hard words, and to-morrow speak what to-morrow thinks in hard words again, though it contradict everything you said to-day. – ’Ah, so you shall be sure to be misunderstood.’ – Is it so bad, then, to be misunderstood? Pythagoras was misunderstood, and Socrates, and Jesus, and Luther, and Copernicus, and Galileo, and Newton, and every pure and wise spirit that ever took flesh. To be great is to be misunderstood.

2.5. SURVIVAL PARADOX

45

There is a distinction to be made between the leader who analyzes disputes in the manner of a chess master and one who approaches conflict with the attitude of a poker player. In both there are well defined rules of engagement, as well as, strategies for gaining position, making sacrifices and attacking. However, in poker there is the added dimension of deceit. Since not all the play is visible to everyone, there is the opportunity to bluff and exploit your opponent’s psychological reaction to uncertainty. In the real world, a “consistently inconsistent” devious player has an advantage over a predictable rule follower. This is where complexity (nonsimplicity) thinking, or a paradoxical frame of mind, can overcome an adversary hampered by the belief that cognitive dissonance is to be avoided. On one hand, an opponent seeks to minimize uncertainty through control and decisions that suppress nonsimplicity, thereby reducing flexibility and cutting down on the number of options available to them. On the other hand, the paradoxical leader’s strategy is to cope with ambiguity and gain advantage by keeping the opponent off-balance through uncertainty; the mathematical models we subsequently discuss strongly favor strategies that incorporate deceit and bluffing into the play, which is discussed in Appendix B.

2.5

Survival Paradox ..If you can meet with Triumph and Disaster, And treat those two impostors just the same; Yours is the Earth and everything that’s in it... — Kipling [134]

In the paradox involving altruism the discipline of sociobiology was developed, in part, to resolve the apparent conflict between biological evolution of the species and social behavior. In that discussion the defining characteristics of the AP were selfishness and selflessness, both observed in the behavior of the same species of animal. Are there other examples of new disciplines constructed to explain similar inconsistencies in nascent scientific theory?

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CHAPTER 2. KINDS OF EMPIRICAL PARADOX (EP)

The answer is a qualified yes. The qualification has to do with applying the EP lens to the new construction, an example of which is in the survival of species and the generation of new species, called speciation. A question of interest to ecologists is how many new species, say of an insect, are contained within a given plot of ground. The number of new insect species is responsive to many environmental factors, most of which are unknown and consequently this number is a statistical and not a deterministic quantity. We shall have a great deal more to say about the importance of statistics, particularly for the resolution of EP, in the next chapter. For the moment we merely observe that one of the better outcomes of an empirical study is the determination of a robust heuristic relation between variables that characterize the phenomenon under investigation. Empirical studies have shown that there is a nonlinear relation, between the average number of new species found within a given area and the variability in that number measured by the variance [229]. This relation between the average and variance generates what is known as a power curve [113]. We propose to use this relation, discussed in Section 3.5, to construct a resolution of the underlying ecological paradox based on statistics. What makes this interesting for the EP question is the mechanism that Taylor and Taylor [230] postulated and which subsequently accounted for the heuristic power curve: We would argue that all spatial dispositions can legitimately be regarded as resulting from the balance between two fundamental antithetical sets of behaviour always present between individuals. These are, repulsion behaviour, which results from the selection pressure for individuals to maximise their resources and hence to separate, and attraction behaviour, which results from the selection pressure to make the maximum use of available resources and hence to congregate wherever these resources are currently most abundant.

2.6. INNOVATION PARADOX

47

Note that it is the conjectured balance between attraction and repulsion, congregation and dispersion, that produces the interdependence of the variance and average number of new species. This could be interpreted, through an information lens as being an imbalance in nonsimplicity, or information, such that there is an information force, resulting from the nonsimplicity imbalance discussed by West [259]. This imbalance in empirical complexity [230] (nonsimplicity) can be interpreted as two kinds of selection pressures, acting in opposition to one another. The more nonsimple species, that is, the increasing number of newer species of beetle moves into the physical region of the simpler, or older, species of beetle and eventually, through natural selection, replaces them. But in the interim the number of new species has increased and may reach intermittently spaced intervals of stasis at these increasingly higher numbers. Applying the EP lens to interpreting the "two fundamentally antithetical sets of behavior", provides a way to quantify the strength of nonsimplicity of a phenomenon through the paradox of competing selection pressures to survive. Increasing nonsimplicity of the ecological web entails the generation of new behaviors many of which are in conflict with one another. Nature’s way of providing relief from the ecologic pressure that such behaviors generate is to favor evolutionary advantage on offspring that have adapted to and are therefore most compatible with the pressure. Consequently, the resolution of the survival paradox is to increase the number of new species with enhanced fitness within the existing evolutionary landscape.

2.6

Innovation Paradox The true sign of intelligence is not knowledge but imagination. — Einstein [77]

One way we tend to separate events in the world, particularly those that affect us personally, is in terms of the opposite extremes of success and failure. Success is typically defined in terms of getting

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the result we desire: an A on the final exam; the raise you were expecting; the love of your life saying yes, and so on. Failure is defined by not getting those things, or is in terms of independent disasters such as getting fired; having the business in which you invested go bankrupt; having your horse, on which you bet the farm, come in dead last. The world comes in two flavors, triumphant success and devastating failure, and you get to choose how you, and you alone, react to them. Farson and Keyes [84], in their discussion of the innovation paradox, argue that how you treat success and failure makes all the difference in what you can achieve. One area of activity where this conventional partitioning into success and failure is known to be a myopic view is scientific research. No carefully designed experiment to test a theory, or theoretical calculation to understand the outcome of an experiment, is either a total success, or a total failure. One reason for this is that only in the simplest system can a prediction be exactly realized by an experiment. But even in the simplest case, the more typical result is one in which the experimental outcomes are each slightly different, but they cluster around a common value, while clearly spreading out into a distribution of values. The statistical distribution of outcomes make the notion of success moot, which is to say, all success is only partial, never complete. Particularly, if success is defined as realizing the exact prediction. In the same way failure is only partial, because even when experimental results deviate sharply from prediction this is an indicator that something new and previously unnoticed is going on. A scientist often learns more from such a failure than they do from a success. Scientific managers, in an attempt to validate their organization, and motivate their workforce, often devise methods to highlight the successes of their scientific staff. They herald a new publication, patent, or widget that does something smarter, faster and more efficiently than it was ever done in the past. Awards are given to demonstrate how much the organization values the contributions made by the individual scientist/engineer. Success is touted, made visible and those that achieve such notoriety are handsomely rewarded. How-

2.6. INNOVATION PARADOX

49

ever, as pointed out by Farson and Keyes, these strategies run the danger [84] of doing the exact opposite of what they were intended to accomplish, that being the winning of the prize becoming the goal, rather than the doing of innovative science or engineering for which the prize was awarded. What is not touted are the multiple failures that precede the lone success. Even though these failures are a crucial part of the creative process they are rarely discussed, in part, because it is the end point that is focused on and rewarded, not the process itself. If it were the success of the process that was being acknowledged there would most certainly be an award for the most failures achieved. The recipient of the Failure Award would be, in all likelihood, the person doing the most innovative research. But the culture would have to be totally transformed to make such an award acceptable, not to say valued. The importance of failure in the achievement of any success is well-documented [84] through the presentation of vignettes and quotations from famous people at the top of their profession, with firsthand experience of both success and failure. The point made over and over by these leaders is the necessity of failure in the pursuit of success, whatever the context. Failure as part of a strategy leading to ever greater levels of success, was succinctly put by the consummate inventor Thomas Edison [76]: When a reporter asked, "How did it feel to fail 1,000 times?" Edison replied, "I didn’t fail 1,000 times. The light bulb was an invention with 1,000 steps." To fail intelligently in the pursuit of understanding a part of the universe is, or ought to be, the motto of the research scientist, whereas to fail intelligently in the application of that understanding to control a part of the universe is, or ought to be, the motto of the research engineer. If you are not failing most of the time, you are not working at the cutting edge. The EP is that the same process that snatches victory from the jaws of defeat, also plucks defeat from the jaws of victory. Success

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and the complacency it often engenders in organizations and in individuals, as well as the reluctance to step beyond the radical innovation that was responsible for that success in the first place, both sow the seeds of eventual catastrophic failure. The business landscape is cluttered with the carcases of companies that were making too much money to modify their business model in order to incorporate what later turned out to be their replacement technology. There is no logical reason why these companies failed to adapt to the changing environment. Quite often they not only possessed the people, but had themselves developed the innovations necessary to succeed, but were incapable of using them to affect the needed change. Those that did make the change and survived, did so by means of a combination of intuition and good fortune, not rationality [84]. The real-time resolution of the innovation paradox requires both intuition and rationality, as well as luck. In summary, the more you fail and survive, the more likely you are to succeed. Success results from continuing improvement, which typically emerges from the devastation of failure. A corollary to this conclusion might be that if you fear failure then you will probably never achieve success. Of course, it is also the case that incremental improvement results in incremental success, which is the result of failing small and often. If your goal is revolutionary success then you must risk and endure catastrophic failure.

2.7

Conflict Paradox By denying scientific principles, one may maintain any paradox. — Galileo Galilei [96]

Psychologists have found that people react more strongly to loss than they do to gain, strength being the intensity of the reaction, not whether it is positive or negative. We vividly remember the embarrassment of a failed exam compared with the temporary pride of

2.7. CONFLICT PARADOX

51

receiving an A; the lingering devastation of defeat in a competition, over the transient glory of coming in first; the lasting frustration of a problem we could not solve, when placed beside the now faint pride of those that yielded to our efforts. The successes all seem to blur together, whereas the failures stand out like markers indicating where we were not up to the task. It is not even that we dwell on these things, it is just that they are the more clearly visible whenever we look back. Or so it seems. Another of these clear retrospective indicators is in the realm of interpersonal relations, the paradox of conflict. This particular form of paradox often precedes a much needed change in one’s life, whether it is the dissolution of a bad relationship, leaving home for the first time, or leaving one good job for another. When a friend is questioned, say over a beer, concerning some life-altering decision they made, they often recall finding it necessary to identify fault with the then existing status quo in order to justify change. They confess to generating conflict whose only resolution was an either/or choice, where they had already determined which side of change was preferred. Conflict is not always of our own making, however. Therefore the either/or resolution is not always the predetermined, or even the most desirable outcome. In point of fact, people are often bewildered at finding themselves embroiled in a conflict, whether at work, or at home. Conflict appears out of the blue, with the people involved having no idea how it started, or how to resolve it. The more nonsimple life becomes, the greater the likelihood that conflict will emerge, without warning, or provocation. As our lives become less solitary and more socially active, social paradox becomes inevitable and conflict is one aspect of this increasing nonsimplicity. As Braathen [48] observed We try to build teams out of (individualistic) experts; trying to explore and innovate while exploiting resources to optimize; thinking globally while acting locally; fostering creativity while we increase efficiency; or trying

52

CHAPTER 2. KINDS OF EMPIRICAL PARADOX (EP) to be in control when letting go of controls seems to be working better.

This paradox has become so common place that a new profession has emerged, conflict intervention, which looks like applied psychology when involving individuals, applied sociology when restricted to organizations and political science when concerned with nations. Conflict paradox and its resolution arises at all levels of the human condition and its anatomy is discussed with remarkable clarity by Mayer [161]. He observes in his book that typically it is in the best interests of those involved in a conflict to reach a resolution that does not involve the total abdication of one position in favor of the other. The apparent mutually exclusive nature of paradoxical alternatives strongly suggests an either/or resolution. But this appearance is based on the assumption that the alternatives are independent of one another, but it turns out that they are, in fact, always interdependent. The result is that in each case the actual resolution is an insightful combination of both poles, the both/and resolution resulting from nonsimplicity. Mayer identifies seven of these paradoxes, whose seemingly contradictory nature he uses to frame his understanding of the conflict paradox: competition and cooperation; optimism and realism; avoidance and engagement; principle and compromise; emotions and logic; neutrality and advocacy; community and autonomy. A number of these will be recognized from the earlier EPs.

2.8

Control Paradox The world is so unpredictable. Things happen suddenly, unexpectedly. We want to feel we are in control of our own existence. In some ways we are, in some ways we’re not. We are ruled by the forces of chance and coincidence. — Paul Auster [196]

2.8. CONTROL PARADOX

53

The control paradox is often first encountered in adolescent romance, or what is euphemistically called young love. When consumed by it one cannot read, write, carry out a conversation, or do anything else that requires concentration, using anything resembling a working brain. In this somewhat demented state adolescents make poor decisions. For example, they want to continually be in the company of the object of their adoration. This is often accompanied by attempts to restrict who the other person is allowed to see. Finally, and this occurs just before the breakup date, they experience jealousy. This truncated version of the pangs of young love is an EP because the more controls the adolescent places on the relationship the greater the chance of its demise. The lesson learned after a typically small number of such affairs is that the only way to extend a relationship is by lessening, not increasing, the number of controls. The fewer the number of restrictions placed on a friendship, love affair, or any other personal relationship in its early stages the greater the chances of its flowering and becoming permanent. The more demands, even those placed for the purpose of prolonging the relationship, the greater the likelihood of the affair coming to an abrupt and predictable termination. This example may strike the reader as frivolous, but it is actually quite profound. Its significance stems from the fact that it captures in a personal way a fundamental truth about how we humans attempt to exert control in complex social situations and fail. The lack of success is clear when the control involves only one other person, but that failure is masked, at least for a short time, when in the face of a changing environment the leader of a large organization imposes an increased number of controls on subordinates to insure the success of the organization. Zelaya [274] phrased this counterintuitive results in the following way: Under uncertain conditions increased control on subordinates increases uncertainty and the possibility of failure.

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CHAPTER 2. KINDS OF EMPIRICAL PARADOX (EP)

Consider a nonsimple organization operating in a nonsimple environment. One characteristic of organizations entailed by nonsimplicity is uncertainty, so that it is never certain if a nonsimple organization will succeed or fail. It is the inability to predict outcomes that accompanies uncertainty that make leaders nervous. It also attracts them to impose controls, whose sole purpose is to reduce nonsimplicity and thereby increase predictability. To simplify the discussion let us assume that there is a goal the organization must achieve in order to survive. If it does not achieve this goal it fails, if it does achieve this goal it succeeds. The large organization discussed by Zelaya [274] was the United States Army, for which he defined failure as, for the sake of argument, to be anything less than 100% success of the mission. However, in every organization there is a set of conditions and tasks, call them states, that must be realized in order to attain the end goal and thereby achieve success. Here we define success to be the realization of 100% of these states and failure is anything less. We assume there are a number of trajectories initiated by the leader’s action in the initial state of the organization, which pass through all the necessary states and ends up in the desired final state. These trajectories are timed and conditioned instructions for subordinate units within the organization; they are control measures intended to constrain the actions of subordinates to completing the necessary assigned tasks. The probability of a trajectory being successful is conditional on each of the subordinate units successfully satisfying 100% of the controls. Consequently if a given set of controls consists of five tasks, each having a 95% chance of being successful, the probability of overall success is 77%. Or stated differently, since there are a substantial number of subordinate units in any large organization, under the same conditions, a little less than one in four of them will fail. Consequently, the more controls placed on subordinate units, under conditions of uncertainty, the greater the likelihood of failure. Because the success or failure of an organization is the responsibility of the leader, once a risk is identified, the system pressures

2.9. PARA BELLUM PARADOX

55

that leader to handle that risk by increasing the constraints on subordinates’ initiatives. It is the rare leader that is able to resist this kind of peer, as well as bottom-up, pressure. The result is, if the leader does yield to the pressure, a loss of innovation due to micromanagement and excessive risk-adverse behavior on the part of the organization overall. Like the love-sick adolescent the overly cautious leader, when confronted with the control paradox, becomes risk-adverse and in the struggle for survival chokes off the creative life blood of the organization that is necessary for survival.

2.9

Para Bellum Paradox One standard argument brought forward by peace activists is for instance that — in order to bring peace in Europe — the first thing to be done is to crush the enemies of peace. In other words, the call to peace is in many cases accompanied by the call to weapons: Si vis pacem, para bellum. — [120]

The Latin phrase Si vis pacem, para bellum is translated as: "If you want peace, prepare for war". In the twentieth century this strategy became known as the arms race, which was mistakenly thought to be of recent vintage. However, the Latin phrase is evidence that this piece of military, or political, philosophy dates at least 2000 years to the Roman general Vegetius and probably much further. It certainly dominated the political search for peace in Europe during the nineteenth century [120]. In any event the peace strategy apparently entails a paradox, that being, the strong advocacy for peace entailing the onset of war. In his book Paradoxes of War, Maoz [154] presents the following summary of [247] on wars resulting from arms races: A total of 99 conflicts involving at least one major power was analyzed. The results showed that 26 of these conflicts escalated into war. Out of the 26 wars, 23 had been

56

CHAPTER 2. KINDS OF EMPIRICAL PARADOX (EP) preceded by an intensive arms race. Out of the 73 conflicts that did not escalate to war, 68 were not preceded by an arms race. Like-wise, out of the 28 cases that could be characterized as intensive arms races, 23 resulted in war, whereas out of the 71 interstate relations that did not involve arms races, 68 conflicts ended in forms short of war. This seems pretty convincing evidence of the arms race-war connection.

This connection between war and the arms race is thought to be counter-intuitive by some, because a major argument for a strong defense is that it deters the aggressive actions of one’s adversaries. A simple formal argument for this counter-intuitive connection between war and the arms race is based on game theory and a modified version of the well-known prisoner’s dilemma game (PDG). The argument given by Moaz [154] goes something like this. Two states α and β have recently gained independence and must decide how to allocate their respective budgets, that is, determine what fraction to devote to defense and what fraction to allocate to internal infrastructure and social services. He discusses the possible outcomes of the four potential choices: state−α chooses not to arm and state−β also chooses not to arm; state−α chooses to arm and state−β chooses not to arm; state−α chooses not to arm and state−β chooses to arm; state−α chooses to arm and state−β also chooses to arm. These alternate choices have different outcomes for the two states and their decisions can be summarized in the form of the matrix indicated in Table 1. The calculus of game theory is briefly discussed in Chapter 4, but we anticipate some of that discussion here and note that under the rules of the game two players (these being the two states) make rational choices of strategy for which they each receive a distinct payoff. The payoff to each player is determined by their joint choices. Consider the matrix given in Table 1. The left hand column corresponds to the choices available to state α and the top row to the choices available to state β. The first entry of the matrix has

2.9. PARA BELLUM PARADOX

57

Figure 2.2: Table 1: The entries in this matrix indicate the choices of states α and β along with the strategic outcome associated with those choices. The first number indicates the payoff to state α and the second number indicates the payoff to state β for the associated choices 4 = best, 1 = worst.Adapted from [154]. ¶ Nash equilibrium. both states choosing not to arm and as a consequence no strategic advantage accrues to either state, but there is the benefit that no resources are committed to arms and are therefore available to social services. The two numbers associate a payoff to each player (state), which in this case are both equal to 3. The second matrix element on the top row has state α still not arming, but state β does elect to arm and thereby receives a strategic edge. The numbers tell the story; state α has a payoff of 1, whereas state β has the greater payoff of 4, reflecting its military superiority. A similar analysis can be carried out for each of the remaining two matrix elements. Maoz points out that the rational choice is for both states to arm, since they are both better off independently of what their opponent elects to do. He goes on to say that it is evident that both states would received higher payoffs by not arming; their resources would have been used to increase social benefits and their relative strategic positions would have gone unchanged. However, if

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the first state making the decision, chooses not arming, the rational choice for the second player is to arm. The winning strategy for the PDG is the arms race. Thus, the PDG is the mathematical rationale for para bellum, which is to say, of you want peace prepare for war. Entering into an arms race concludes the first phase of the argument, because once this path is embarked on their is no turning back. One state cannot stop, because in doing so would cede strategic advantage to its opponent, which it would not do if an alternative were available. The second phase connects the arms race to fighting a war. The decision to fight a war was not an alternative in the first phase because neither state had the capability, but over time the arms race provides that capability. Maoz argues that eventually a state will realize that the money being spent on arms could be better spent of social services for its own people and given a reasonable chance for success would consider the option of a war in order to defeat and disarm their opponent. The second reason for war is the fear that the opponent state would attack following the arguments in reason 1. Thus, it would be prudent to use the element of surprise and adopt a preemptive strike. The game theory argument is extended beyond the PDG in the second phase using a logic tree, which is reduced to the second 2 × 2 matrix depicted in Table 2. Here again the matrix summarizes the final form of the decision making process as a game. The ARMARM outcome is the result of the ’winning’ (sic) strategy in the arms race game and collapses into the first element in the arm or fight game. In this choice both states continue the arms race indefinitely, until one, or both, exhaust their resources. It appears that this strategy lead to a successful outcome of the cold war for the United States, with the Soviet Union throwing in the towel during the Reagan administration. Cutting to the chase, here again we have the ’winning’ (sic) strategy being that of FIGHT-FIGHT; a highly destructive and costly war carried out by two states that have been escalating their military capabilities for a long time. Both states wanted peace and

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Figure 2.3: The entries in this matrix indicate the choices of states α and β along with the strategic outcome associated with those choices. The first number indicates the payoff to state α and the second number indicates the payoff to state β for the associated choices 6 = best, 1 = worst.Adapted from [154]. ¶ Nash equilibrium. only entered into the arms race to avoid war and yet this abundance of caution has led them directly, if belatedly, into war. Maoz [154] points out that this game theory model of decision making clearly identifies the arms race as being one of the underlying causes of war. This is the para bellum paradox.

2.10

What have we learned?

Is there a common theme running through the eight examples of paradox discussed in this chapter? We would say yes, but then that opinion could be countered with the observation that we both selected and presented each of them. Guilty as charged. However, in our defense we would say that the many books quoted on paradox of various kinds do arrive, by multiple paths, at the same conclusion, that being, the EP emerges from the nonsimplicity of the phenom-

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enon being investigated. In addition, the explanation of paradox with which we opened this chapter that being that EP is a consequence of the fundamental incompatibility of simple models of reality has been born out by the empirical evidence. Let us reiterate. As scientists we follow the dictum of Einstein, carved in stone beneath his statue in front of the National Academy of Science building in Washington DC, and make our models of reality as simple as possible, but no simpler. This is prudent advice, as long as the phenomena being modeled are separate and distinct. As technology has advanced, however, our instruments have become more refined, and we are able to measure metaphenomena and consequently our theories must explain data not previously available to us. Such metaphenomena are composed of two or more of the simpler phenomena on which the simple models are based and so the metamodel must also be compatible with the simpler models. However, the simple models were not constructed to be mutually compatible and often are not. It is this incompatibility that entails EP. New theory, often of a revolutionary kind, is constructed to put into a larger context the metamodel required to explain these new data in addition to previous data sets. In other words, the new theory must synthesize the previous models even when they are in conflict with one another. The greater the conflict the greater the potential for more new knowledge generated through the synthesis and the greater the likelihood that this new knowledge will be of a revolutionary kind. Such conflict was irrelevant until it became necessary to think about the piece of the world being explored in a both/and way. The revolutionary advances in science are most apparent to the authors, being physicists, through the resolution of physical paradox. The world we can experience through our five senses is described by the deterministic laws of Newton. When we look at the very small through a microscope we see the restless middle world, whose explanation leads to the irreversibility paradox in the macroworld that was partly resolved by statistics. The resolution of the wave-

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particle paradox, concerning the nature of light, destroyed the idea of continuity and lead to Newtonian mechanics being replaced with quantum mechanics, with even a different kind of statistics. Each new theory produced a disruptive change in how we understand the world. We expect that consequences will be no less disruptive with the resolution of each social and psychological paradox. The remainder of this essay is devoted to the exploration of the potential resolution of these latter paradoxes.

Chapter 3

Thoughts on Nonsimplicity A preliminary explanation of EP is given in this chapter. A brief history of how science has come to understand nonsimplicity through the use of statistics is presented. The first successful attempt to understand the divergence from the simple to the complicated was provided by the central limit theorem and normal statistics, characterized by a mean and variance. However, the conditions for the central limit theorem, the theoretical foundation for the Machine Age, are violated by the truly nonsimple phenomena of dramatic importance in the Information Age and are quite often described by Pareto’s IPL statistics. The Industrial Revolution of the nineteenth and early twentieth centuries argued against the existence of paradox, just as it was not accepting of uncontrolled variability. On the other hand, the Information Revolution of the late twentieth and early twenty-first centuries accepted all manner of variability and subsequently even 63

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3.1

Some background Peering into the future can be scary and surely is humbling. Events unfold in complex ways for which our brains are not naturally wired. Economic, political, social, technological, and cultural forces collide in dizzying ways, so we can be led to confuse recent, dramatic events with the more important ones. It is tempting, and usually fair, to assume people act “rationally,” but leaders, groups, mobs, and masses can behave very differently– and unexpectedly–under similar circumstances. – National Intelligence Council [173]

One thing we have learned from the study of nonsimple systems is that the microdynamics do not control the macrodynamics, whether we are discussing the particles making up the fluid in a physical system, the cells making up the membrane in a biological system, or the individuals making up the group in a social system, the behavior at the macrolevel need not be explicit in the specification of the microdynamics. There is nothing in the nature of isolated water molecules that would suggest the existence of the critical states of water vapor, water and ice. It is the complexity of the interaction within a collection of such molecules that entails criticality and the emergence of universal properties, such as scaling. But through the insight of some remarkable scientists we subsequently learned how to anticipate the collective behavior and emergent properties of phase transitions by recognizing that “more is different” [12]. In the cited article the physics Nobel Laureate P.W. Anderson explained that the macroproperties of interest in a physical system can be, and often are, completely independent of the details of the microdynamics. This is the lesson of the renormalization group theory explanation of phase transitions [267].

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The borders separating the distinct phases of water are sketched in Figure 3.1 where the lines indicate sharp demarcations between the phases of water. The boundary lines indicate the values of pressure and temperature where water may transition from, say liquid to solid, with the corresponding change in molecular structure from the weak bonding forming liquid water to the crystalline structure forming solid water (ice). This is the remarkable change from short-range interactions between the molecules of a liquid to the long-range interactions of the crystals of a solid. We stress this way of organizing the familiar properties of water, using a phase diagram and the language of phase transitions, because we subsequently use this view of nonsimplicity to organize our thinking about reaching consensus within social groups and what they entail about EP. This lesson, first learned in the physical sciences fifty years ago, has been shown by experiment, theory and computation over the past decade, to be equally true in the social sciences. One major difference between a social and physical collective is the existence of awareness on the part of the basic element, that being a person. This obvious difference enables individuals to take advantage of the emergent properties of the collective to modify their behavior. Experiments show that individuals contribute at an intellectual level (as well as other levels), when they are part of a group, which is far above their typical capability when functioning alone. We are now beginning to understand this social phenomenon theoretically and to see how it may work in other individual-group actions, such as realizing Norbert Wiener’s prophetic vision of Cybernetics [265] to optimize the output of human-agent collectives, regardless of the task. The mystery of phase transitions was solved in physics by finding a formal way to discuss the difference between the behavior of an individual and that of a large group of interacting individuals. The first approach to shed light on this difference was through the invention and application of statistics to collections of identical particles. This is the simplest of all models since it represents complex individuals as identical featureless points, or as two-valued entities. A

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Figure 3.1: This is a sketch of a two-dimensional phase diagram for water with the lowest amount of free energy. There are two control parameters for water: pressure and temperature. At a given value of these two parameters water is in one of three phases, solid, liquid or gas, except at the tricritical point (•) where the three phases coexist. Things become more complicated to explain to the right of the solid lines separating the various phases.

partial success in the understanding of paradox was achieved using the same strategy, as we show in this chapter. We opened the first chapter by introducing the capability paradox and suggested that we could understand how that came about, even though on its face the resolution itself seemed counter-intuitive. We need to understand how a given distribution of athletic abilities within a large group can yield a large majority of randomly sampled people that are below average. Consequently, we explain this EP in terms of the contrast between how we map the world’s charac-

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teristics within our brains, including how we treat uncertainty and the gaps in knowledge, and how properties are manifest in the real world. Cognitive maps are typically constructed under the belief that the distribution of capabilities among people are statistically normal, which is to say, the variability in capabilities within a large population are determined by the mode and width of a bell-shaped curve. The interpretation of a statistical distribution in terms of quantities measured in the physical sciences was first conceived at the turn of the nineteenth century. This was the normal distribution; proven to be universal, as long as each variation was small, linear, and independent of every other variation. For the moment we ignore the fact that most phenomena in the real world are large, nonlinear and interdependent and therefore cannot have normal statistics. We adopt this misleading perspective since most scientists did so with impunity for over a century; and by impunity we mean that other scientists nodded their heads in agreement, or looked away, while these assumptions were being made and implemented. We often forget that science is a social activity and having the agreement of one’s colleagues is a strong motivating factor for doing anything, either correctly or incorrectly.

3.2

Uncertainty and Empirical Law I think that when we know that we actually do live in uncertainty, then we ought to admit it; it is of great value to realize that we do not know the answers to different questions. This attitude of mind - this attitude of uncertainty - is vital to the scientist, and it is this attitude of mind which the student must first acquire. — Feynman [87]

Why is it that some kinds of variability are acceptable and we adapt to them, whereas other unanticipated changes are experienced

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as being abnormal, or bizarre, resulting in our being shocked to the point of being momentarily unable to respond? The key idea here is that some kinds of unexpected change are familiar, having occurred in the past and are expected to be encountered at unpredictable times in the future. These random changes occur frequently enough that they are represented by the normal curve. The implication of such familiarity is that other kinds of change, those being encountered for the first time, are not typical and take us completely by surprise. For the sake of clarity we examine the notions of normal behavior and typical variability more closely. Let us think back three hundred years, to a time before the introduction of statistics into science. Statistics had not been introduced into the modeling of the world prior to this time because it had only then been invented as a mathematical discipline; this being a time when gentleman wined, wagered and wenched, with the doctrine of chance determining the likelihood for a given outcome of a bet, not an experiment. It was a time when the erratic changes, or fluctuations, in experimental data were thought to be the result of poor experimental design, inferior equipment and/or mistakes by the experimenter. In the worst situation these fluctuations were thought to be generated by ‘fundamentally’ unknowable causes of randomness. Many of the top scientists of the day asked the same question in response to statistical arguments: How can a collection of measurements, each one of which is individually wrong, lead to a result that is correct in the aggregate? It came as a welcome, if surprising, result to scientists that random, that is, uncorrelated and unexpected, fluctuations in data follow an empirical law in the physical world, the law of frequency of errors (LFE). At the turn of the nineteenth century, two mathematicians, Gauss [99] in Germany and Adrian [1] in the United States, independently proposed the same resolution of the mystery as to why physical experiments produced such erratic data. Each offered equivalent mathematical arguments to explain the confounding observation of the variability in the measurements, which subse-

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quently became known as the LFE. The empirical law gives a precise quantitative measure of the potential inaccuracy of each and every measurement. The name is unfortunate, because error connotes mistakes, or deviations from the right answer, consequently this often leads to an improper interpretation of what the law actually means. The academic discussions in science regarding the nature of uncertainty did not change how farmers planted their crops, nor did it modify how soldiers fought in battle, since both the farmer and the soldier were already intimately acquainted with uncertainty. But in the cities where invention and innovation were flourishing in anticipation of the Industrial Revolution, the intelligentsia was listening. The idea that randomness obeys a law, in the same way that mechanical phenomena obey laws, captured the imaginations of the nineteenth century Natural Philosophers. Consequently, they began looking beyond the physical sciences to find application areas for this exciting new knowledge and they found such in the social and life sciences. A couple of years after the distribution of Gauss and Adrian was introduced into science, the physicist/mathematician Marquis Pierre-Simon de Laplace presented a mathematical proof of the central limit theorem. His proof, which was the first of many, establishing that the validity and applicability of the LFE is much broader than anticipated by its originators. The Marquis showed four conditions on which the bell-shaped curve rests, which expressed in the language of errors are: 1) the errors are independent; 2) the errors are additive; 3) the statistics of each error is the same and 4) the width of the distribution is finite. These four assumptions were either explicitly, or implicitly, made by Gauss and Adrian in their separate discussions of the statistical fluctuations of experimental data and Laplace showed them to be necessary and sufficient properties to obtain the LFE, which is the familiar bell-shaped curve. It is worth emphasizing again that even though the variation in individual measurements may be capricious, when all the measurements are taken together the collection follows a deterministic law. The LFE unexpectedly constrains the mysterious variability ob-

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served in the measurements in all physical experiments and whose explanation had eluded generations of scientists. The mathematical form for this statistical effect has the one humped shape of a bell depicted by the solid curve in Figure 3.2 and became known as the Gauss distribution in the physical sciences, and the normal distribution in the mathematical and social sciences. Eventually, the LFE led to the mistaken impression, on the part of working scientists, that all the world’s uncertainties are typically of this form, which is devastatingly far from the truth. In fact, the statistics that influence the way we live, such as the distribution of income, the number and kinds of words we use in talking to one another, and how we respond to profit versus loss in our investments, look nothing like the LFE, which we subsequently make use of. On the other hand, a simple variable, such as height, has a distribution of values over a large collection of similar individuals, say middle class white males in the United States in the early 20th century. These data, when ordered into equal sized bins from smallest to largest, form a bell-shaped distribution, from which it is possible to determine an average height, which is typical of the collection, and a standard deviation, which quantifies how well the average characterizes the distribution of heights. The heights of members of this group are empirically determined to be normally distributed as depicted in Figure 3.2. Presumably, the normal distribution does so well in this case because an individual’s height is determined by stringing together a large number of bones, each of which is statistically independent of the others, but each having a typical length. However, the distribution of heights does not carry over to other, less obvious, characteristics, such as the distribution of athletic capabilities. The evaluation of the performance of individuals was, for a long time, mistakenly based on the normal distribution. It was a mistake because the normal curve is predicated on the assumption that an average can characterize how athletic ability is spread over a collection of young enthusiasts, with nearly 70% of the capability falling within two standard deviations of the mode, that is, the peak of the curve. True excellence would be anomalous in such a

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Figure 3.2: A histogram of heights of young adult males in the United States early 20th century is fit with a normal distribution (solid curve).

distribution and the record breaker would be an unwelcome outlier. It is therefore not surprising that the distribution of performance looks nothing like the distribution in Figure 3.2. In fact the bellshaped curve in the figure artificially restricts the number of those that would excel, and when used in an evaluation process promotes mediocrity. The universal curve depicted in Figure 3.2 captures the variability of all experimental data arising from a relatively simple phenomena. Of course, the bell-shaped curve did not start out as universal, but depended on whether the measurements are the heights of a homogeneous population of individuals, the spacing of vehicles in a convoy on a typical highway, the variations in the distance between points of impact of meteors striking the surface of the moon, and so on. The distribution in the measurements for each of these phenomena has a different average value and width, depending on what is being measured. The universal form is obtained by subtracting the average value from each data point and then dividing the result by

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the standard deviation of the data. Consequently, the bell-shaped curve peaks at zero (average removed) and the width of the universal distribution is unity (the units of the data have been divided out by the standard deviation). The LFE is universal in terms of the shifted and normalized variable, which is to say, the standardized variable. We have come to accept that the average is the best representation of the data and the standard deviation provides a measure of how well the average characterized the data. The universal curve depicted in the figure replaces the notion of making a single prediction, with data, never repeating, always varying. As observed by West and Arney [262], statistics and its adaptation to the interpretation of experimental data revolutionized, not only how scientists think about the world, but how nonsimplicity and uncertainty are understood. Uncertainty was shifted from the domain of the unknowable unknown to the knowable unknown, and the magnitude of that unknown can be quantified. The greater the nonsimplicity of the phenomenon being measured, the greater the uncertainty, as measured by the width of the LFE. In this world view there exists a correct value for the outcome of an experiment and although this interpretation of uncertainty may be accurate for simple physical systems, generally it imposes an unrealistic constraint on how we understand the world of experience. A good prediction has a narrow distribution, where the anticipated future is not too dissimilar from the one previously experienced; the future is mostly like the past. Unless of course, there is a catastrophic event, such as an earthquake, or war. On the other hand, a poor prediction has a broad distribution, where the experienced future can be quite different from the predicted one. The latter is filled with surprises. The LFE encountered relatively little resistance in its adoption into science, in part, because it presented a theory of uncertainty that was consistent with Newtonian mechanics. The theory that determines the inertia of planes, trains and automobiles, as well as, how celestial bodies orbit one another. The universe is understood to be determined by clockwork mechanical processes and therefore

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variables ought to be quantifiable, measurable and predictable, even those referring to an individual’s life and to society. Thus, even when uncertainty enters our understanding of the world, where randomness blurs what is expected, the scientist believes there ought to be a proper value that characterizes the process and the engineer believes there ought to be a way to control the expected outcome. From this perspective random fluctuations do not invalidate the mechanical world view; rather they complement it, often making predictions only slightly less certain than the ticking clock. The universal normal curve describes in a compact way how we deal with the world’s complications in order to keep us from becoming frozen in the grip of uncertainty, unable to make a decision. But as we show, the LFE applies to complicated, not nonsimple, phenomena; truly nonsimple things require something else altogether.

3.3

Man, Machine & Management Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry. — Feynman [87]

The world of the LFE is linear, with fluctuations that are weak and additive, but primarily the average value is king. This view was eventually adopted by the lion’s share of the scientific community over the nineteenth and twentieth centuries. The adoption of a linear world view had less to do with the properties of the data being recorded than it did with the simplicity of the models scientists were able to construct using this assumption. Linear models could be solved and therefore led to predictions that could be tested; small initial changes in a process produces relatively small changes in its final behavior. This perspective was so attractive that quite often when the experimental result deviated significantly from the linear prediction, a scientist might inventively construct an ever more complicated linear model, rather than accept the more defensible conclusion that it was linear thinking that was at fault.

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The side benefits of having a linear world include the applicability of Aristotelian logic; complicated phenomena can be controlled in a straight forward way; the world is stable and the appearance of instability is an illusion, not a reality. However, this attractive linear model is not the world that produced Michelangelo and Machiavelli, Madame Curie and Bonnie Elizabeth Parker, as well as, Hitler and Ghandi; the world in which we live. But if linearity is so far from reality why is it so important? Linearity provides insight into complicated, if not nonsimple, phenomena. It provides a way to understand the elementary laws governing the dynamics of the underlying elements. A complicated process can be separated into its elementary components, which can be individually understood. These components can be reassembled to reconstitute the original process and the gestalt can be understood in terms of the understanding of the separate fundamental elements. In physics this is the principle of superposition, used to explain everything from music to munitions. In other words, like the LFE, the world is assumed to consist of networks of linear additive processes, but unlike that probability law, these parts are inherently knowable. The linear world is predictable; it is a simplified version of Newton’s mechanical clock. A view that helped to transform Westerners from cowboys and farmers into businessmen and industrial workers. The normal curve captured and held captive the imagination of scientists throughout the world for over two centuries. The bellshaped curve was accepted, in part, because it allowed scientists in the various disciplines outside the physical sciences to quantify the uncertainty in their phenomena in a manageable way. A little uncertainty was not only tolerable, but welcome, particularly because it also reinforced the assumption that the phenomenon being studied could be controlled, as well. The foundational ideas implicit in LFE found fertile ground in the manufacture of widgets and in the operation of the devices of the Industrial Revolution. From the tolerance of a crank-shaft to the quality control of widgets coming off the assembly line, the ar-

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tifacts of the Machine Age lent themselves to quantification by the normal curve. Gathering and analyzing data for two centuries has taught us the properties of systems described by the normal curve. At the top of the list of properties is that the strength of the response is proportional to that of the stimulation. A 5% excitation produces a 5% response, or maybe a 10% response, but nothing too outrageous; such systems are linear. The system does not become unstable when it is stimulated, so that its subsequent behavior can be predicted. More accurately, the distribution of possible responses can be predicted. Second on the list of properties is additivity. If two stimuli are simultaneously applied to a system the total response is the sum of the responses obtained if the stimuli were applied separately. Such systems can be reduced to fundamental elements, which weakly interact and recombine after being perturbed to reconstitute the overall system. Consequently, linear additive phenomena that are stable when stimulated describe the world of Adrian, Gauss and Laplace. The uniformity of the machines that populate our world are the result of the normal curve dominating engineering design to reduce variability in manufacturing outcome. We have come to expect that refrigerators, dishwashers, automobiles, rifles, computers and all other machines of a given make and model are identical. But not just machines, the clothes and shoes we wear share a similar uniformity. Variability can be reduced to the point of extinction, by the process of manufacturing, under strict quality control. The underlying assumption in manufacturing is that the variability is bad (unwanted) and standardization is good (wanted), but more to the point, it can be achieved. This anticipated controllability has permeated the psyche of Westerners and reinforced the normal curve description of variability leaving the impression that the world can be controlled, or it could be, with just a little more effort. The history of the success of the Industrial Revolution reinforces the linear world view and conspires with the basic desire of human beings to control their life to make suppressing variability attractive. But is this strategy for attaining predictability indicative of a deep

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truth, or does it merely impose an apparently desirable property that is useful in some contexts, but destructively harmful in others? Reducing variability to achieve a predictable outcome certainly simplifies manufacturing, as long as human beings are excluded from the process. Keep it linear, keep it simple, keep it working. However, the culture has devised even more insidious ways to reinforce the linear view of the world, such as grading on a curve. Every college student in the US, who has taken a large freshman class has, for the good of the professor, been graded on a curve. This curve provides a transformation from the A on your paper to the raw score of 47% you actually received. Actually, it is the other way around, but that is a detail. The distribution of grades within the class is forced to conform to the normal curve depicted in Figure 3.2. Grading on a curve means that 68% of the students in the class receive Cs, another 27% are equally divided between Bs and Ds, and the final 5% share the As and Fs. It has become an article of faith in the academic world that the distribution of talent or academic ability, whatever the subject, in a large class will follow the LFE. Consequently, the scores on the test ought to reflect this distribution. After taking a number of these classes students recognize what is expected of them and adjust their behavior accordingly. They generally accept the academic reality, at least on an unconscious level, that something as nonsimple as the interactive activity of teaching and learning, can be represented by a linear additive process. This is a gross distortion of the learning experience, but it seems that the education establishment cannot be weaned from such grading practices. The data for education/learning of scientific disciplines has been shown to support a heavy tailed distribution, which is contrasted with the bell-shaped curve in Figure 3.3. But like many other norms in society, it is not a matter of the truth of the position, as it is a matter of its perceived utility. It is a much more efficient use of a professor’s time to grade on a curve, than it is to devise examinations that reflect the student’s absolute knowledge of a subject, or so it seems to the professors. The eventual watering down of the

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subject being taught/learned is an inconvenient, but inescapable, consequence of this methodology that not only goes undiscussed, but unacknowledged.

Figure 3.3: A sketch of the Normal and Pareto distributions are presented on log-linear graph paper. The sketch is intended to emphasize the inverse power law nature of the Pareto distribution. The Pareto Law is also referred to as the 80/20 Law, because in many applications 80% of the results are obtained from the efforts of 20% of the people. A more complete discussion is given in the text. In the world view of the Machine Age, everyone should make the same amount of money for the work they do. Some might make a little more, and others a little less, but the average salary ought to be a good indicator of the state of the economy. In such a world everyone would be considered a mediocre artist; play a musical instrument poorly; all sports would result in a tie and there would be no heroes. No one would be outstanding and everyone would be nearly equal and equally uninteresting. But that is also not the world in which we live. In our world there is Michael Jordan and Shaquille O’Neil, who

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destroyed the competition in basketball; Pablo Picasso and Salvador Dali, who made introspection a visual experience; Frank Sinatra, who captured the soul of generations with his singing; as well as, people who make a staggering amount of money as income, whether deserved or not. In the real world there are market crashes and economic bubbles, earthquakes and brain quakes, as well as, peaceful demonstrations and deadly riots. There is extreme variability everywhere, which would be considered outliers by the normal curve, but are, in fact, no different statistically from the rest of the population. They are the disrupters of the status quo, Taleb’s Black Swans [224]. The point is that we live in a nonsimple world [60], but our understanding and way of thinking about that world has its roots in the simple models formulated to function in the Machine Age. The distribution of income is not a bell-shaped curve, with a well-defined average and standard deviation. It is a curve that has a very long tail, as shown in Figure 3.3. This distribution, first discovered at the end of the nineteenth century by the engineer turned sociologist Vilfredo Pareto [184], is one more appropriate for the Information Age. It captures the extreme variability that defines not only the boundaries of our life, but the unexpected events, both large and small, that take us by surprise.

3.4

Nonlinearity and Contradiction In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual. — Galileo Galilei [96]

The distribution of performance looks more like the long-tailed distribution given in Figure 3.3, where the median of the distribution, the point where 50% of the weight is above and 50% below this point, is skewed by the tail of the distribution, since there is no mode. This distribution with a long tail is named after its inventor, the nineteenth century engineer turned sociologist, Vilfredo Pareto,

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who discovered this behavior in his analysis of the distribution of wealth within European City-states. How well does the Pareto distribution explain the capability paradox? The paradox is a consequence of the logic of linear systems including those with normal statistics being inappropriately applied to athletic capabilities. In the linear paradigm a limited number of individuals excel and should be rated highly. In school this is the top 2.5% and would be the recipients of the A’s and right behind them the 13.5% of B’s in academic rankings. In turn, these individuals would be followed by 68% of the class that receive a satisfactory, or average score of C, and bringing up the rear are the 16% of D’s and F’s. But according to a recent study by O’Boyle and Aquinis [178], this is not what the empirical distribution of capabilities is observed to be. O’Boyle and Aquinis conducted 5 studies involving 198 samples, including over 600,000 entertainers, politicians, researchers, as well as, amateur and professional athletes. Their results are consistent across industries, types of jobs, types of performance measures, and time frames, and indicate that individual performances follow a Pareto, not a normal, distribution. Thus, these results have implications for all theories and applications that directly, or even indirectly, address the performance of individuals, including performance measurement and management, training and development, personnel selection, leadership, and the prediction of performance. The Pareto distribution is skewed towards the extreme, indicating that certain members of the cohort are more capable than would be expected based on the bell-shaped curve. Consequently, the wealthy are observed to make not just a little more money than the average person, instead their income is profoundly more than the average income, indicating that wealth is a measure of a person’s ability to make money and little else. But returning to the capability paradox, it is true that I am likely to be less capable than my friends on average, being a member of what O’Boyle and Aquinis called the ‘rest’, based on the analysis of

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athletes [178]. Unless, of course, I am one of the athletic leaders, a member of what they called the ‘best’, out in the tail of the distribution, which I can assure you I am not. Yes, a significant majority of those being measured are below average; the size of the majority depends on what ability is being assessed. Therefore, it is not a mystery that when the average is thought to be representative of the group a paradox results; a paradox that is a consequence of the empirical imbalance. Pareto labeled this disparity between what is observed and what is expected to be true, The Law of the Unequal Distribution of Results, and referred to the inequality in his distribution more generally as a predictable imbalance, resulting in social inequality as explained by West [258]. One measure that is of particular concern to academics is the impact their published research has on other researchers. Historically, this impact has been quantified using the number of times a particular paper is cited by other scientists within a given time interval. The number of scientists having a paper receiving a given number of citations versus the number of citations within a given year is depicted on log-log graph paper in Figure 3.4. The distribution in the number of citations is seen to be given by a heavy-tailed, or inverse power-law, distribution, where the first 35% of all papers published have zero citations, the next 49% have one citation, and so on. The average number of citations is between three and four and the average appears near the point at which 96% of all published scientific papers occurs. Consequently, no more than 4% of all scientists in the world have published papers that are cited four times, or more, in a given year [72]. Consider a young faculty member who is being evaluated for tenure. The tenure committee finds that her publications have received the average number of publications and one person on the committee argues that tenure should not be awarded. The argument is that the impact the candidate’s research has had on the scientific community is just average. This failure to understand that having an average number of citations puts the candidate’s paper beyond that of 96% of her competitors, places her tenure in peril.

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Figure 3.4: The number of scientists versus the number of citations their paper has received on log-log graph paper. Note that the average number of citations per year is 3.2 so that on log base ten graph paper this is 1.16, which occurs after 96% of all scientific papers have been exhausted. Numbers taken from pg.105 of the de Sola Price’s book [72].

Fortunately, there was a committee member that explained the implications of the Pareto distribution to the rest of the committee and pointed out that, in this case, being average was truly exceptional. She got tenure. But it is not just at the level of the individual that paradox is encountered. The paradox explained by an IPL PDF is also seen within organizations to be deeply embedded in the bureaucracy, having significant implications for the organization’s operation. Martin [157] observed that the rule governing how human institutions behave is paradoxical and he conjectured that the way devised to handle paradox in the social domain is through politics. Politicians are often

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excoriated for being liars, or for abandoning their principles, which they all too frequently do. However, such behavior is not always capricious, or arbitrary, but is often a necessary step in resolving a paradox in order to reach an accommodation; a compromise that requires seeing the value in each of the conflicting points of view, simultaneously. But the paradox is an apparent irreconcilable contradiction that occurs because the observer is viewing the scene through the bell-shaped lens of linear thinking. One thing that makes the research of Pareto so remarkable is that it contradicts the understanding of the nonsimplicity within social and biological phenomena, developed in the nineteenth and early twentieth centuries. Prior to his studies it was believed that data sets gathered from social phenomena have a normal distribution, such as measured by the English scientist and eccentric Sir Francis Galton, a cousin of Charles Darwin. He believed that the frequency with which an event occurs and the duration of its persistence reflect the nature of the underlying causes [97]. Statistics was a new way of thinking about complex social phenomena that was being championed by social scientists such as Quetelet, among others. As Cohen [59] pointed out, one way of gauging whether the new statistical analysis of social phenomena was sufficiently profound to be considered a revolution was to consider the intensity of the opposition to that way of thinking. One of the best-known opponents to statistical reasoning was the philosopher John Stuart Mill, who did not believe that it was possible to construct a mathematics that would take the uncertainty that prevails in the counting of social events and transform that uncertainty into the certainty of prediction required by science [166]. Of course there are many that today share Mill’s skepticism and view statistics as an inferior way of knowing. But what does the statistical distribution mean? Even though members within a population vary, here the population is a sequence of measurements; the characteristics of the population as a whole are themselves stable. The fact that statistical stability emerges out of individual variability has the appearance

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of order emerging out of chaos and has inspired a number of metaphysical speculations. On the other hand, nonsimplicity science does show that randomness is lawful in the scientific sense. It constitutes a shift of focus from the individual, who can be quite variable, to that of the system, which can be stable even when consisting of aggregated unstable elements. Without going into the mathematical details, it is clear from Figure 3.3 that at the extremes the probability for a normal variable drops precipitously, as a consequence of the exponential nature of the normal curve. On the other hand, the IPL of Pareto follows the more gentle downward slope depicted in the figure. The Pareto distribution corresponds to those in the upper few percent of the income distribution and this income imbalance is also found in the distribution of wealth in the twenty-first century within Western Society. The Pareto distribution reveals a disproportionately small number of people, claiming a disproportionately large income. These are the individuals out in the tail of the income distribution. The imbalance in the distribution was identified by Pareto to be a fundamental social inequality and he concluded, after much analysis, that society was not fair [184, 260]. The Pareto world view, with its intrinsic imbalance, is very different from the balanced view of the normal curve. The Pareto imbalance is a fundamental feature of the Information Age and is a manifestation of the modern world’s nonsimplicity. The nonsimple phenomena of biological evolution, letter writing, turn-taking in people talking, urban growth, and making a fortune, are more similar to one another statistically and to the distribution of Pareto, than they are to the normal curve of individual heights within a population. The chance of a child, selected at random from a large group, becoming famous or rich, is very small, but it is still much greater than the chance of that same child becoming tall. A person’s height is a hard ceiling determined by nature, genetics, whereas a person’s wealth or fame is a much softer upper limit that allows for the dedicated to overcome the social barriers that impose artificial limitations. But no matter how much you stretch

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you cannot become taller. On the other hand, many racially biased people have attributed characteristics to minority groups, such as to new emigrés, that were, in fact, socially determined, such as an innate inability to read or do arithmetic. Consequently, once the social barriers were eliminated, or at least lowered, the apparent inability disappeared. I do not bring up individuals such as George Washington Carver, because they would be dismissed by these same people as being anomalies. I am equally sure that such people would have found more than sufficient reason to stop reading long before reaching this point in the essay. The Pareto distribution burst into the general scientific consciousness at the turn of this century, with the realization that it described the connectivity of individuals on the Internet, social media and of computers on the World Wide Web (WWW). This was followed by a flood of applications, showing that such IPL, scalefree networks could model the failures of power grids; explain consensus formation during group decision making; describe how small groups collapse into groupthink; characterize the collective neuron discharge in brain quakes; clarify the turn-taking in conversations; make transparent the mechanism of habituation; quantify the variability in stride intervals during normal walking and a broad range of other phenomena in the social and life sciences, see e.g. West and Grigolini [255] for a list of references to such phenomena. The understanding has been growing that it is the network structure of social organizations that manifest their nonsimplicity in the non-intuitive statistics of Pareto. Results are non-intuitive because IPL statistics are so different from that of the LFE. For example, it is often the case that in real data sets, not under laboratory-controlled conditions, that the second moment diverges, consequently the standard deviation is ill-defined and the traditional measure of the mean as the best way to characterize a data set breaks down. This can be observed in the bursty behavior of a time series, which is a manifestation of Pareto statistics, such as depicted by the exemplars in Figure 3.5.

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Figure 3.5: Depicted is the bursty behavior in naturally occuring time series of nonsimple phenomena: turbulence, solar radiation, EEG and hypoxia.

What caught the attention of many, who were attempting to interpret the bursty behavior of network time series, was the fact that scale-free networks are robust against attacks for which they had been designed, but are fragile when confronted with unanticipated challenges. A common feature of all these nonsimple networks is their dependence on information and information exchange, rather than on energy, or energy exchange. The implication is that the mechanical forces generated by energy gradients that dominated the physical modeling of the Machine Age may no longer be primary. The new Information Age dynamics entail a new way of thinking, because of the IPL PDF and the associated nonsimplicity-induced information imbalance. Understanding the implications of this transition from energy-based to information-based forces [259], is a challenge for the modern day policy maker.

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A word of caution is in order before we continue. IPLs arise in a wide variety of venues and although they all have the imbalance noted by Pareto, the causes for the imbalance may be as varied as the phenomena in which they arise. There is one road to the normal curve of the Machine Age and it is straight and narrow, if somewhat steep. However, there are many roads to the Pareto distribution of the Information Age; some are wide and smooth, others are narrow and tortuously convoluted, but they are all the result of complexity (nonsimplicity) in its various guises [258].

3.5

Statistics and Taylor’s Law All truths are easy to understand once they are discovered; the point is to discover them. — Galileo Galilei [96]

Taylor’s law [229] is known as the power curve in the ecological literature [113]. It is a nonlinear power-law relation between the average of a variable of interest, say μ = N , where N is the number of new species, with the brackets denoting an average and the vari2 ance of the number of new species is σ 2 = N 2 − N , resulting in the relation: σ 2 = aμb .

(3.1)

In this equation a and b are empirical constants that determine how the average and variance are related to one another. We tend to think of a process as having a single mean and single variance, but that turns out to be a naive view of nature’s subtle variability. A nonsimple phenomenon could very well generate processes whose statistical behavior depends on the scale at which they are viewed. Such statistical processes were named fractal by Benoit Mandelbrot [153], the mathematician that introduced them into the scientist’s lexicon. Consider a computer experiment to generate the appropriate data and then fit Eq.(3.1) to that data set. For the sake of ar-

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gument we use a computer package implementing Normal statistics, having a mean of one and a standard deviation of two to generate 106 data points. The results of these calculations were presented elsewhere by West [253], but I will repeat pieces of that discussion here for clarity. In Figure 3.6a a computer generated data set is used to plot the variance versus the average for the aggregated data set. In Figure 3.6b is plotted the logarithm of the variance versus the logarithm of the average, for the same aggregated data set. At the extreme lower left of both figures the first dot denotes the value of the variance and average obtained using all the million data points. Moving from left to right, the next dot we encounter corresponds to the variance and average for the time series with two (m = 2) adjacent elements added together, yielding a data set of half a million data points. The next point is the mean and variance calculated with one-third of a million data points, obtained by aggregating every three (m = 3) sequential data points from the original data set. Each of the successive dots in the figure, moving from left to right, corresponds to the ever-greater aggregation number m. Consequently, this process of aggregating the data is equivalent to decreasing the resolution of the time series, and as the resolution is systematically decreased, the adopted measure, the relationship between the mean and variance, reveals an underlying property of the time series that was present all along. The linear increase in the logarithm of the variance with increasing logarithm of the average as the aggregation number increases depicted in the figure is not an arbitrary pattern. The relationship indicates that the aggregated uncorrelated data points can, in fact, be interconnected. In fact, if we take the logarithm of Eq.(3.1) we obtain log σ 2 = log a + b log [μ] ,

(3.2)

which is the algebraic form of the solid curve fitting the data in Figure 3.6b. The original data points are not correlated, but the ad-

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Figure 3.6: (a) Computer-generated data that satisfies Taylor’s law is plotted. (b) The same data used in (a) is plotted on log-log graph paper. It is evident that the log-log graph yields a straight line when the average and variance satisfy Eq.(3.1). Taken from [261] with permission.

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dition of data through the aggregation process induces a correlation, one that is completely predictable. The induced correlation is linear, that is, the slope of the curve is unity so that b = 1 in Eq.(3.1) if the original data is uncorrelated, but it is not linear (b = 1) if the original data is correlated. Thus, the qualitative nature of an empirical data set can be determined by the value of b. The power curve was first constructed by Taylor [229] in an ecological context to quantify the number of new species of beetle discovered in a given plot of ground. His method of collecting data was to section off a large field into squares and in each square sample the soil for the variety of beetles that were present. This enabled him to determine the distribution in the number of new species of beetle spatially distributed across the field. From the distribution he could then determine the average number of new species and the standard deviation around this average value. After this first calculation he partitioned his field into smaller squares and redid his examination, again determining the mean and standard deviation in the number of species at this increased resolution. This process was repeated a number of times, yielding a set of values of averages and corresponding standard deviations. Taylor was able to exploit the curve in Figure 3.6 in a number of ways using the two parameters. If the slope of the curve and the intercept are both equal to one, a = b = 1, then the variance and the mean are equal. This equality is only true for a Poisson distribution, which, when it occurred, allowed him to interpret the number of new species as being randomly distributed over the field with the number of species in any one square being completely independent of the number of species in any other square. If however the slope of the curve was less than unity (b < 1), the number of new species appearing in the squares was interpreted to be quite regular. The spatial regularity of the number of species was compared with the trees in an orchard and given the name evenness. Finally, if the slope was greater than one (b > 1), as shown in Figure 3.7, the number of new species is clustered in space, like disjoint herds of sheep grazing in a meadow, or flocks of sparrows breaking

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up to avoid a predator. This spatial clustering is analogous to the intermittency observed in the time series for nonsimple signals, as pointed out by Taylor and Woiwod [231].

Figure 3.7: The power curve is a graph of the logarithm of the variance versus the logarithm of the average value. Here three such power curves are sketched in which the three ranges of values for the slope determine the general properties of the underlying statistical distribution of new species. Adapted from [253] with permission. As pointed out in Section 2.5, Taylor and Taylor [230] used the language of mechanical forces and postulated that it is the balance between the attraction and repulsion, migration and congregation, that produces the interdependence of the spatial variance and the average population density. West [261] reinterpreted their proposed mechanism as being an imbalance in complexity, or information, such that there is an information force, resulting from an imbalance in nonsimplicity, here interpreted as selection pressures, acting in opposition to one another.

3.6.

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Paradox and Uncertainty It is change, continuing change, inevitable change, that is the dominant factor in society today. No sensible decision can be made any longer without taking into account not only the world as it is, but the world as it will be. — Isaac Asimov [18]

Previously we explained the emergence of EP qualitatively, using the difference between the LFE and Pareto’s law. In that interpretation paradox is a consequence of the misperception that the erratic data is generated by nonsimple phenomena, which are properly described by Pareto distributions, are instead interpreted as if they were merely complicated and consequently described by bell-shaped distributions. The failure to take into account the full range of variability contained in the Pareto distribution of a process, due to this misconception, is what ushers one into EP and is a consequence of using linear thinking rather than the more appropriate criticality thinking. The distinction made between simple and nonsimple (criticality) thinking, at the policy maker level within a bureaucracy, is equally valid at the level of the individual within the same organization. It is criticality thinking that resolves paradox. In fact, given that the individual has less control, s/he must be more adept at resolving paradox by innovative thinking than is the policy maker. The advice from other policy wonks and panels of subject matter experts, followed by thoughtful critical analysis by economic leaders, all enter into the determination of new policy to resolve an intellectual conflict, or EP, within the bureaucracy. The individual employee does not have the luxury of such on-site support groups. A new strategy is required to directly confront EP, either alone, or in the company of a small band of peers, who, with more or less the same level of expertise, must resolve the EP in a relatively short time frame or they will be unemployed, if not worse. It is useful to have in mind the various ways EP can come about

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in organizations. Computational social science relies on the assumption that criticality, such as reaching consensus, is a consequence of self-organization. The same can be said for complex biological, physical and sociological phenomena. A number of investigators have devised mathematical models by which to quantify critical phenomena in the physical, social and life sciences. Such models, as reviewed by Sornette [212], describe the phase changes of solids, fluids and gases; the frequency and magnitudes of earthquakes; the flocking of birds; the crash of markets; the onset of brain quakes; the transition from demonstrations to riots; the list goes on and on. These models can be widely separated into two groups. The first group models phenomena that achieve criticality through the external manipulation of a control parameter, such as the changing of temperature in physical phenomena to induce phase transitions. The second collection models phenomena in which criticality emerges due to internal dynamics, without adjusting an external control parameter, and has acquired the name self-organized criticality (SOC) [249]. West et al. [263] generalized SOC models, using a two-level dynamic network model of decision making, the self-organized temporal criticality (SOTC) model, having criticality in time as an emergent property. In the context of the AP the SOTC model drives the selfish behavior of individuals within a social group to generate cooperation and simultaneously achieve maximum social wellbeing. Previous models attempted resolution by placing the good of the individual and the good of society in competition, with a winner and a loser, an either/or contest. The SOTC model, as we subsequently show, reveals that an individual can consistently act in their own self-interest, while simultaneously attending to the social benefit of their decisions, independently of any specific physical, social or biological mechanism, thereby replacing the either/or with both/and resolutions. As discussed subsequently, the mathematics demonstrates that the intellectually conflicting notions of selfishness and selflessness, are not logically inconsistent when treated dynamically. The dynamic model reveals that what is good for society need not be purchased at the cost of what is good for the individual.

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We emphasize that the SOTC model extends evolutionary game theory, to explain the resolution of EP. The resolution of paradox is a consequence of scaling and criticality in the nonsimple decision making process. In the SOTC model criticality is not forced upon the network by externally setting a system parameter to a suitable value. The critical value of the parameter strength is spontaneously reached, without artificially raising or lowering its level within the network from the outside. Instead, criticality is dynamically attained by having each individual within the network select the value of their individual interaction strength that results in maximum benefit to themselves at every point in time. Individuals weigh each and every decision they make, using the two networks of intuition and cognition, to adapt to the changing composite-network behavior. The SOTC model has the intensity of the imitation strength, when using the DMM, selected by the individuals on the basis of increasing their own benefit, rather than as a form of blind imitation. It establishes the emergence of cooperation through the use of a PDG payoff, thereby connecting the evolution of global cooperation with the search for agreement between individuals and their nearest neighbors. This balancing of the tension between intuition and cognition not only resolves EP, but gives pleasure in overcoming the apparently impossible. More generally, within a bureaucracy the role of leadership is to support opposing internal forces and harness the continuous tension between them, enabling the system to not only survive, but to continuously improve. Of course, this is the ideal case, all too often bureaucracies become stuck at the level of survival and need to be jolted from their complacency.

Chapter 4

Two Useful Models In this chapter we introduce a network-based model of decision making that is used as the basis for subsequent understanding of EP. The detailed mathematics underlying the model is provided in Appendix A, so that the mathematical equations do not interrupt the narrative flow. A number of calculations that reveal the interpretive content of the model are presented graphically in the text. The decision making model (DMM) assumes that the intuitive component of decision making is based on individuals imitating the opinions of others and leads to collective effects. These effects were suggested by, but are distinct from, physics-based models of phase transitions. The DMM shares a number of the mathematical properties of the physics models such as scaling, criticality and universality, but has severed its connections to its physics heritage and relies solely on psychological/sociologicalbased assumptions. The DMM mimics fast decision making without rational thought. A second modeling strategy discussed in this chapter addresses rationality in decision making and this is game 95

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CHAPTER 4. TWO USEFUL MODELS theory, or more properly, it is evolutionary game theory (EGT). This modern form of game theory incorporates dynamics into the decision making process, allowing individuals to modify their behavior over time in response to the payoffs others in the network receive for how they behave. The review of EGT in this chapter is brief, given the large number of excellent books and the extensive literature on the subject of game theory, however, we do compare and contrast DMM with an evolutionary game model (EGM).

4.1

Modeling Sociology Since all models are wrong the scientist cannot obtain a "correct" one by excessive elaboration...Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity....George E. P. Box [47]

A significant part of everyday life consists of decision making; both small, deciding what to have for lunch, and large, whether to move to a new city. Many personal choices are matched with collective decisions, made within a group of friends, or as part of different types of committees and boards. Acting within a group offers unique advantages to an individual, in that it allows that person to accomplish tasks that could not, or would not, be completed alone. Moreover, groups often act intelligently when selecting among an array of alternatives, while working to complete a complex task. By pooling imperfect and partial information available to its members, individual erroneous decisions tend to cancel each other out, leading to an improved collective opinion, in a phenomenon known as the wisdom of the crowd [218]. The idea that decision accuracy increases monotonically with group size is suggested by both theoretical and experimental re-

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search, with large groups making more reliable judgements than do smaller groups. The uncertainty of the environment experienced by the individual is mitigated by forming groups, which tends to enhance fitness by developing such specific behaviors as optimal foraging strategies [208]. Groups of people demonstrate a general collective intelligence that has been used to explain group performance on a wide variety of tasks, which does not correlate with the individual intelligence of group members [188]. However, as observed by Kao and Couzin [130], many social animals make decisions in relatively small groups, apparently not capitalizing on the advantages of large group size. Such strategy is often explained by the need to balance the costs and benefits of group membership, where larger group size increases competition for resources, thus affecting chances of survival. With the help of numerical modeling, Kao and Couzin [130] demonstrate that in nonsimple environments, characterized by some degree of correlation and overlap in the information available to group members, classical wisdom of the crowd phenomenon is not observed. Instead, small finite groups maximize the accuracy of decision making process. Economic and logistic factors naturally restrict the size of groups of people, particularly in professional settings, such as academic or government committees, executive boards, or even juries. These small groups are not free from displaying certain negative aspects of cooperation [167] even though they operate under the assumption of improving the accuracy of decision making. The extremely negative outcomes of small group decision making process were recognized by Janis [124, 125] as groupthink, which he argued promotes corruption [252], dishonesty and unethical behavior. The groupthink phenomenon denotes instances when a small group, of otherwise intelligent individuals, when working together, reach consensus on what is arguably the worst possible decision available; one that no member of the group would have reached working alone. Factors such as the tendency to minimize conflict and reach a consensus decision without lengthy critical evaluation of alternative ideas, or points of view, increase the likelihood of the groupthink mechanism impacting the

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decision making process. Since the primary consequence of groupthink is the loss of independent thinking and individual creativity, a possible countermeasure is to create constructive conflict within the group. West and Turalska [264] explored the impact of a devil’s advocate on the effectiveness of small group’s decision making process. They assumed that individuals within a group make decisions by adopting the fundamental mechanism of imitation. At the dawn of the twentieth century it was argued qualitatively [228] that imitation was the mechanism by which the phenomena of crowd formation, fads, fashions and crime, as well as other collective behaviors, could be understood. It was also argued [29, 30] that imitation theory developed out of the mental development of the child, resulting from imitation being a basic form of learning. Currently, imitation remains an important concept in social sciences, being pointed to as a mechanism responsible for herding, information cascades and many homophily-based behaviors. This basic assumption might appear overly simplistic, but the significance of other-than-rational mechanisms of decision making is strongly emphasized in the literature on decision making. Wittmann and Paulus [269] point out that factors such as satisfaction and impulsivity can under certain conditions, such a drug dependency, become of equal, or higher, importance than rationality in the decision making process. Similarly, two popular books [15, 128] discuss how rational individuals often make irrational and illogical decisions. In light of these observations, reducing decision making to a process of imitating the behavior of close acquaintances becomes a reasonable assumption. Imitative behavior is captured by the DMM [235, 257], which consists of a network of individuals who imperfectly imitate each other. This dynamic model gives rise to cooperative behavior, induced by the critical dynamics associated with a social phase transition, and has been shown to be a member of the Ising universality class. However, unlike the physical Ising model, the fluctuations observed in social dynamics are a consequence of the finite size of the network, not the result of thermal agitation, as it would be in

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a physical system. Previous applications of the DMM concerned examining the consensus-seeking behavior of large networks [235], the dominance of committed minorities in changing the behavior of large social groups, and the influence of large groups on the behavior of the individual [257]. Herein, we briefly examine the dynamics of a small network to gain insight into the possible mechanisms that determine the decision making behavior of small groups. This is far from the thermodynamic limit of the Ising model in which there are essentially an infinite number of interacting individual units. For emphasis, let us interpret the opening quotation of this chapter to mean that models never exactly replicate the thing they are purported to represent in the real world, so we must strike a balance between simplicity and fidelity. However, in defense of even the poorest models, we concede that they do enable us to organize our thinking about difficult subjects, sometimes providing deep insights into phenomena we seek to understand. We adopt this strategy in the present chapter and discuss a number of apparently disjoint aspects of nonsimple dynamic networks. First, we exploit the DMM, which we mathematically define in Appendix A [257], as a way to make rapid decisions. Second, we discuss an EGM as a way to describe slower deliberative decisions. How the two interact and compliment one another is discussed, but the details of a new and extraordinary mode of interaction are postponed until the next chapter. Social patterns that persist over time are often notoriously difficult to explain. Consequently, when Stanley Milgram [165] clarified the familiar experience of the small world problem, using a brilliant social experiment, it came as a surprise. He based his explanation of the small-world experimental results on the probability of two strangers being indirectly connected through a small number of individuals directly linked by mutual acquaintances, which subsequently became known as the phenomenon of six-degrees-of-separation. Milgram identified this chain of connections as a mathematical structure within society and his experiment confirmed the convergence of opinion that the world was in fact becoming more interconnected. This view was subsequently theoretically vindicated by Watts and

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Strogatz [250] though the development of the small world dynamic network model. Small world theory introduces relatively few interconnections between random individuals, each within its own densely internally connected, but isolated cluster, and thereby shortens the average separation distance between randomly selected individuals and is depicted in Figure 4.1.

Figure 4.1: A regular network with nearest neighbor and next nearest neighbor interaction is on the left. On the right is a random network with randomly assigned interacting units. In the center the regular network with local interactions is supplements with randomly assigned long-range interactions, constituting a small world network. Small world theory initiated an avalanche of research studies into the nonsimple nature of social networks, over the past two decades. It also introduced, to a generation of scientists, a network calculus that enabled an understanding of how and why group properties can be so different from the properties of the individual members of the group. For example, although a group may have reached consensus, the opinion of any given individual may fluctuate in time. Consequently, the dynamics of the group is more than the aggregate behavior of the individuals belonging to the group and is strongly dependent on the interaction between and among group members.

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The emergent behavior observed in a variety of animal groups by naturalists, including, the swarming of insects [271], the schooling of fish [131] and the flocking of birds [53] can be related to the collective and cooperative behavior observed in human society studied by psychologists and sociologists [257]. In this chapter we examine the collective structure of social networks and determine that they are not static, but dynamic and under well-defined conditions rapidly fluctuate into and out of one consensus state, without necessarily making a transition into another. We refer to this random losing and regaining of consensus, in very short time intervals, without making a transition between consensus states, as commitment flicker. We identify commitment flicker as a mechanism responsible for the flexibility in the rapid response capability of many large groups. It is the time during which a very small internal group may take advantage of the transient loss of organization to redirect the behavior of the significantly larger group of which it is a part. It is also the time during which the larger group is susceptible to the external influence of a small, but committed, external group. The analysis of group behavior is based on the DMM [257], as well as, the EGM.

4.2

Decision-making Model (DMM)

Our research group [257] adopted a small number of assumptions about how to model the behavior of individuals interacting within groups, say a neural network, a gatherings of animals, or a society, and from this set of assumptions constructed the DMM, which among other things emulates how weakly interacting individuals within groups reach consensus. In our formulation of the DMM we adapted a technique that had been used successfully in nonequilibrium statistical physics to model the dynamics of the probability of occurrence of an event. The time rate of change of the probability in the physics literature is determined by what is called a master equation, which describes how the probability of being in

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a specific state changes over time. This determination would be given by a birth-death equation in ecology, but with the probability replaced with an animal population. The DMM explicitly assumes that the interactions between the individuals in a nonsimple social network is represented by a coupled web of two-state master equations, one for each member of the population. The mathematical details of the DMM are collected together into Appendix A. A parameter is introduced to characterize the strength of imitation between members of the network. When this parameter is vanishingly small the network members behave independently of one another, and consequently exercise free will. We assume the existence of two states that the individuals must choose between; whether to become a Democrat or Republican, to accept or reject the new job posting, to be or not to be, and so on. As the magnitude of the imitation parameter increases, an individual’s behavior depends increasingly on the behavior of the other people in the network, the network’s dynamics, which some interpret as an expression of a collective intelligence. We assume imitation to be the underlying social interaction on which to base the DMM. At least one thing not apparent at the time we first made the echo response hypothesis was the extent of the social implications implicit in this apparently benign assumption. Consequently, uncovering these implications constituted a major part of a research monograph [257]. Emphasizing the social implications, Tarde [228] argued that imitation was the fundamental mechanism by which crowds, fads, fashions and crime, as well as other collective behaviors, could be understood. However, no mathematical demonstration was provided prior to the DMM in support of his arguments and there was ample criticism of his assumption in the literature [80]. The early discussions on the limitations of the imitation hypothesis failed to address the fact that no copy is perfect, so that imitation introduced uncertainty into the social model. No process ever yields exactly the same result twice, so every copy is unique and contains deviations from the original, that is, they contain uncer-

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tainties. Consequently, the social model we introduce is in terms of probabilities of observing a given behavior and not in terms of the behavior itself.

Figure 4.2: In this photograph we see society being reborn within the child through the simple process of imitation, unencumbered by analysis. Mirrors and echoes are physical imitators; the former is produced by reflected light and the latter by reflected sound. In an analogous way people imitate one another through reflected behavior, as does the small boy, depicted in Figure 4.2. But as stated the imitation is always an imperfect echo, as dramatized in the cartoon of a funhouse mirror depicted in Figure 4.3. We have elsewhere [257] referred to this imperfect imitation in the context of the decision making model, as the echo response hypothesis. The distortion in the reflected behavior can be large or small, static or dynamic, however it can never be exactly zero. The accumulated distortion in the reflection process measured at the receiver

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Figure 4.3: Imperfect imitation is like the distorted image from a funhouse mirror. All the elements of the original are there, but they are deformed and not quite where they are supposed to be. From [257] with permission.

is determined by a stochastic function and is referred to as noise. It was never our intent to develop a theory or model of imitation, but rather it was to adopt the concept of imitation as a basic social/psychological mechanism that is part of what it means to be human. The DMM adopts the hypothesis that imperfect imitation (echo response) is fundamental to human behavior and implements the hypothesis in mathematical form in networked decision making. The resulting dynamics were used to determine what the echo response entails about behavior that is not implicit in the initial statement of the model, and to determine if that behavior corresponds to the cooperation observed in social interactions [256]. Subsequently,

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in Chapter 5, we develop a modified form of DMM to explore implications regarding EP in its many forms. The model society we discuss is assumed to consist of N individuals, each of which can switch between the two values, +1 and −1. The choice of the interaction strength between individuals is the heart of the model and is a consequence of the echo response hypothesis. Mathematically, the assumptions made are manifest in N coupled two-state master equations. This is the DMM in which each individual is represented by the time-dependent probability of making a transition from one state to the other [257] and an individual’s decision to change their opinion, or not, is strongly influenced by what others in the network do. The observable used to characterize the behavior of the society is the average opinion held by its members. If the opinion held by individual j is sj (K, t) = ±1, with a strength of the imitation parameter K, the global variable, or mean field, is N

ξ (K, t) =

1 N1 (K, t) − N2 (K, t) sj (K, t) = . N j=1 N

(4.1)

Note that the mean field has a value in the range [−1, +1] and is the instantaneous difference in the fraction of the total number of individuals in the two states, which changes in steps of 2/N as individuals change their opinions.

4.2.1

Criticality

As discussed elsewhere [236] and briefly in Appendix A the DMM and the Ising model have a number of similarities in their emergent properties, such as criticality, even though the origins of their interactions are quite different. Stanley et al. [213] explain that the constant interaction in the Ising model of a ferromagnet is relatively uncomplicated. Of particular interest is the fact that when two neighboring spins are parallel there is a negative contribution to the total energy. Consequently, the lowest energy of the magnet

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system is the configuration having all spins parallel. This relatively simple model has been extremely successful in helping to formulate two of the most powerful concepts in many-body physics: scaling and universality. Both were developed in the study of dynamic nonsimple physical networks in the vicinity of critical points, where, as they emphasize, an effective long-range interaction between elements appears that was not included in the short-range interactions of the original Ising model dynamics. This effective long-range interaction is manifest when the temperature, which is the external control parameter in many physical systems, is at its critical value. The same qualitative behaviors occur in the social network, including criticality, scaling and the emergence of effective long-range interactions, but for different reasons. Since the opinion of each individual is changing, with a time-dependent probability, the global opinion of the society is also flickering in time. Typical time series for three values of the echo parameter, selected for a typical individual in the DMM, are depicted in Figure 4.4a and for the mean field in Figure 4.4b. One can clearly see a shift from the behavior dominated by the lack of consensus for K in the subcritical regime K < KC , shown in the upper panel of Figure 4.4b, to a state in which the majority of individuals share the same opinion at the same time, that is, in the critical and supercritical regimes K ≥ KC in the lower two panels of Figure 4.4b. These time series clearly indicate a transition from an aggregation of independent opinions to a time-dependent shifting consensus, as the echo parameter is changed from its subcritical to its critical value KC and beyond. The statistics of an isolated individual is prescribed to be a Poisson process, but when individuals are coupled together, as within a social group, with echo strength K, the statistics of the group, as measured using the mean field, is not obvious. To make matters worse, an additional source of uncertainty is the random fluctuations that persist even in the absence of transitions between states. These fluctuations are a consequence of the finite size of √ the network N , and are determined to have an amplitude of size 1/ N . Elsewhere

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Figure 4.4: (a) The transitions between two states for the single person sj (K, t) in the network; (b) the time series of ξ (K, t); the top, middle and bottom curves correspond to K = 1.50, 1.70 and 1.90, in that order. Lattice size is N = 50 × 50 nodes, calculations done with periodic boundary conditions, for g = 0.01. The critical value of the imitation parameter is KC = 1.72. (c) Dashed line denotes an exponential distribution, with the decay rate g = 0.01, K = 0. (d) Gray dashed line denotes an IPL function with exponent μ − 1 = 0.50. The three solid curves are the survival probability Ψ (τ ) corresponding to the subcritical, critical and supercritical time series moving from left to right, for a typical individual in c) and the entire network in d).

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we have determined the statistics of the network variables under a variety of conditions, both numerically and in some limiting cases analytically [257], see also Appendix A. The survival probabilities displayed in the Figures 4.4c and 4.4d are calculated by recording the times at which the time series considered passes through zero. The time interval between sequential zero-crossings constitutes a surrogate data set for the switching times from one state to the other. A histogram is constructed from these time intervals to obtain an empirical PDF between successive switchings, which is the waiting-time PDF ψ (τ ). The probability of not switching states in a time τ is given by the integral ∞

Ψ (τ ) ≡

ψ (t) dt,

(4.2)

τ

which became known as the survival probability due to its use in Extreme Value Theory. It is the probability that the process being considered has survived in a given state for a time τ without changing states, say from normal operation to failure. This is also the rational for using the zero-crossing time intervals as a surrogate data set. The probabilities displayed in Figure 4.4c are obtained from the time series for a single individual depicted in Figure 4.4a. It is apparent from the shape of the curves that the individual survival probability does not change very much from that of the isolated individual. This is an illusion of scale, however. At intermediate times the individual survival probability is an IPL, which asymptotically has a sharp exponential termination. Compare these probabilities with those displayed in the Figure 4.4d, which are obtained from the mean field time series depicted in Figures 4.4b. The mean field survival probability varies dramatically from that of a collection of non-interacting individuals indicated by the dashed curve. The IPL behavior of the mean field survival probability extends from three (subcritical) to five (critical) decades of time, having an IPL index of μ = 1.5.

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The DMM dynamics change with increasing values of the echo response parameter, until at a critical value the network undergoes a phase transition, analogous to that which occurs in physical systems having solid, liquid, and gaseous phases, see the phase diagram in Figure 3.1. The phase transition, or tipping point as it is known in the sociology literature [103], is indicated in Figure 4.5 using the time average of the absolute value of the mean field: ξ eq (K) = |ξ (K, t)| ,

(4.3)

where the brackets here denote a time average. In the all-to-all (ATA) coupling case, where each individual is connected to all the other individuals in the social group, the critical value of the imitation parameter is KC = 1.0, as seen by the location of the rapid transition of the solid curve in Figure 4.5a. At this value of the imitative coupling the elements in the network cease to be independent entities and instead coordinate their behavior. We saw this coordinating, or synchronizing, behavior in the dynamics depicted in Figure 4.4. A second form of the coupling within the DMM that we consider herein includes dynamics on a two-dimensional lattice, where only the behavior of nearest neighbors influence the individual. On a lattice network criticality again occurs, but in this case the critical value of the echo parameter is KC = 1.7, as determined by calculation and indicated by the dots in Figure 4.5a. At criticality the dynamics of both the ATA and nearest neighbor coupled DMM networks become synchronized, which is to say, the individuals within each of the networks reach consensus. Consequently, the once independent dynamics of the separate individuals become synchronized due to their imitative behavior, independently of how the members of the network are connected to one another. Recall that only the four nearest neighbors are allowed to interact with an individual on a simple lattice. The influence of these local interactions becomes global at the critical point, which suddenly induces long-range correlations through the amplification of

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Figure 4.5: a) The phase diagram for the global variable ξ eq (K) = |ξ (t)| in both the ATA case (solid) and the lattice (dots). The dashed line is the Onsager theoretical prediction [180] for the twodimensional regular network of infinite size. The dots corresponds to the global states observed for the DMM on a regular lattice (N = 100 × 100) and g = 0.01. Periodic boundary conditions were applied in the DMM calculations. The lower panel displays the spatial distribution of opinions, yes-white, no-black for echo parameter values, b-dot K = 1.0, c-dot K = 1.7 and d-dot K = 2.00.

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fluctuations. The dots in Figure 4.5a are the result of a DMM lattice calculation compared with the analytic calculation of the phase transition at the critical value of the control parameter, for the Ising model by Onsager [180]. Consequently, the social network begins to look suspiciously like a physical phase transition, where a reasonable individual in isolation can, through interactions with a social group, become part of an economic bubble, a riot, or a lynch mob. This apparent loss-of-identity phenomenon is captured in the remarkable book, Extraordinary Popular Delusions and the Madness of Crowds, where in 1841 Mackay [147] observed: Men, it has been well said, think in herds; it will be seen that they go mad in herds, while they only recover their senses slowly, one by one. Real social networks have a finite number of individuals, and as evident in Figure 4.5a the numerical calculations for large, but finite sized, networks overlay the phase transition predictions of Onsager [180] for an infinite network of spins. This is evidence of the DMM being a member of the Ising universality class and as one would expect it shares a number of scaling properties in common with the kinetic Ising model [238]. For values of the imitation parameter corresponding to the disorganized phase K < KC , individuals are only weakly influenced by the decisions of their neighbors and change their state with probability only slightly faster than the decoupled rate g. This appears in Figure 4.5b as an approximately equal number of individuals expressing both opinions randomly positioned on the lattice. Note the clustering in the distribution of opinions in Figure 4.5c at the critical point. Finally, there is a clear majority of the white over the black opinion in Figure 4.5d when the echo parameter exceeds the critical value. Of course, this latter configuration could also have been just a clear majority of white over black. The distribution in the locations of individuals with a specific opinion on the lattice cannot be determined from the mean field. The time series for the three values of K discussed in regard to

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Figure 4.5 are depicted in Figure 4.6. It is evident that for K < KC the fluctuations of the mean field ξ (K, t) are characterized by small amplitude, very fast, oscillations about the zero-axis, as shown in Figure 4.6a. For K > KC , the interaction between individuals gives rise to a consensus state, during which a significant number of agents adopt the same opinion at the same time, as shown in Figure 4.6c. Figure 4.6b depicts the DMM phase transition at the critical value of K, under the condition of nearest neighbor coupling on the two-dimensional lattice, where ξ eq (K) is used as a measure of the organization of the network. In Figure 4.6d the colored dots denote the values of K on the phase transition curve for which the mean field time series on the left and the survival probability below it are calculated. Figure 4.6e depicts the probability that the mean field will not transition out of its present state for a given time interval denoted by the waiting time. This is the mean field survival probability calculated, as mentioned above, using the statistics of the time intervals between successive zero-crossings as a surrogate for the statistics of the time interval between switching states. The dynamics depicted in Figure 4.6c is of most interest to us here, because it reveals a kind of behavior where consensus is lost, but only for a relatively short period of time, what we referred to earlier as commitment flicker. The mean field is seen to assume a value that passes through zero, the point at which there are an equal number of individuals in both states and therefore consensus is lost. However, the mean field often snaps back to the upper state without completing the transition to the lower state. It flickers. Why is that?

4.2.2

Control of Transitions

To clarify the above observations regarding the relation between the mean field, approaching criticality and the onset of long-range correlations, we compare in Figure 4.7 the behavior of a large group of individuals interacting on a lattice, at the instant of crisis, with

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Figure 4.6: These are the same time series calculations from which Figure 4.5 was constructed. Panels (a—c) depict the temporal evolution of the mean field for increasing values of the control parameter, (a) K = 1.50,(b) K = 1.62, (c) K = 1.66 and KC = 1.72. (d) Phase transition diagram for the amplitude of the mean field parameter ξ eq (K) as a function of the echo response parameter K. (e) Mean field survival probability Ψ (t) versus the waiting time for the three dynamic configurations; left curve first dot, middle curve second dot and right curve third dot. Lattice size is N = 100 × 100 nodes, and transition rate g = 0.01; all the calculation were done with periodic boundary conditions. Adapted from [238].

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those before and after a transition. We observe that the correlation function corresponding to the moment of crisis, in this case a transition, becomes significantly more extended than it is either before or after the transition, as depicted by the red curve in the lower panel of Figure 4.7. It is this extended influence that enables the network to transition from one state to the other, as shown in the upper panel. Note that the three inserts in the middle panel of Figure 4.7 display the spatial distribution of the white and black states on the lattice, at the three locations of the mean field time series. The white is the upper and the black the lower opinion state. It is apparent that in either of the majority states, the green square pre-transition and the blue square post-transition, the spatial distribution is predominately of one state randomly distributed, with a relatively small number of the other interspersed. It is also apparent that the transition between states is not instantaneous, but has structure, that is, fluctuations exist even in this non-committal region. The spatial distribution in the middle square, corresponding to the transition region, is quite clustered, while maintaining an essentially equal number of white and black states. The typical correlation between elements in a network separated by a distance r is exponential: C(r) ∝ e−λr ,

(4.4)

which is consistent with the notion that the percentage decrease of influence of one element on another decreases in direct proportion to the separation distance between them. The correlation length 1/λ is a characteristic of the network such that for separations exceeding this value the correlation function becomes negligibly small. The correlation between elements in the DMM is no exception in this regard and is found to decay exponentially for the control parameter away from its critical value. Note that the correlation length is not the same in the two majority states in the vicinity of the transition. This difference is due to the fact that the state of the lattice at the

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Figure 4.7: The mean field time series is depicted for a N = 100×100 size lattice network and is taken from Figure 4.6c. The up state is white, the down state is black, and the status of the group is shown pre-transition, during transition and post-transition. The spatial distribution of opinions is depicted by the three squares sampled at the indicated points of the mean field time series.The correlation function averaged over all individuals separated by a distance r in the three points of the time series indicated are shown in the bottom panel, the lower two are exponential, left and right boxes, and the upper one is inverse power law, middl box. The relative order of the two lower curves is not significant. Adapted from [238].

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two times is not the same and the correlation length is obtained by averaging over the spatial heterogeneity to determine the correlation between individuals separated by a distance r. At the critical point the correlation function in the DMM, just as in the Ising model, becomes an IPL of the separation distance [256]: 1 , (4.5) rμ as opposed to exponential. In the physics literature the IPL index μ is known as the critical exponent and is often used to characterize the physical process being investigated. This is the source of the scale-free nomenclature used to identify the properties of nonphysical nonsimple networks, when such networks do not possess a characteristic scale, see, for example, [213]. The critical exponents are found to be insensitive to the detailed dynamics of the non-physical network being studied, just as they are in physical phase transitions. Stanley et al. [213] explain that in the Ising model of a ferromagnet the IPL decay is a consequence of the multiplicity of interacting paths that connect two spins on a lattice of dimension two or greater. The intuition that has been developed over the years is that although the correlations along each path decreases exponentially with path length, the number of paths available increases exponentially as criticality is approached. The IPL is the optimal compromise between these competing effects. The fact that the numerical results obtained using the DMM give rise to the same scaling properties as that for the Ising model, suggests that the physical intuition may be adopted for understanding the decision-making process in social groups, as well. The lower panel in Figure 4.7 displays the autocorrelation function in each of the three tested regions of the mean field. The autocorrelation function is exponential in the tested regions of the two cooperative states. The correlation is seen to be vastly extended when there is a global transition between states, depicted in the middle square, where it is IPL as given by Eq.(4.5) and not expoC(r) ∝

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nential. Thus, more information transfer is required to induce a transition than is necessary to maintain an established consensus. This is a time of crisis, formally defined by ξ (t) = 0, the point at which there is no dominant number of individuals in either state. It is during such times of crisis that the collective nature of criticality is manifest, where a random excess of either state can be amplified to induce a transition in the direction of the excess, resulting in commitment flicker.

4.2.3

Committed Minorities

To investigate network properties in a time of crisis we take advantage of commitment flicker, and add to the DMM lattice a committed minority — a small group of randomly selected individuals, which have the character of being zealots. A zealot is a person who is 100% devoted to a certain issue, and such people do not change their opinion by interacting with their neighbors, or indeed with all of society. They are unreasonable, irrational, and generally annoying, or so I am told. A small group of zealots constitutes a committed minority that always holds the opinion YES (or NO). Since the committed minority always holds the same opinion it is just a matter of time until a crisis arises in the network dynamics into which they can exert their influence. A crisis is defined here as the social condition where the influence of the committed minority can be amplified by means of the long-range interaction within the internal network dynamics at criticality. In times of crisis it is only the zealots that have a uniform opinion and consequently they may take over, using the critical nature of the network’s dynamics to further their agenda. This same effect is observed in a flock of birds in their coordinated flight to avoid a predator. In this latter case the zealots are lookout birds at the periphery of the flock. When the lookout birds spot a predator, they communicate a signal to change the flock’s direction of flight. It is only at the time of crisis, however, when the flock is essentially instantaneously unstable and sensitive to pertur-

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bation, that the long-range interaction transmits this information about the predator and the flock responds in unison [242]. Thus, the optimal interaction strength to ensure the flock’s survival is the critical one, where there is ample opportunity for crisis to emerge and be controlled.

Figure 4.8: Temporal evolution for the global variable in the absence of the minority (upper) is compared with its evolution once a committed minority (lower) is present. Fluctuations of the global variable ξ(K, t) for K = 1.65 and the two-dimensional lattice of size N = 100 × 100 nodes are compared with the behaviour of ξ(K, t) once 1% of the randomly selected individuals are kept in state yes at all time. Adapted from [238]. To emphasize the influence of the committed minority, we compare the dynamic behavior of the mean field for a network, without a minority, to that with a committed minority in Figure 4.8. It is not that the minority group is strong, only 1% of the members of

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this 100 × 100 lattice are zealots and they are randomly distributed throughout the network. It is more how the zealots triggers the collective behavior of the group to act independently of what they have decided. The committed minority leverages the group’s long-range interactions during times of crisis and that is their strength. During these short time intervals the network may undergo an abrupt change of opinion and the correlation length is sufficiently large to make it possible for the committed minority to influence the social network to adopt an arbitrary view. As a consequence, during the time interval over which the minority actions can be significant, it imposes its opinion over the entire network [238]. This is the tipping point so often discussed qualitatively in the social literature [103]. Note that other investigators, using different mathematical social models, have also found this profound influence of minorities as well, but it typically occurs at percentages significantly higher than the 1% of zealots within the network used in the present discussion [94, 159, 168, 270, 275]. We conclude from comparing the present calculations with those of other investigators that the particular percentage of zealots necessary to dominate the opinion of a social group is model-dependent. However, in all cases the percentage is rather small, being 10% or less, depending on the model. The committed minority need not be static however, as has been assumed in other mathematical models; it need only be unresponsive to the dynamics of the social network it influences. One way to achieve this situation is by having two intrinsically identical nonsimple networks, but with different values for their echo response parameters, so they can be in different dynamic states when not in contact with one another. A small number of individuals from one dynamic network, say the mother network M, volunteer to replace individuals in the other dynamic network, say the host network H. The replacement individuals interact with their nearest neighbors in the same way the individuals did from the host network that they replaced. However these M-individuals do not respond to the H-individuals, they are only influenced by their mother network. In the top panel of Figure 4.9 the mean field of the social network

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Figure 4.9: Th upper panel depicts the mean field of a DMM host network H, with subcritical dynamics. The middle panel depicts a second DMM social network M in a critical state. The lowest panel depicts the host network H with 1% of the individuals replaced with individuals from the mother network M. H is depicted. It is clear that the echo response parameter is below its critical value in this case and so the opinion of the social network is erratic and no consensus is ever reached. This is prior to exposure to the influence of the M network. In the middle panel of Figure 4.9 the mean field of the social network M is depicted. It is equally clear that the echo response parameter is slightly above the critical value in this case and so the opinion of the social network does reach consensus. After some random time interval the mean field switches to the other consensus state and so on. The fluctuations around the consensus states are modest; this is the dynamic network M in isolation. The lower panel depicts the mean fields of the two networks when 1% of the individuals within H are replaced by those from M. It is clear that the host network becomes completely subservient to the

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dynamics of the mother network. One might mistakenly conjecture that this subservience depends on which states the mother and host networks are in. We have done the calculations and it does not. The dynamics of the host network are invariably enslaved by those of the mother network and this is accomplished without a noticeable change in the behavior of the host individuals, or a change so slight that it goes unnoticed. One further thing worth mentioning is that although the host network tracks the dynamics of the mother network, its mean field still retains a higher level of fluctuations than does the mean field of M. The difference in the level of fluctuations arises because the fluctuations in the mean field of M are the result of the finite size of the M network, whereas the fluctuations in the H network are a consequence of its own internal dynamics. The H network does not have the imitation strength of the M network, although that would appear to be the case, based on its observed dynamics. This enslavement of the dynamics of the second network by the first is a consequence of the information being transferred from network M to H, through selective interactions.

4.2.4

Groupthink and the gadfly

To model the phenomenon of groupthink we concentrate on the dynamics of a network that is relatively small in size. Despite the limited size, it is determined that such small networks share most of the scaling properties of the DMM observed for large networks. With increasing strength of the echo response parameter K between individuals, the opinion change rates decrease, leading to the elongation of intervals during which an individual holds a given opinion. On the time scale of the mean field, this in turn results in the prolongation of intervals during which the majority of group members are of the same mind. Thus, the cooperative behavior that arises within the group is of a transient nature, due to the fluctuations in the value of the mean field. Recall that the noise present in the social network is generated by the finite number of individuals within

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√ the group, with an amplitude that decreases as 1/ N . Due to this scaling with the network size, fluctuations observed in the global dynamics on a small lattice are strong when compared with a large network, resulting in a less steep phase transition curve. The DMM dynamics of a small network clearly demonstrates that small groups are able to reach consensus, given sufficiently strong imitation strength between its members. This strength of will to cooperate comes in parallel with group cohesion and a willingness to minimize conflict, the main factors underlying the groupthink phenomenon. Subsequently, we investigate the efficiency of an individual gadfly in inhibiting such strong collaboration; a collaboration which might lead to groupthink behavior. Consider a DMM initially consisting entirely of cooperators, individuals favorably disposed towards one another and therefore rapidly reaching consensus. A gadfly plays the devil’s advocate, an individual who rather than imitating others, holds opinions contrary to those of other members of the group. Such behavior is herein modeled by changing the sign of the echo response parameter for such a gadfly to −K, resulting in a new definition of transition rates between the two states for individual r. Due to their nature, in a neighborhood of the lattice where a cooperator would be enticed to change an opinion to conform to the majority behavior, the gadfly would typically be reluctant to do so. Similarly, where the cooperator would prefer to persist the gadfly would flout change. The contrarian nature of a gadfly is visible in the behavior of individuals, where other individuals, directly and indirectly interacting with it, tend to adopt the opposite opinion to that of a gadfly. Additionally, the global opinion demonstrates less cohesiveness and persistence. The phase transition is not completely suppressed, but it is evident that it occurs at significantly larger values of K. This result implies that the dynamic behavior of a small collegial group, say a tenure committee in a typical university, will not reach consensus when such a curmudgeon is on the committee. That is, the committee members must work harder to reach consensus than they would in the absence of the gadfly. The failure to undergo a

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phase transition, except for larger than normal values of K, supports the interpretation that a gadfly does not need to possess any special skills, nor have a charismatic personality, to disrupt consensus formation in a small group in which uniform agreement would easily be reached by congenial people. As illustrated in Fig. 4.10 a gadfly impacts the dynamics most strongly when members of the group are strongly coupled. As reference we plot an exponential distribution as the blue dashed curve, which depicts the behavior of a single individual in isolation. In this case the parameter g describes the rate at which an isolated individual changes his/her mind. The comparison of the behavior of individuals within a system, full of cooperators, and one containing a single gadfly, demonstrates symmetry breaking. In the case of nine cooperators all individuals are equivalent, while introducing a gadfly separates the individuals into three different classes: those that are adjacent and interact with the gadfly directly; those that do not interact directly with the gadfly and therefore experience its influence indirectly; and the gadfly. In all cases the effect of the gadfly is to reduce the survival probability to a value below that of the gadfly-free network as indicated by the dashed black line, but to increase it above that of an isolated individual. Thus, a gadfly not only inhibits global consensus, but modifies the behaviors of the individuals forming the group. The nature of the DMM network indicates that groups, large and small, reach consensus based on the criteria of imitation, that is, consensus can be based on irrational imitation and not necessarily on rational decision making. Consequently, in the DMM network, the dynamics of reaching consensus and that of groupthink are no different. In the theory of groupthink it was hypothesized that introducing an acerbic individual into the small group would inhibit consensus. This hypothesis was explored in the context of the DMM, by introducing a gadfly into the dynamics. It was determined that a single cantankerous individual not only discourages the formation of small group consensus, but also pushes the other members of the group to reach their decisions more quickly than they would in a

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Figure 4.10: The left dashed curve denotes an exponential distribution. The calculations are done on a lattice of size N = 3 × 3, and g = 0.01.The three bundles of curves denote 500 realizations of the gadfly in the center; directly coupled individuals, to the right, and indirectly coupled individuals to left. The dashed curve on the right has no gadfly.

completely cooperative network. These quantitative results support the qualitative historical argument [125, 162] that groupthink can be postponed, if not avoided altogether, by introducing a skeptic into the group making the decision.

4.3

Evolutionary Game Model (EGM)

As pointed out in [239] the mechanistic approach to decision-making can be contrasted with the functional approach, in which we ask

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125

what is the value, or function, of a particular behavioral strategy. The basic assumption is that a given behavior can be rationally evaluated in terms of costs and benefits, which allows for an objective comparison of alternative strategies. In mathematical terms the functional approach aligns with the basic assumptions of game theory. In a recent book Gintis [102] attempted to unify anthropology, biology, economics, political science and psychology, stressing that game theory was necessary, but not sufficient, to realize this intellectual goal. The strategy was to capitalize on the success achieved by Nowak and May [175] in using their concept of network reciprocity, a special condition where spatial structure favors the emergence of cooperation. As Mahmoodi and Grigolini [150] point out, the state of global cooperation is achieved in spite of the fact the participants play the PDG , with an incentive to defect [23]. Actually, the players update their state in each time step by using unconditional imitation to select the strategy of the player with the maximum payoff in their neighborhood, including themselves. This choice of largest payoff is not influenced by an inclination to either cooperate of defect, implying that the consensus state of cooperation emerges as solely the result of network reciprocity. Originating in the theory of games of chance, such as gambling, game theory reached into behavioral sciences with the introduction of an utility function by Daniel Bernoulli in 1730 [37]. In doing so Bernoulli resolved the famous St. Petersburg paradox [207], demonstrating that a rational strategy could be based on the subjective desirability of a game’s outcome, rather than being proportional to the expected value of the game. The suggestion that the value of things to an individual are not simply equivalent to their monetary values reached its full articulation in the voices of von Neumann and Morgenstern [245] in their seminal work on game theory and economics. More recently game theory was introduced to study the emergence of cooperative behaviors, as a way of obtaining insight into this evolutionary puzzling phenomenon. The work of Nowak and May [175] for the first time extended game theory principles into

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spatial networks, and demonstrated that the introduction of spatial structure between players lead to spatially and temporally rich dynamics. Contrary to the well-mixed case, where the non-cooperative behavior is favored, the well-known PDG performed on a square lattice promotes cooperation over time. In game theory an individual is associated with a particular strategy, which for historical reasons are labeled either C for cooperator, or D for defector. In the two-player game there is the payoff matrix

C D

C D R S T P

,

(4.6)

which under the ordering of payoffs T > R > P > S, and 2R > T +S results in the PDG. This latter game was originally introduced as a metaphor for the problems affecting the emergence of cooperation in social groups [21]. Traditionally the PDG consists of two players, each of whom may choose to cooperate or defect in any single encounter. If both players select the cooperation strategies, each of them receives the payoff R and their society receives the payoff 2R. The player choosing the defection strategy receives the payoff T . The temptation to cheat is established by setting the condition T > R. However, this larger payoff is assigned to the defector only if the other player selects cooperation. The player selecting cooperation receives the payoff S, which is smaller than R. If the other player also selects defection, the payoff for both players is P , which is smaller than R, but larger than S. The EGM used herein is a generalization of the PDG and was discussed in detail elsewhere [239]. Nowak and May [175] considered a two-dimensional lattice over which this game was played in a sequential fashion, where at each time step every node was able to change its strategy (defect or cooperate) depending on the outcome of the game played with its neighbors at the previous time step. It is evident that for a player, say #1, the choice of defection condition is always the most convenient, regardless of the choice

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made by the other player, say #2. In fact, if the player #2 selects cooperation, player #1 receives R, but the better payoff is T if that player selects defection. If player #2 selects defection, player #1 receives the payoff S if that player selects cooperation and the larger payoff P if defection is selected. However, the whole society receives the largest payoff, 2R, if both players select cooperation, a smaller payoff, T + S, if one selects defection and the other cooperation, and the smallest payoff yet, 2P , if both players select defection. Axelrod and Hamilton [21] noted that if the PDG is played only once, no strategy can defeat the strategy of pure defection. If the game is played more than once however, reciprocity may make the choice of cooperation the winning strategy. Nowak and May [175] substantiated this concept with their model of network reciprocity. The players are located at the nodes of a regular two-dimensional lattice and each one can interacts with their nearest neighbors. The players are initially randomly assigned either the cooperation or the defection strategy. After each play, but before the next play, they are left free to update their strategy; selecting the strategy of their most successful nearest neighbor. Since the environment of the cooperators, as noted above, is wealthier than the environment of the defectors, it is possible that the most successful nearest neighbor is a cooperator, rather than a defector. This is a rational form of imitation that may lead to the survival of cooperators.

4.3.1

Choice of strategies

Consider the EGM network dynamics in which the cooperators and defectors are placed on the nodes of a two-dimensional lattice and interact only with their four nearest neighbors. In each generation every individual plays a deterministic game defined by the payoff matrix given above, with all its neighbors. The payoff gained by each individual at the end of each generation is determined by summing payoffs of 2 × 2 games with each of its neighbors. The scores in the neighborhood, including the individual’s own score are ranked, and in the following generation the individual adopts the

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strategy of the most successful player. In the case of a tie between the scores of cooperating and defecting players, the individual keeps the original strategy. Thus, the adopted evolutionary strategy is to act like the most successful neighbor.

Figure 4.11: The equilibrium fraction of cooperators present in the EGM lattice of N = 20 × 20 nodes with periodic boundary conditions in the calculations. The indicated regions locate the prisoner’s dilemma (PD) game; the shag hunt (SH) game; the snowdrift (SD) game; the leader game (LD);and the battle of the sexes (SX) game. From [239] with permission. Even this simple and completely deterministic situation leads to an array of behaviors. Figure 4.11 depicts the equilibrium fraction of cooperators present in the EGM game as a function of the two parameters S and T . Without loss of generality, we assume that R > P and normalize the payoff values such that R = 1 and P = 0.

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The initial configuration of the game consists of 50% cooperators and defectors, distributed randomly on the lattice. It is evident that in the domain T ≤ 1 for almost all values of S that the equilibrium state is dominated by cooperators. These regions are determined to have between 5% and 10% randomly distributed defectors asymptotically. Whereas for T ≥ 1 and S < 0, the region of the PDG, defectors dominate, with the network having between 5% and 10% randomly distributed cooperators. The remaining regions have differing levels of cooperators at equilibrium. The traditional games for which there is a substantial literature are marked and are not addressed here in more detail. Turalska and West [239] noted that the parameter values enabled the determination of the outcome for the two-layer network and thereby determine the mutual influence of the DMM and EGM dynamics and the relative influences of imitation and payoff on decision making in all these cases. A new version of the two-level network is taken up in the next chapter.

4.3.2

Some general observations

Natural selection inhibits the evolution of cooperation, as pointed out by Rand and Nowak [197], unless balanced by specific countervailing mechanisms. They identify five such mechanism as having been proposed in the literature: direct reciprocity, indirect reciprocity, spatial selection, multilevel selection, and kin selection. In their paper they discuss the bridge between the theoretical models that have proposed these mechanisms and the empirical evidence supporting their existence in situations where people actually cooperate, using evolutionary game theory. Direct reciprocity is a dynamic concept referring to the same two individuals interacting over the same link multiple times and consequently the history of the choices made by an individual influences their future choice [22]. Although many subtleties exist, 50 years of experiments show the power of repetition in promoting cooperation in pairwise interactions.

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Indirect reciprocity, as its name implies, concerns the transmission of information within the group about direct reciprocity behavior via indirect means such as gossip. The resulting person’s reputation influences, either positively or negatively, the strategy adopted by the people with whom the player subsequently interacts. Experimental evidence for this mechanism has been found in markets where such information regarding individuals can be bought and sold [190]. Spatial selection recognizes that individuals interact with those nearest to them and subsequently cooperators form clusters, islands of stability, or fitness in a sea of defectors. Dynamic networks, unlike static networks, have been shown to successfully promote cooperation in laboratory experiments [85, 126]. Multilevel selection is the mechanism postulated by Darwin to resolve the altruism paradox (AP), on which we will again focus in the next chapter. Multiple experiments have established that cooperation increases significantly with intergroup competition [194, 227]. Kin selection is the favorable behavior bestowed by an individual on those identified as family members. This is the least empirically studied of the five mechanisms and although thought to be important has little experimental evidence to support it. Rand and Nowak point out that these various mechanism have historically been studied in isolation, but it is important to acknowledge that their interplay can be very important. However, to date there is no unifying theory that incorporates the five mechanism to test the strength of their mutual interaction during evolution, but in Section 4.4, as well as in Chapter 5, we examine some of the surprising behaviors arising when one is modified and two of them are included in a single model.

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131

Exploring simple level coupling

A finite number of individuals located at the nodes of a regular two-dimensional lattice are assumed to play a game based on the combined action of two distinct levels; one making decisions based on DMM and the other based on EGM. At the first step of the game, half the players are randomly distributed across the lattice and assigned the cooperation state. The remaining half are assigned the defection state. At the lower level 1 the strategy choice is made on the basis of imitation according to the echo response hypothesis discussed in Section 4.2. This level may be interpreted as generating a form of collective intelligence, which makes the network sensitive to the criteria determining the strategy choice adopted at the upper level. The individuals, when the upper level is activated, play the PDG and update their strategy either by selecting the strategy of their most successful nearest neighbor, the success model (SUM), or strictly on the basis of the criterion of the best financial benefit to the individual making the choice, the selfishness model (SEM). The mathematical details of these two models are given in Appendix A.2. The intelligence emerging from imitation-induced criticality leads in the former model (SUM) to the extinction of defection and in the latter model (SEM) to the extinction of cooperation. The SUM case is interpreted as a form of network reciprocity enhanced by the imitation-induced criticality and accelerates the evolution towards global cooperation (altruism). We anticipate the convergence of the SEM towards a global defector state and to test the robustness of this convergence we perturb the SEM with a form of moral pressure. This pressure is exerted by introducing a psychological reward of parametric strength λ for cooperation, to establish the sensitivity of collective intelligence to morality. We find that when λ attains the critical value λC , exceeding the temptation to cheat, the network makes a transition from the supercritical defection state to the critical regime, with the admonition that excessive moral pressure 1 The

distinction between upper and lower levels is arbitrary and is not intended to imply any relative importance of one network over the other.

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may annihilate the criticality-induced resilience of the system.

4.4.1

Joining the two models

Evolutionary game theory was the first mathematically rigorous attempt to theoretically explain how and why the selfish actions of single individuals may in fact be compatible with the emergence of altruism and cooperation. As mentioned previously the concept of network reciprocity is a condition whereby spatial structure favors the emergence of cooperation, which is to say altruism, in spite of individuals playing the PDG [23] that was expected to favor defection. Actually, since an individual, with a strategy identifying them as either a cooperator C or a defector D is replaced by the strategy label of their neighbor with the largest payoff [175], which is equivalent to updating the strategy of each player adopting that of the most successful neighbor. A cluster of cooperators has the effect of protecting individuals within the cluster from the exploitation of defectors. The recent review paper of Wang et al [248] illustrates that the research work in the field of multilevel networks focuses on the topology of networks, particularly on explaining the pattern formation through the production of clusters protecting cooperators from the exploitation of the defectors, so as to favor the survival of cooperators. In the present section and Appendix A.2 we adopt this multilayer perspective, but using dynamical rather than topological arguments in such a way as to be as close as possible to the unification of behavioral sciences, recently proposed by Grigolini et al. [106], as an attempt to address the challenge of Gintis [102]. Furthermore, the multiple layers do not correspond to different individuals, but refer to different levels of human behavior for a single individual, the social, the financial, as well as, the spiritual (moral). The lower layer of the composite network is based on the observation that the individuals within a network, playing the DMM, are members of human society and are expected to be strongly influenced by imitation [117, 188, 257]. The individuals in this network

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make a choice between cooperation and defection, without any tempering by cognition. This choice is not determined by a desire to maximize personal benefit through imitation of the most successful neighbor [177], or by the self-serving choice of an immediate payoff, but rather by imitation that is as blind as a bird’s tendency to select their flying direction on the basis of that of their neighboring birds [243], with no consideration of direct or indirect personal benefit. The imitation strength is taken to be a parameter denoted by K1 . A critical value of the imitation parameter K1 = K1C exists, making it possible for the flock of birds to fly as a single entity. Although the action of single individuals within the lower network does not require any form of cognition for its activation, the onset of criticality generates a form of collective intelligence. This collective intelligence is characterized not only by criticality-induced long-range correlations, but more importantly by temporal complexity [36, 146, 148, 220], a condition making the nonsimple network flexible and resilient. It is important to stress that the supercritical condition is characterized by fluctuations around a non-vanishing mean field. These fluctuations are as random as those around the vanishing mean field of the subcritical regime and thereby lack the flexibility and resilience of the interdependent fluctuations at criticality. At the upper layer individuals play the PDG at each time step and exert an influence on the lower level, choosing their strategy according to either the success or selfishness criterion. In the SUM for most of the time steps the choice of strategy is determined by the lower level, but from time to time individuals are allowed to select their strategy by adopting that of their most successful nearest neighbor [175]. SUM has the impressive effect of annihilating the emergence of the defector branch, as depicted in Figure 4.12, when the network adopting this choice manifests intelligence by imitationinduced criticality. It is reasonable to believe that this effect affords a rigorous explanation of the evolution of the overall network towards cooperation, or altruistic behavior. Very large values of K1 lead to the extinction of defectors. On

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Figure 4.12: The mean field of the DMM with 1% of the L time steps randomly selected to update the strategy of each individual to adopt the strategy of the most successful nearest neighbor. The lower branch is suppressed when the 1% is added. the basis of the results obtained by our research group on the DMM dynamics we make the plausible conjecture that this effect is independent of the topology of the adopted network. In fact, moving from one topology to another has the singular effect of reducing the intensity of the effort necessary to achieve consensus. The condition K1 = 1 represents the ideal topology requiring the weakest effort necessary to reach consensus [257]. We compare the results of SUM to those of SEM, where the player does not adopt the strategy of the most successful nearest neighbor, but she makes her choice only on the basis of her personal benefit. She evaluates the financial benefit derived from the defection choice and the financial benefit that she would obtain from the cooperation choice, giving greater weight to the maximal profit. The

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benefit of a given choice is realized assuming that the individual can play with equal probability with all her neighbors. The selection of a convenient strategy is influenced by a second imitation parameter K2 , with the ratio ρK = K2 /K1 establishing if the links within the upper layer are stronger, ρK > 1, or weaker, ρK < 1, than the links within the lower layer. Switching on the interaction of the lower with the upper layer has the effect of extinguishing cooperation with a corresponding loss to the benefit of society. This eradication occurs even if, as shown in Appendix A.2, K1 > 0 yields imitation-induced clusters of cooperators, with financial benefit for society, this being the reason why SUM, for K K1C leads to the extinction of defectors. The strategy choice determined by the criterion of maximal personal benefit, rather than by the choice of the strategy of the most successful nearest neighbor [175], on the contrary, yields the extinction of cooperation, even with a very small value of ρK , as an effect of imitation-induced intelligence, see Figure 4.13 for typical results. To complete the picture of the role of criticality-induced intelligence we model the influence of morality on the dynamics of the SEM, showing that as an effect of imitation-induced intelligence the network becomes so sensitive to morality as to make a psychological reward that only moderately exceeds the temptation to cheat, sufficiently robust to prevent the collapse of the social system into subcriticality disorder. In principle the influence of morality on the network should be established by the interaction of the network with an additional layer. For the sake of simplicity we modify the conventional PDG [102] by introducing a psychological reward of strength λ for the choice of cooperation, as discussed in some detail in Appendix A.2. The psychological reward affords a simple way to describe the influence that an additional layer, concerning morality may have on the SEM. We find that, when the lower layer operates at criticality, a crucial value λC exists, with the effect of preventing the extinction of cooperators and of recovering the criticality-induced temporal complexity that is essential for the healthy behavior of the social

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Figure 4.13: The mean field for the SEM as a function of K2 for a few values of K1 to provide some intution for the dependence on ρK . Ten realizations are calculated for each value of ρK . network. This is the Asbiyyah effect [129], an arabic world meaning group feeling, namely the natural tendency of human beings to cooperate. This natural disposition can be enhanced by religion and it has the eventual effect of increasing social prosperity, but, in accordance with the observation of Ahmed [11], we find that λ > λC , the super-Asbiyyan condition, may be as destructive for the social network as the lack of social cohesion.

4.5

Conclusions and observations

In this chapter, along with Appendix A, we have included many of the results from research regarding the properties of a new social model of decision making, the DMM, which is based on a two-state

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master equation description of the dynamics for each of the individuals in a nonsimple network structure. Although initially motivated by the Ising model for a ferromagnet, the DMM is separate and distinct from its dynamic cousin, since its interaction strength is a measure of the influence individuals have on one another due to imitative behavior and fluctuations are due to the finite size of the network, not thermal excitation, which is to say there is no fluctuation-dissipation theorem. Given these differences and others the DMM does however share the properties of having a phase transition to consensus and is a member of the Ising universality class. A number of other differences, including the sensitivity of large network response to minority group influence, small network groupthink response to the disruption of a gadfly, were also discussed. The DMM is argued to be a useful model of the irrational component of decision making. In a similar way the rational component of decision making is modeled using a dynamic version of game theory. A brief introduction to an EGM is given, one specifically targeted at PDG is used to model the cognitive component of decision making. The two aspects of decision making, the rational and irrational, are then joined by means of an interaction to explored how the two components could work together to achieve some of the more elusive properties of real decision making. These arguments lay the groundwork for the new results discussed in the next chapter. We have computationally established that imitation-induced criticality can have multiple effects, including enhancing the phenomenon of network reciprocity in the EGM. The strategy of adopting the behavior of the most successful nearest neighbor in SUM not only protects the cooperator from extinction, as in the pioneering work of Nowak and May [175], but, at criticality, it annihilates the dynamic branch having the majority of defectors. If instead, we adopt the selfishness criteria of SEM for the choice of strategy, the imitation-induced criticality has the opposite effect of favoring the extinction of cooperators. Under the influence of a morality stimulus, however, this latter imitation-induced criticality has the opposite effect of producing the extinction of defectors. However,

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the temporal nonsimplicity of the network is lost for the morality parameter λ being either larger or smaller than its critical value, indicating, in accordance with the reasoning of Ahmed [11], that a condition of super-Asbiyyan is as detrimental to human society as is the complete lack of Asbiyyah. It would certainly be of interest in today’s world of mutual suspicion to devise experiments to test this remarkable prediction.

Chapter 5

Self-Organized Temporal Criticality This chapter pulls together the two modeling strategies of the previous chapter to model a new kind of thinking, using a two-level dynamic network. In this composite model the two networks, one modeled by the DMM and the other modeled by an evolutionary PDG, are coupled together in a unique manner that enables the internal dynamics to spontaneously converge on a critical state that simultaneously satisfies the self-interest of each member of society, while also achieving maximum benefit for society as a whole. The dynamics of this two-level network are what we call the self-organized temporal criticality (SOTC) model [150, 262], which is self-adjusting through its internal dynamics and does not require the external fine tuning of a parameter to reach criticality. The SOTC model is also used to show that the bottom-up basis of democracy is intrinsically more stable and resilient than a top-down control exerted by a power elite. 139

140CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY

5.1

The Inadequacy of Linear Logic The old saying of the two kinds of truth. To the one kind belongs statements so simple and clear that the opposite assertion obviously could not be defended. The other kind, the so-called “deep truths”, are statements in which the opposite also contains deep truth. – Bohr [42]

In this chapter we develop the idea that an EP is a deep truth and that paradox results from the mistaken application of linear logical thinking to such nonsimple dynamic phenomena that manifest criticality. For example, computational social science relies on the recognition that criticality, such as reaching consensus, is a consequence of the self-organization of social nonsimplicity. Criticality generates an interdependency between a statement and its negation. Consider the statement that humans are law abiding and considerate of one another, along with its negation that humans are drawn to anarchy and selfish behavior. We have all seen both aspects of humanity and in this chapter we develop a mathematical model that shows at least one way these mutually exclusive views of humanity can be held simultaneously. But how can people who are firm believers in the rule of law and the importance of order within society find themselves to be part of a riot that destroys property and may ultimately become a lynch mob? People who believe themselves to be moral have been overwhelmed by emotion and often do not understand how they could have done such things. Is a riot the result of a social tipping point that awaits a single incident to initiate the transition from tranquility to turmoil? Does every pattern in a nonsimple network contain the seeds of its own anti-pattern within it, or does it require more? The shameful actions of the mob may have a certain structural similarities to the collective behavior in certain physical and biological phenomena, but do the regularities run any deeper? A number of

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investigators, us included, have devised mathematical models with which to quantify critical phenomena in the physical, social and life sciences. The patterns observed in data throughout the various scientific disciplines are what scientists use to wrestle understanding from nonsimplicity and confusion, and when the same pattern is extracted from physical and social data, every attempt is made to identify those variables in the two very different venues that produce those patterns. The position we have adopted herein is in the tradition that paradox is a consequence of nonsimplicity, or perhaps stated even more strongly, paradox is entailed by nonsimplicity. The literary device of conjoined contradictions into an oxymoron may be the clearest example of this entailment. When authors want to convey intrinsic, but deeply held, conflict in a dramatic way they devise expressions such as idiot-savant, beautifully-ugly, dramatically-mundane, and so on. In fact, an oxymoron has been called a collapsed paradox. This literary device is intended to jolt the reader beyond the simple image constructed in their mental map of the behavior and understanding of human beings and to incorporate more elaborate ways of behaving, knowing and reacting to the world. When a system is simple, independently of the discipline in which it is found, we determined in Chapter 3 that the variability in the underlying process is linear, additive and weak, so that kind of confusion is captured by the central limit theorem and normal statistics. However, the variability of a truly nonsimple system, again independently of the discipline in which it is found, can be nonlinear, can be additively and/or multiplicatively stochastic, and can be devastatingly strong. The last characteristic is particularly true for critical phenomena, which we argue is one of those properties that transcends the discipline in which it is observed and its emergence is a manifestation of true nonsimplicity. The image of a person acting solely in their own self-interest is typically an overly simple characterization of human behavior; a caricature. However, it might occur when a person is isolated and functioning at the level of survival, that is, when individuals have

142CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY isolated themselves and focused their behavior on realizing a single goal, say avoiding starvation. If, however, such a person reestablishes a personal interaction with society, they typically adopt additional goals, say forming a friendship, or getting a job, as well as eating regularly. As life becomes more complicated, and more goals are adopted, it is unavoidable that some of these new goals come in conflict with one another and produce stress. To resolve the stress it is necessary to reconcile the conflicts. Phase transitions and critical phenomena occur frequently in nature and have been widely studied in physics, as we have previously discussed. The work horse of such discussion is the Ising model, originally introduced to explain ferromagnetic phase transitions in a metal and whose exact solution was found by Onsager [180] for the occurrence of phase transition in the two-dimensional case. This analysis is widely recognized as an example of outstanding theoretical achievement. In the last decade a number of scientists have used the Ising model outside the confines of physics to shed light on biological and neurophysiological collective processes [55, 90, 133, 169]. More precisely, Mora et al. [169] used the Ising model to explain the collective behavior of biological networks and other researchers [55, 90, 133] adopted it for the purpose of supporting the hypothesis that the brain works at the edge of criticality. However, they did so without establishing a clear distinction between externally induced criticality and self-organized criticality (SOC) [27]. Finally, we have to mention that the Ising model is frequently used, see for instance [205, 232], to model neurophysiological data subject to a maximal entropy constraint. The term criticality is used to denote the physical condition corresponding to the onset of a phase transition, generated by the adoption of a suitable value of the control parameter K, such as the temperature in a physical system. In other words, criticality itself is the object of study, whether identified in the shift from an informal gathering to that of a student protest, the devolution from normal sinus rhythm to a heart attack, or in the transition from water to ice. As mentioned, mathematical models have been developed to capture the essential features of

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criticality in each of these disciplines, and it turns out that these mathematical models can be widely separated into two categories. The first category of models contains those phenomena that achieve criticality through the external manipulation of a control parameter, such as the tuning of temperature to induce phase transitions among gases, liquids and solids, see Figure 3.1, or it modifies the magnetic properties of a piece of metal. This critical behavior was first successfully described by a complete theory using renormalization groups, see, e.g., the works of Wilson [267], which established that the dynamic properties of a physical system near a phase transition are universal, as expressed through scaling relations. It was subsequently shown that such scaling relations have little, if anything, to do with the system’s detailed microdynamics, but relies primarily on its level of nonsimplicity, as manifest in the emergent behavior of its macrodynamics. These same scaling properties are observed in the macrodynamics of nonsimple social networks, such as in the DMM discussed in Chapter 4 [235, 257]. In the case of the DMM the criticality condition is obtained by tuning the echo response parameter to the theoretical value that in the limiting case of an infinitely large network is determined by an Ising-like prescription, since the DMM used is a member of the Ising universality class [257]. Criticality entails long-range correlation among the members of the society, even those communicating solely by means of nearest-neighbor interactions. Such criticality has been interpreted as a form of global intelligence, also identified as swarm intelligence [242], a phenomenon that may be shared by microbial communities and mechanisms of carcinogenesis [203], as well as, by neural systems [118]. In the specific case of individuals playing the EGM as a generalized PDG, the criticality-induced swarm intelligence enables the members of society to become aware of the benefits of network reciprocity, and thereby biases their interactions to favor, rather than disrupt, this network property [150, 151]. The second category of models contains critical phenomena that emerge through internal dynamics, along with scaling behavior, without externally adjusting a control parameter. As previously men-

144CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY tioned, this autonomous driving of a nonsimple network to a critical state by its own internal dynamics has acquired the name selforganized criticality (SOC), see [249] for a review of the four-decade history of this modeling strategy. Here again the universality of criticality makes the properties of the system independent of the microdynamic details. The manifesto of computational social science [61] relies on the assumption that criticality is a consequence of self-organization, and interprets this to mean that social criticality is a form of SOC. A 25-year review of the concepts and controversies surrounding SOC [249], emphasize that SOC occurs in open, extended, dissipative dynamical systems that over time are attracted to the critical state. This is distinct from a continuous phase transition where at a critical point correlations become long-range and are characterized by an IPL PDF. In order to arrive at the critical point an external control parameter K, such as temperature, must be fine-tuned to its critical value. By way of contrast, SOC of network dynamics occurs universally where fine-tuning is accomplished by means of its internal dynamics [26]. This independence from an external tuning is the defining property of SOC phenomena. Bak observed in his book how nature works [28], which traces the history of the ’science’ of SOC, that nonsimplicity is a consequence of criticality. He was able to relate at the level of nonsimplicity the properties of physical processes involved in earthquakes, starquakes and solar flares, to those of the mass extinctions in the punctuated equilibrium of macroevolution, to the characteristics of the very different, but no less nonsimple, biological dynamics of the human brain and consequently the economics behind stock market variability and collapse, as well. The emergence of SOC is typically signaled by the births of anomalous avalanches, see Zapperi et al. [272], or Martinello et al. [158] for more recent work. In [151] we illustrated a generalized form of SOC based on the spontaneous search for the critical value of the echo response parameter K, which is selected by the network through a bottom-up dynamic process, that is, through the dynamic

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behavior of the individuals and is not externally imposed. The main signature of self-organized criticality is the time interval between crucial events, with a non-exponential waiting-time PDF, a property referred to as temporal complexity in earlier work [236]. We therefore refer to the form of SOC developed therein as self-organized temporal criticality (SOTC). The crucial events are defined by comparing the network variable A(t) to its time average A and are identified with the intensity variable ζ(t) = A(t) − A changing sign. The scaling of the intensity variable is subsequently be shown to scale as an IPL with system size N . In this chapter we sketch this generalization of SOC, using a two-level dynamic network that gives rise to SOTC [151]. The universality of criticality in the SOTC model is used to develop the idea that conflicting characteristics may be made compatible over time through dynamic self-regulation. We also establish the notion that an EP can be resolved through the judicious application of nonlinear dynamics. But recall that the wave-particle duality resolution in physics established the fundamentals of quantum mechanics; resolving the paradox was only the beginning of an in-depth understanding of microscopic phenomena, not the end, as witnessed by the overwhelming scientific activity in quantum physics over the last century. We emphasize that the form of SOTC is fully compatible with the spirit of the game theory perspective of Axelrod and Hamilton [21]. In fact, the payoff of the choices made by the individuals withtin the composite network is established using an EGM, without neglecting the incentive to defect. The choice of the strategy to adopt is determined by the individual’s imitation of the choices made by their nearest neighbors. The individuals only decide to increase, or decrease, their tendency to imitate these choices according to whether, on the basis of the last two payoffs, this imitation increased, or decreased, the benefit to them as an individual. This indirect and apparently blind strategy choice does not disrupt the beneficial effects of network reciprocity [175], but it is a way of efficiently establishing the reciprocity condition hypothesized by Axelrod and Hamilton [21].

146CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY Linear logical thinking cannot typically be applied to a nonsimple nonlinear dynamical situation, or more precisely, if applied, such thinking leads to inconsistencies when the phenomena being addressed have emergent properties, that is, the emergent properties engender paradox. Consider the transition from independent individual behavior within a group to consensus, in which local interactions result in long-range correlations at criticality. Conclusions reached based on the reasoning about implications of the dynamics of local interactions are invalidated when criticality is reached. The linear logic of local interactions, say within quiet jury deliberations, can be strongly modified, if not outrightly contradicted, by the collective behavior of groupthink infused by imitative behavior and not by a charismatic jury foreman. So here we address the reasons why human behavior is replete with paradox and find that is the way we think and by extension, the way in which we formulate successful organizations and governments.

5.2

Two Brains, Two Networks Intelligence is not only the ability to reason: it is also the ability to find relevant material in memory and to deploy attention when needed — Kahneman [128]

The remarkable book review Two Brains Running by Jim Holt appeared in the New York Times shortly after the 2011 publication of Daniel Kahneman’s Thinking, Fast and Slow [128], from which we have drawn inspiration for the interpretation of the coupling of our mathematical models.1 From its provocative title to its laudatory conclusion Holt’s book review focusses on the iconoclastic interpretation that Kahneman used to understand the myriad results of 1 It is interesting to note that two of us had determined the basic mechanism for SOTC in an unpubished manuscript [149]. But its full implications only began to reveal themselves through discussions over time [150, 151]. With the most recent being the one given herein in terms of the two-level model of the brain.

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the groundbreaking experiments conducted in collaboration with his friend and colleague Amos Tversky; that human beings think with two separate and distinct brains, as the cartoon shows in Figure 5.1. One brain (System 1, in his book) is irrational, forms the basis of intuition, and works almost instantaneously. The other brain (System 2, in his book) is rational, requires time to put all the pieces of a rational argument together, and is therefore significantly slower than System 1. Moreover, contrary to common belief, the rational brain is not always the more important of the two. This separation into two brains is, of course, not just a metaphorical separation rather it is a psychosocial model of how the brain functions, if you will.

Figure 5.1: This figure is an adapted version of one found on the internet. The internet figure contains mistaken percentages quantifying the relative use of intuition and rationality and attributes the source of the figure to Daniel Kahneman. I know these are mistakes because I contacted Professor Kahneman and he assured me the figure was not his and that he certainly would not use the percentages quoted. In spite of the fact that Kahneman is a psychologist, he was

148CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY awarded the 2002 Nobel Prize in Economics. He received the award, which could not be shared with Tversky because his friend and collaborator had died six years earlier, essentially for establishing that people do not always act rationally in making decisions, contradicting much of accepted economic theory. Their experiments, explained by their Prospect Theory, established that the ’rational man’, which had been the foundational basis of economic theory for over a century, was not only unnecessary theoretically, but was non-existent empirically. A behavioral economist, Dan Ariely, in his engaging book Predictably Irrational [15], using decades of experimental data, also argues for the overlooked importance of the irrational component of decision making. It is now evident that the two-level decision making model discussed in the last chapter is a natural mathematical instantiation of the two brain idea of Kahneman and Tversky, even though this did not occur to us while the SOTC model was first being developed. At the time we did not mention the fact that the DMM and the EGM could be viewed as two interacting subnetworks of a composite network, since we did not then appreciate the eventual benefit such a perspective would provide. Here we emphasize that perspective, using Figure 5.1, where System 1 has a natural representation in terms of the DMM, whereas System 2 lends itself to EGM. In this chapter we consider how the two brains may exchange information in a dynamic model that can address the resolution of EP in general and the AP in particular. To accomplish this task we consider two networks, each one consisting of N dynamic individuals (actors, agents, elements, or units). The two networks constitute two levels of a single composite network that work together to make decisions, using separate and distinct sets of criteria. On the one hand, individuals formulate their decisions based on the perception of actions and appearances of their neighbors, adopting imitative behavior. This behavior is captured by the DMM [235, 236] network discussed in Chapter 4 and demonstrates, among other things, the cooperative behavior induced by the critical dynamics, leading to a tipping point and subsequently criticality. The

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DMM constitutes one of the two networks making up the two-level decision-making network. The intuitive DMM behavior is counterbalanced by the rational behavior of the second network using an EGM. Thus, while personal decisions are influenced by the desire to be liked and accepted, individuals also weigh the effect of certain potential relations on their long-range goals, balancing the costs against the payoffs of such relations. This latter behavior is captured by the rational and deterministic game theory rules in the spirit of the original approach of Nowak and May [175]. The decisions made by individuals are consistent with the criterion of bounded rationality [155], which was expanded by Kahneman [128], and more recently discussed from the perspective of EGT [197, 198]. Rand and Nowak [197] acknowledge the tension between what is beneficial for the individual and what is beneficial for society when the two do not coincide. Moreover they discuss that tension in the language of EGT. Without reviewing the long history of studies on the nature of cooperation and defection, we note the meta-analysis of 67 empirical studies of cognitive-manipulation of economic cooperation games by Rand [198]. He concluded from his meta-analysis that all the experimental data could be explained using a dual-purpose heuristic model of cooperation, a model containing a balance between deliberation and intuition. Deliberation is considered to be a rational process that always favors noncooperation, whereas intuition is viewed as an irrational process that can favor cooperation, or non-cooperation, depending on the individual and/or the circumstances. Herein we adapt these linked concepts of intuition and deliberation by constructing a dynamic two-level network model [151]. The level based on the DMM leads to strategy choices made by the individuals under the influence of the choices of their nearest neighbors. The other level uses the EGM to measure the success of these choices in terms of payoffs associated with each of them. The interaction between the two levels is established by making the imitation strength K increase or decrease, according to whether the average difference

150CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY of the last two payoffs increase or decrease, producing corresponding changes in K. Although each individual’s imitation strength is selected selfishly, which is to say the individual choices of imitation strengths at the time are made in the best interest of the individual making the decision, the social system is driven by the resulting internal dynamics towards the state having a high level of cooperation, which has the greatest social benefit. In this way the AP is resolved by using the internal dynamics of the composite network.

5.2.1

Intuition

The intuitive mechanism proposed by Rand [198] is realized through the dynamics of one network modeled using the DMM [257]. The DMM on a simple two-dimensional lattice is formally based on individuals imperfectly imitating the majority opinion of their four nearest neighbors as discussed in Chapter 4 and more extensively in [257]. The echo response strength biases the probability of individual r deciding to transition from being a cooperator (C), to being a defector (D): (r)

(r)

gCD = g exp −Kr

(r)

MD − MC M

.

(5.1)

We use the notation made popular by proponents of game theory, (r) so that MC is the number of nearest neighbors to individual r that (r) are cooperators; MD is the number of nearest neighbors that are defectors, and each individual on the simple two-dimensional lattice has M = 4 nearest neighbors. In the same way the probabilistic (r) transition rate from being a defector to being a cooperator gDC is obtained from Eq.(5.1) by interchanging the cooperator and defector indices. The unbiased transition rate is g = 0.01 throughout the calculations, and 1/g defines the time scale for the process. In the present two-level model the individual imitation strengths Kr can all be different [151, 152]; a dramatic departure from early studies [257] in

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which a given network was characterized by a single echo response parameter K. This decision making process is consequently fast, emotional, idiosyncratic and does not involve any reasoning about a payoff. In Appendix A the formal manipulations and numerical integration of the two-state master equations that were used to obtain the results discussed in Chapter 4 are described. The original DMM assigns to all the individual imitation strengths Kr the same value K, an imitation parameter that has been shown [257] to make this theory undergo critical phase transitions. Consequently, as mentioned a number of times, the DMM is a member of the Ising universality class in which all the members of the network can act cooperatively, depending on the magnitude of K. Consequently, this early version of decision making dynamics is consistent with the class of criticality models in which the passage to criticality is externally controlled. In the present two-level model the Kr can all be different and this is an all-important difference. This difference changes the criticality from being externally controlled to having the potential for being internally controlled. The DMM takes cognizance of the fact that in making decisions people are strongly influenced by the opinions of others, which can even overcome rationality. Vilone et al. [244] point out that experiments in economics are providing a firm scientific foundation for the irrational component of human decision making. They explore the relative influence of what they refer to as social imitation and strategic imitation using game theory. Thus, the interaction mechanism present in the DMM adopts the idea of social imitation, which is hypothesized to be a basic social/psychological mechanism. Herein the strategic imitation is modeled as a deliberative process using a second network, as we subsequently show. On the other hand, Acemoglu et al. [3] emphasize that it is normal for disagreement among individuals within society to persist on issues both large and small and that consensus is an exception. Moreover, most disagreement persists even though people communicate and even change opinions. In the DMM both consensus and

152CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY disagreement are manifest in the dynamics, as are the fluctuations in opinions that occur as a consequence of the finite size of social groups.

Figure 5.2: Temporal evolution of: (a) a single element sj (K, t) and (b) of the global order parameter ξ(K, t); for the DMM realized on a toroidal lattice of N = 50 × 50 nodes, with g = 0.01 and K = 1.70. To illustrate the concept of crucial events we mark the time intervals τ between two consecutive events, the crossing of ξ = 0. Notice the differences in time scales between (a) and (b). Reprinted with permission from [236]. In Figure 5.2a the random transition between states for a single individual is shown; whereas in Figure 5.2b the mean field for the network ξ (K, t) is depicted for a slightly subcritical value of the imitation parameter. Note √ the fluctuations in the mean field, whose magnitude is given by 1/ N . An event is the passage of the mean field through the value zero and the time interval between successive

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events are statistically independent of one another. Consequently, the events are determined to be a renewal process, but more importantly with the determination of IPL time intervals between events they are determined to be crucial events.

5.2.2

Deliberation

As observed previously, Rand and Nowak [197] acknowledge the tension between what is good for the individual, what is good for society and they discuss the tension between the two in the language of EGT. As also mentioned, game theory is a mathematical model representing the behavior of individuals in a social setting. In a social situation there are a given number of individuals that interact to makeup the social network. Each player has a number of choices available during their interactions with other players. Subsequently, a payoff is given to each player that depends on the choices she and each of the other players make. Evolutionary game theory generalizes the single choice made in the original game theory to a dynamic set of choices over time. In a biological context the dynamic payoffs are interpreted in terms of individual fitness, with the systematically greater payoff being strategies resulting in survival in the long term, whereas the lower payoff constitute strategies resulting in eventual extinction. In a sociological context the dynamic payoffs have a somewhat more immediate interpretation in terms of the rise and fall of civilizations. They [197] also discuss alternatives to Darwin’s multilevel selection explanation of altruism, not considered in the critique of Wilson and Wilson [268] addressed in Chapter 2. We reviewed some of the fundamentals of game theory in Chapter 4, in part, because we use these ideas to develop a specific network model of deliberation. Here we note the idea that when two or more people play a game there are rules, strategies, payoffs and costs. The rules determine how individuals treat payoffs and costs during the play through the choices they make, and strategy, an overarching plan of action under conditions of uncertainty, is how they decide to play under various

154CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY configurations of costs and payoffs. The connection with self-interest is consistent with Kahneman’s thinking slow mechanism [128] and is established by the second network, which determines the payoffs for the choices made using game theory. To define the payoffs we adopt rules based on the PDG [102] so that this network becomes a realization of Rand’s deliberative mechanism within the two-level network model. Recall from Chapter 4 that two players interact and receive a payoff from their interaction, adopting either the defection, or the cooperation, strategy. If both players select the cooperation strategies, each of them receives the payoff R and their society receives the payoff 2R. The player choosing the defection strategy receives the payoff T . The temptation to cheat is established by setting the condition T > R. However, this larger payoff is assigned to the defector only if the other player selects cooperation. The player selecting cooperation receives the payoff S, which is smaller than R. If the other player also selects defection, the payoff for both players is P , which is smaller than R. The game is based on the crucial payoffs T > R > P > S, which defines the PDG, along with T + S < 2R. Note that their choices are made continuously as the network dynamics unfold. In the SOTC calculations subsequently shown we use R = 1, P = 0 and T = 1.5 or T = 1.9.

5.2.3

Criticality and some extensions

Before addressing how we couple our models of the two brains together it will be helpful to take a step back and attempt to identify threads linking the organizing and operating principles of biological/sociological systems to well-studied physical phenomena. We are talking about phase transitions and critical phenomena, which appear to provide one such link [95, 251]. The signatures of criticality are, of course, long-range correlations, lack of robustness and great sensitivity to external perturbations, which have been observed in multiple experiments involving the inner workings of sociological systems [64]. However, even if further observations and experiments

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determine that sociophysical laws are not fundamentally related to critical phenomena, an in-depth examination of the criticality hypothesis and its implications may retain practical value for the purpose of understanding the emergent behaviors within social groups of various sizes and consisting of various species. As observed elsewhere [221], a conceptual difficulty that arises when connecting critical phenomena with biological, or sociological, systems is that critical points are rare in parameter space, and it is not clear how natural systems might come to be operating there, whether by social or biological evolution, or other mechanisms, acting on shorter time scales. As mentioned a number of times previously, there are, in principle, at least two methods for making critical phenomena more robust that are plausible for biology and/or sociology: self-organization (causing the parameters of the system to adapt toward the critical point [119]), and extended criticality (causing the region in parameter space associated with critical behavior to expand beyond the singular point [24, 143, 144]). Here we focus on extended criticality, which is a new concept and has not received much attention to date. The finite-size of social networks tends to blur critical points due to the increasing amplitude of fluctuations at smaller system sizes in such a way that the near-critical region becomes more extended as the system size decreases [114, 137, 200]. Despite this relaxation of criteria to include near-critical points, alternative means to further extend the parameter space over which the system displays critical behavior would be useful, especially for establishing closer contact between criticality observed in physical systems and the extended criticality of social systems. Elsewhere we [221] review phase transitions theory for interacting two-state stochastic oscillators and discuss the DMM as an exemplar. We also propose transition rates that depend on two control parameters that extend the critical behavior from a single critical point to be a line of such points in parameter space. The critical line we interpret to be an extended critical phase that includes both an ordered and a disordered critical phase, the details of which were

156CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY given in Chapter 3; we need not repeat them here. In the analysis of criticality we adopt the nomenclature K → K1 and introduce a second control parameter K2 into the model, thereby making the phase transition diagram two dimensional in general, which in a physical system might be the temperature and pressure. This extensions turns the critical point λ1 (K1 ) = 0, which is the coefficient of the linear term in the expansion in powers of the mean field of the DMM potential, into the critical curve λ1 (K1 , K2 ) = 0. Less trivial than this general result is the relationship between the empirical control parameters and the mathematical parameters (i.e., expansion coefficients) characterizing the dynamics, which in this context involves the mapping from the two control parameters to the two lowest-order expansion coefficients: (K1 , K2 ) ↔ (λ1 , λ3 ),

(5.2)

where λ3 is the coefficient of the cubic term. The actual structure of this map depends on the way in which the control parameters are introduced into the transition rates, that is, how each control parameter couples to the field ξ (K, t) and K = (K1 , K2 ). Note that this extension of criticality does not supersede self-organization, but rather calls for a combination of multiple control parameters and self-organization mechanisms acting within a finite sized system. This realization of extended criticality, through the independent action of the control parameters, could be beneficial in allowing the system to access the critical regime while retaining the ability to adapt other parameters over a wide range of values to cover different effective critical operating regimes. The advantage becomes clearer when comparing to the case where the parameter K2 couples to all powers of ξ. Then K1 and K2 are interdependent, and this complicates the realization of criticality, i.e., K1c = K1c (K2 ) or K2c = K2c (K1 ). Such interdependence between control parameters would make it more difficult for a system to adapt its critical state, or fine tune it with respect to the tricritical point, as it requires simultaneous organization of both parameters. However, the trade-off

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lies in the added difficulty of engineering or evolving such a peculiar type of control that acts selectively on nonlinear terms. Taking some liberties, the control parameter K1 might be associated with the morphology of an individual’s behavior within a group, while K2 (or a collection of remaining parameters) with internal control, or the dynamics of the brain. Then a specific morphology, which has been optimized through evolution, could keep the system close to the critical state while the higher-level internal control is used to adapt this near-critical state and improve beyond what had been done by evolution alone. A few words are in order to emphasize that in the following [221] we have presented only one strategy for introducing a second control parameter into the dynamics. Other ways exist. For example, we [151] devised a form of self-organized criticality (SOC) based on the spontaneous search for the critical value of the control parameter K1 , which is selected by the DMM network through a bottom-up process, that is, through the dynamic behavior of the individuals and is not externally imposed. The main signature of SOC used in that paper is the time interval between successive crucial events, with a non-exponential waiting-time PDF, yielding SOTC. In the context of extended criticality we can identify a second control parameter as being defined by coupling the DMM network to a second network, whose dynamics are determined by a dynamic form of game theory [151]. In this way the variable control parameter for DMM individual r is denoted by K1,r (t) = K1,r (t − Δt) + K2,r (t − Δt) , and the coupling of this individual to nearest neighbors is updated by the payoff from nearest neighbors in the previous time step, as determined by the second network and here parameterized as K2,r (t − Δt). We discuss this in some detail in the next section. We stress that K1,r (t) is not a conventional fine-tuned control parameter that is artificially fixed to make the network achieve criticality, but is freely selected by the dynamics of the network itself to

158CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY spontaneously achieve criticality. Numerical calculations [151] show that increasing the dependence of the individuals on the strategic choices of their neighbors has the effect of increasing their payoff. Imitation of the choices of their neighbors in the SOTC model is a form of social interaction that is made at the level of the individuals and is not forced upon them in a top-down process. The bottom-up process is shown to not only be in the best interest of the individual, but to provide optimal payoff for society, as well. The discussion presented here on two-state stochastic units with two control parameters is intended to contribute toward a better understanding and eventual theory of extended criticality in complex dynamic systems. Analyzing the effects of multiple control parameters (i.e. where the dimension of parameter space exceeds the dimension of the state space) is necessary for closing the gap between the simplest, well-understood models in statistical physics and the nonsimple behaviors displayed by social/biological systems. The two-parameter models we discussed, with a critical line and tricritical point, exhibits what is perhaps the simplest form of extended critical behavior. Theories of extended criticality are expected to have applications to adaptive social groups that have access to a rich bifurcation structure and are able to switch between different critical states in response to a nonsimple environment.

5.3

SOTC model of two-level brain

We now have a primitive understanding of the two-brain model of cognition, but this is not the way the real brain functions. These two brains work together to form what we refer to as the two-level brain (TLB), capable of thinking fast and slow, just as Kahneman and Teversky determined the real brain functions. To construct the composite network model requires that we specify how the fast and slow components of thinking influence one another. How do they compliment and interfere in preforming their respective functions? A couple of mathematical choices for how the components of such a

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brain might function were considered towards the end of Chapter 4, but these choices did not reflect the rich nonsimplicity observed in the behavior of human beings. It was more to familiarize ourselves with the potential mathematical behaviors of the model. Here we take a first step in the direction of modeling how the levels in the TLB might exchange information to generate the observed nonsimplicity in decision making necessary to carry out nonsimple tasks. Perhaps, not surprisingly we find that in order to think properly requires social interactions. The calculations discussed adopt the choice of game theory parameter values in the EGM made by Gintis [102] and set R = 1 , P = S = 0 and T − R = 0 .9 . We evaluate the social benefit for the single individual, as well as, for the community as a whole as follows. We define the payoff Pr for element r as the average over the payoffs from the interactions with its four nearest neighbors. If both players of a pair are cooperators, the contribution to the payoff of the individual r, is Br = 2. If one of the two playing individuals is a cooperator and the other is a defector, the total contribution to the payoff of r is Br = T . If both players are defectors the contribution to the payoff of r is Br = 0. The payoff Pr to individual r is the average over the four Br ’s. In the SOTC network simulation we work with a society of N individuals. On the global scale, the mean benefit to society of all the individuals is given by the average over all the individual payoffs Pr : 1 N Pr (t), (5.3) Π(t) = N r=1 whereas the mean imitation strength is given by the average over all the individual imitation strengths Kr (t): K(t) =

1 N Kr (t). N r=1

(5.4)

This completes the inventory of dynamic parameters with which the SOTC model of the brain is described.

160CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY

5.3.1

Crucial events, swarm intelligence and resilience

The SOTC model of spontaneous organization [151] is found to be based on the close connections between information transport and resilience, betwixt resilience and consciousness, between consciousness and criticality, and finally between criticality and crucial events. We may provide an intuitive interpretation of crucial (nonsimple) events, using the example of a flock of birds flying in a given direction, as an effect of self-organization. A crucial event is equivalent to a complete rejuvenation of the flock that after an organizational collapse may freely select any new flying direction. An external fluctuation of even weak intensity can force the nonsimple system to move in a given direction, if it occurs at the exact instant of commitment flicker of the SOTC model system. It is important to stress that the organizational collapse is not the fall of an elite, which will be discussed subsequently, because the flock self-organization occurs spontaneously and is not instigated by the action of a leader. The choice of a new flying direction is thus determined by an external stimulus of even weak intensity occurring at the same time as the collapse, thereby implying the property of complexity (nonsimplicity) matching between the perturbed and the perturbing nonsimple networks [257]. As mentioned earlier, the crucial events maximize the transport of information from one nonsimple network to another [146]. Crucial events are generated by criticality and consequently the transport of information between nonsimple networks becomes maximally efficient at criticality [148]. However, criticality may also be the Achilles’ heel of a nonsimple network, if criticality is generated by a fine tuning control parameter. In fact, committed minorities acting when a crucial event occurs in the case of DMM can make the dominant behavior of the system jump from the state C to the state D [238]. Herein we show that this lack of resilience is not shared by the bottom-up approach to SOTC modeling, in fact, starting from the bottom generates a very

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resilient social organization. The model of [151] is a form of spontaneous transition to criticality, revealed by the emergence of events with the temporal properties of crucial events, thereby explaining the adoption of the name SOTC to define it. We show that the bottom-up SOTC modeling is resilient and that the top-down SOTC modeling is not. We believe that the SOTC model may positively contribute to the discussion of Haidt’s sociological issues [108]. In fact, Haidt emphasized that the political conflict between conservatives and liberals is due to cultural and religious influences that have the effect of creating divisions. We believe that the top-down SOTC approach may be used to model these cultural influences. This is an extremely difficult problem, made even more difficult by the philosophical controversies over the definition of morality [101]. According to the brilliant word-picture painted by Haidt, the philosophy of Hume and Menciu may be compatible with the bottomup origin of cooperation, while the hypothesis that morality transcends human nature, an interpretation that started with Plato and reached full maturity with Kant [108], may justify a top-down perspective. We make the extremely simplified assumption that the top-down SOTC, which undermines social resilience, may explain the fall of elites, if they represent only limited groups, a phenomenon that may be explained by noticing that “our minds were designed for groupish righteousness" [108]. The source of social conflict seems to be that cultural evolution differs from life evolution. These culturally-induced conflicts may, in some instances, overcome the biological origin of cooperation.

5.3.2

Influence flows bottom-up

It is important to notice that Kr , the value of imitation strength adopted by the generic unit r to pay attention to the choices made by its four nearest neighbors about selecting either the cooperation or the defection strategy, is not necessarily adopted by its four nearest neighbors. In other words, the imitation strength Kr (t) is

162CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY unidirectional and it goes from r to all its nearest neighbors. The imitation strength Kr (t) changes from individual to individual, as well as in time, and it is consequently very different from the echo response parameter K of the conventional DMM phase transition processes, where K has a single unchanging value throughout the entire network. Each member the network is assigned a vanishing initial imitation strength to initiate the computation, corresponding to a total independence of the choices made by its nearest neighbors. At each time step the individuals play the game and they independently change their echo response strength carrying out the explicit assumption that the increase (decrease) of their individual payoff in the last two trades makes it convenient for them to increase (decrease) their present imitation strength. More precisely, they adopt the rule encoded in Eq.(5.5). As stated earlier, time is discrete and the interval between consecutive time events is Δt = 1. The imitation strength of individual r changes in time according the individual choice rule:

Kr (t) = Kr (t − Δt) + χ

Pr (t − Δt) − Pr (t − 2Δt) , Pr (t − Δt) + Pr (t − 2Δt)

(5.5)

noting that when Pr (t−Δt) = Pr (t−2Δt) = 0, which is equal to the initial condition, we set Kr (t) = 0 in Eq. (5.5). Furthermore, the parameter χ determines the intensity of interest of the individuals to the fractional change in their payoffs in time. Consequently, the imitation strength of the individual in the DMM network is responsive to the recent history of the payoffs determined by the EGM network, through this coupling. Note that in the limit of vanishing time intervals that Eq.(5.5) relates the time rate of change of an individual’s imitation strength to the time rate of change of the logarithm of the local payoff to that individual. In this way the individual’s DMM dynamics is not responsive to the absolute level of the payoff received, but instead responds to the rate of change in the relative payoff over time.

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The internal dynamics is between the two networks and is generated by the shift in the echo strength interaction determined by Eq.(5.5). This modification of the individual echo strength drives the network toward criticality as manifest by the average imitation strength, as well as, the mean social benefit reaching the fluctuating plateau values discussed in the next section. The emergence of maximum social benefit corresponds to the network evolving towards a majority of cooperators, or altruists, thereby requiring the definition of the mean field ξ(t) given by Eq.(4.1). The condition ξ(t) = 1 corresponds to a social group consisting entirely of cooperators. Bottom-up Results: In the absence of interaction with the PDG, the DMM for the case of a regular two-dimensional lattice with N = 100 would achieve criticality at the value KC ≈ 1.45. At criticality the mean field ξ(t) fluctuates around zero and the time interval between consecutive zero-crossings is described by a markedly non-exponential waiting-time PDF ψ(t), with IPL structure ψ(τ ) ∝ τ −μ ,

(5.6)

where it is determined by numerical calculations that μ = 1.5. Away from criticality the waiting-time PDF is quite different. In the sub-critical regime K < KC the interval between consecutive zero-crossings is exponential, as are the intervals in the super-critical regime K >> KC . Criticality generates non-Poisson renewal events characterized by an IPL PDF. Critical behavior is manifest through crucial events, which have been shown to generate phase transitions, modeled by members of the Ising universality class, in the DMM [257]. The occurrence of a phase transition in a DMM network, with a finite number of interacting individuals, occurs at a critical value of the individual echo response parameters Kr = KC = 1.45. When χ = 1 the calculation is done for a two-dimensional regular lattice (with periodic boundary condition) having N = 100 units, g = 0.01 and T = 1.9, with the mean social benefit, mean imitation

164CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY strength and mean field, all starting from zero. The mean field of the TLB network is driven toward criticality by its internal dynamics, where the time averaged value of the mean field, ξ, does not vanish, due to the fact that criticality in this case generates a majority of cooperators. To stress the occurrence of crucial events in a social system we adopt a method of event detection based on recording the times at which the mean variable crosses its time averaged value. Thus, there are fluctuations around ξ and the inverse power-law structure for the PDF is obtained by evaluating the distribution of time intervals between consecutive re-crossings of ξ. It is important to stress that in addition to ξ(t) that the variables K(t) and Kr (t) are also characterized by the same property, namely, the waiting-time PDF of the time interval between consecutive crossings by K(t) of K and by Kr (t) of Kr . These PDFs are graphed versus time on log-log graph paper in Figure 5.3, and yield an IPL index close to that of ξ(t), which has an inverse power-law index of μ ≈ 1.3. Notice that the regime of intermediate asymptotics [31] for K(t) is as extended as that for ξ(t), while the regime for the individual Kr (t) is somewhat shortened. This is a consequence of the fact that the behavior of the single individual is characterized by frequent collapses to zero and even negative values of Kr (t). On the basis of the definition of the transition rates we can interpret these rare events with negative values as individuals turning into contrarians, e.g., see the interpretation of the gadfly in the discussion of groupthink in Chapter 4. The results of the calculation depicted in Figure 5.4 are used to establish the bottom-up origin of altruism, rather than interpreting it, as it is frequently done, to be the result of a religion-induced topdown process. As we will see, the calculations show that the topdown process generating altruism weakens the system’s resilience, whereas the genuinely bottom-up approach makes the emergence of altruism robust against external perturbation. Figure 5.4 shows that the time evolution of the individual social sensitivity Kr (t) is characterized by abrupt jumps that from time to time may also bring the single individual back to a behavior which is independent

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53

3 2.png

Figure 5.3: The top, middle and lower curves on the left are the waiting-time PDFs of the time intervals between consecutive crossings of ξ (t) , K(t) and Kr (t) with their average values, respectively.

of the choices made by the nearest neighbors. This is a healthy social condition that has the effect of making the global variables ξ(t), K(t) and Π(t) host crucial events favoring the transmission of information between different nonsimple social networks, either countries or people. To stress the occurrence of crucial events in a social network resting on the bottom-up emergence of altruism, we have to extend the method used for criticality generated by the fine tuning of the universal control parameter K. In that case, at criticality the mean field fluctuates around a vanishing value and the crucial events correspond to the occurrence of this vanishing value [36, 240]. We follow [186] and evaluate the fluctuations around the proper non-vanishing mean value of K = 1.5. To explain this choice notice that in the conventional case of criticality, generated by the choice of a proper

166CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY

Figure 5.4: The bottom-up SOTC model: Time evolution of the average social benefit Π(t) top, the average imitation strength K(t) middle, the mean field ξ(t) bottom and the imitation strength of one of the units Kr (t) plotted versus time, also in the middle. From [152] with permission.

control parameter K, with N = 100, K = 1.5 is the value at which the onset of phase transition occurs. This is the value making the mean field ξ(t) of the conventional DMM fluctuate around ξ = 0 with nonsimple fluctuations, thereby generating criticality-induced intelligence [237, 242]. In the case of the SOTC model this condition of criticality-induced intelligence, with fluctuations of K(t) around 1.5 is spontaneously generated. When the criticality condition is reached the nonsimple fluctuations of ξ(t) no longer occur around ξ = 0, but instead around a positive value on the order of 0.8. The time intervals τ between consecutive crossings of the 1.5

5.3. SOTC MODEL OF TWO-LEVEL BRAIN

167

level are monitored and the corresponding waiting-time PDF ψ(τ ) is illustrated in Figure 5.3. The quantity Kr depicted in Figure 5.4 is the value of the echo strength adopted by a randomly identified individual r, under the influence of the choices made by its four nearest neighbors, concerning their selecting either the cooperation, or the defection strategy. This value is not necessarily adopted by its four nearest neighbors, however, as distinct from network reciprocity assumption of Nowak and May [175]. Consequently, the dynamics of the SOTC model network never reduces to the dynamics of the Nowak and May network. In other words, the imitation strength Kr (t) is unidirectional and determines the interaction of r with all its nearest neighbors, but is not necessarily reciprocated. The lack of reciprocity is a consequence of the fact that each of the neighbors experiences a different set of nearest neighbors. The time-dependent imitation strength Kr (t) changes from individual to individual, as well as in time and is depicted in Figure 5.4 along with the erratic time-dependence of the mean field ξ(t), the mean social benefit Π(t) and the mean imitation strength K(t). Figure 5.4 shows the self-organization of the social network as a result of individual choices of the separate individuals. The mean imitation strength K(t) moves very quickly from its zero initial value, corresponding to no social interaction, towards an average value of K ≈ 1.4. Notice that the mean social benefit Π(t) at the top of this figure, which results from averaging over each of the individual payoffs obtained from the play of the evolutionary PDG with their four nearest neighbors, also moves very quickly from its negligible initial value to a fluctuating plateau. The rate of the transition to SOTC is controlled by the parameter χ of Eq. (5.5) and it does occur, either sooner or later, for any positive value of the parameter. Thus, a smaller parameter indicates a weaker response to the change in payoffs, thereby slowing the transition to the plateau of the asymptotic state. In the case of brain dynamics there is wide consensus on the connection between consciousness and criticality, see, for instance,

168CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY [55, 181, 182, 240] and the recent review paper [57]. The electroencephalogram (EEG) signals are characterized by abrupt changes, called rapid transition processes (RTP), which have been proved [8, 9] to be renewal non-Poisson events, with μ ≈ 2. We interpret this to mean that the awake state of the human brain generates crucial events. The crucial events are responsible for the information transport from one system at criticality to another system at criticality [146]. Furthermore, the emergence of crucial events requires that the size of the complex system N to be finite. The intensity of the fluctuations of the mean field ξ(t), as well as other SOTC model variables, obey the general scaling prescription Δζ ∝

1 , Nν

(5.7)

where we express the intensity as the difference between a generic variable A(t) and its time average A : ζ (t) = A(t) − A.

(5.8)

The intensity of these fluctuations is defined by the square root of the variance (standard deviation) Δζ ≡

V (ζ),

(5.9)

and the variance is calculated by the integral over the time series of length L: L

1 V (ζ) = L

2

ζ (t) dt.

(5.10)

0

When working with DMM at criticality, A(t) is the mean field ξ (t), with ξ = 0, resulting in ν = 0.25 [36]. In the case of SOTC model, with A = K, we find ν = 0.5. These criticality-induced fluctuations, becoming visible for finite values of N , are referred to as an expression of temporal complexity.

5.3. SOTC MODEL OF TWO-LEVEL BRAIN

169

Presently we do not have an analytic theory to determine ν for the SOTC model variables, but it is interesting to notice that the numerical calculations illustrated in Figure 5.5 show that ν = 0.5, making fluctuation intensity of Δζ more significant than they are in the case of the ordinary criticality [36]. The fluctuations are here determined by the crucial events and their nonsimplicity constitutes the information transferred from one self-organizing network to another. Increasing the intensity of these fluctuation favors this transport process, but, as we subsequently show, there exists a crucial value of N , below which no signs of the IPL properties of temporal nonsimplicity remain.

Figure 5.5: Individual case: The square root of the fluctuation variance, Δζ of Eq. (5.9), as a function of N . In this case ζ ≡ K(t) − K. We adopted the values: T = 1.5, χ = 4. The red curve is the fit to the data points with slope yielding ν ≈ 0.50

170CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY

5.3.3

Influence flow top-down

In the bottom-up SOTC case the individual imitation strength was generated by nearest neighbor interactions at each point in time. This would, in some crude sense, simulate a democratic way of reaching consensus. An alternative social model is reminiscent of Orwell’s Big Brother, wherein a central figure, or small power elite, determines what is best for society and dictates that behavior for all its members. The latter case is the top-down SOTC model in which we assume that all the units share an universal echo response parameter strength K (t), which changes in time according to the global choice rule: K(t) = K(t − Δt) + χ

Π(t − Δt) − Π(t − 2Δt) . Π(t − Δt) + Π(t − 2Δt)

(5.11)

The global payoff Π(t) is evaluated by making a sum over all possible pairs (i, j), as defined by Eq. (5.3). In the global case we select as the initial condition K(0) = 0.5. The implicit rationale for constructing Eq. (5.11) is that the social community, acting as an entity, makes the same assumption as do the individuals of Eq. (5.5). The assumption is that a payoff increase (decrease) in the last two time increments, before setting the value of the imitation strength to adopt at time t, suggests its increase (decrease) to be convenient. This condition requires a top-down process, a decision made by a leader on the appropriate imitation strength that the single units must adopt for the overall benefit of society. Figure 5.6 depicts the self-organization of the social network as a result of the global choices with all individuals sharing the same value of echo response strength K(t). The qualitative behavior is similar to that of the bottom-up choice, thereby suggesting that the choices of the interacting individuals are characterized by the same kind of intelligence as that of the leader driving the global choice. In fact, the top-down case is tacitly based on the assumption that the collective payoff is communicated to the individuals, who are then forced to share the same imitation strength. Whereas the bottom-up

5.3. SOTC MODEL OF TWO-LEVEL BRAIN

171

choice is based on the more realistic assumption that each person is aware of his/her own individual payoff, without requiring any information transmission from a global social leader.

Figure 5.6: The top-down SOTC model: time evolution of, from the top to the bottom: the average social benefit Π(t); the average imitation strength K(t); the mean field ξ(t). In the calculation the network values are: T = 1.5, χ = 4, and N = 100. From [151] with permission. Thus, we are led to the conclusion that the SOTC model of decision making should be interpreted as a spontaneous emergence of the swarm intelligence that in earlier work was based on externally tuning a control parameter K to a critical value [242]. The comparison between Figure 5.6 and Figure 5.4 leads us to an even more interesting observation. We notice that the global, or top-down,

172CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY choice yields intermittent behavior asymptotically that has the effect of significantly reducing the social benefit, even if, in qualitative agreement with the individual, or bottom-up, choice rule, the system moves toward cooperation. The individual choice rule is more efficient than the global choice rule and is not affected by the strong fluctuations that intermittently reduce the social wealth in the topdown case. For this reason we are inclined to identify the society leader, implementing the global choice, with the benevolent dictator discussed by Helbing and Pournaras [115]. Under the leadership of a power elite, as depicted in Figure 5.6, the imitation parameter K(t) reveals a behavior totally different from that depicted in Figure 5.4. The collapses to small values of K(t) in the top-down configuration are interpreted as the fall of elites [185]. The subcritical condition, as well as the supercritical one, is characterized by a lack of collective intelligence, since the social network is far from the intelligence condition that according to widely accepted scientific opinion [55, 57, 181, 182, 222] requires criticality.

5.3.4

Resilience vs vulnerability

As pointed out by Nowak and Sigmund [176] the EGM is a leading metaphor for the evolution of cooperative behavior in populations of selfish agents. They introduced a cooperator, with a Pavlov strategy, into the EGM, which they determined to be more robust in establishing cooperative stability than the leading alternative strategies. The modification of the imitation strength in the SOTC model, incorporates a memory dependence of the behavior strategy of the individual, based on the utility of prior payoffs. This utility strategy is related to, but is not the same as, the Pavlov strategy, however it does achieve similar stabilizing results in the bottom-up case. Another question of interest is how stable the SOTC network is to the inclusion of a given number of committed defectors. A given fraction of the population is assigned committed defector status and

5.3. SOTC MODEL OF TWO-LEVEL BRAIN

173

the dynamics are allowed to play out, in the same way the influence of a committed minority was calculated in the last chapter. In Figure 5.7 the asymptotic mean field is depicted for a bottom-up SOTC network having a random distribution of a fraction of the total size of the population being committed defectors. It is clear from the figure that it requires nearly 20% of the total population to be uncompromising defectors to achieve a society with 50% cooperators and 50% defectors, where the mean field is zero. Consequently, it is clear that the bottom-up SOTC model is more robust, that is, it is less sensitive to the influence of committed minorities, than is an isolated DMM.

Figure 5.7: The asymptotic mean field level obtained for a halfdozen realizations of the SOTC model is plotted versus the fraction of the total population that are randomly distributed pure defectors. Adapted from [152] with permission.

174CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY The SOTC model does not lead to the unrealistic condition of having a society with 100% cooperators, asymptotically. Note the fact, as shown by the blue curve in Figure 5.4, that 10% of individuals remain defectors. The mean field increases to 0.8 and then fluctuates around this time averaged value. Some individuals may be converted to cooperation, but all individuals are equivalent and the conversion to cooperation is not permanent. As a consequence, 10% of each individual’s time is spent in the defection condition. The red curve of Figure 5.4 shows that while the mean value of the average imitation strength K (t) generates weak fluctuations around a mean value K ≈ 1.4, the imitation strength of the single individual (black curve) may undergo collapses to a condition of total independence of the others (commitment flicker). The bottom-up SOTC model is based on the assumption that individuals can change imitation strengths, so as to maximize individual payoff and this extension of the DMM realizes a spontaneous transition to temporal criticality, meant as a sequence of unpredictable crucial events, which are non-ergodic and non-Poisson. These are the properties of living systems that must be made compatible with the traditional approaches to statistical physics, based on the adoption of stationary correlation functions [249]. The SOTC model successfully realizes this important task. The resolution of AP made using the SOTC model is a consequence of scaling and criticality dominating the nonsimple decision making process. In the SOTC model criticality is not forced upon the network, as it would be by externally setting the imitation strength K to a critical value. The critical value of the imitation strength is spontaneously reached without artificially enhancing the level of altruism within the network, and is dynamically attained by assuming that each individual selects the value of Kr that assigns maximum benefit to themselves at any given time. Individuals are assumed to reach every decision they make using the two networks to adapt to the changing SOTC network behavior. Note that the value of K used in Chapter 4 was interpreted as a form of blind imitation [257]. But the SOTC model leads us

5.4. THE SURE THING PARADOX

175

to interpret Kr , the intensity of which is decided by the individuals on the basis of their own benefit, as the origin of cooperation, or altruism, rather than a form of blind imitation. The SOTC model does not require us to adopt the network reciprocity argument [175] to prevent the infiltration of defectors into cooperation clusters, but instead establishes the emergence of cooperation by the use of the evolutionary PDG payoff, thereby connecting the evolution of cooperation with the search for agreement between individuals and their nearest neighbors. The dynamics of the SOTC model establishes the kind of dynamic steady state that balances the tension generated by the conflicting characteristics of an EP, in this case the AP. This tension is inherent and persistent within an EP typical of complex organizations, as discussed qualitatively by Smith and Lewis [210] for organizations within a business context.

5.4

The Sure Thing Paradox A businessman contemplates buying a certain piece of property. He considers the outcome of the next presidential election relevant. So, to clarify the matter to himself, he asks whether he would buy if he knew that the Democratic candidate were going to win, and decides that he would. Similarly, he considers whether he would buy if he knew that the Republican candidate were going to win, and again finds that he would. Seeing that he would buy in either event, he decides that he should buy, even though he does not know which event obtains, or will obtain, as we would ordinarily say. — Savage [204]

In this section 2 we prove that temporal complexity can also shed light on the paradox generated by the violation of the sure 2 This

section can be skipped by those unfamiliar with elementary quantum mechanics without loss of continuity. But try it, you might like it.

176CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY thing principle (STP), as articulated above by Savage. This rather benign statement can be more formally stated in the following way: a subject selects A over B under both condition X and the complementary condition X, in addition the subject is expected to select A even when it is not known whether condition X, or condition X, is realized. It is apparently obvious that since the selection is the same whether a given condition is met, or the negation of that condition is met, that a wager as to the choice of outcome is a sure thing, even if the person does not know the status of X. This principle was experimentally tested by Tvesky and Shafir [241], with the result marked as an impressive violation of the STP. This important finding is at the origin of the concept of bounded rationality that captured the attention of a significant fraction of the psychological community with the work of Kahneman [127]. In summary, the theory of SOTC [151, 152] addresses the problem of bounded rationality by simultaneously contributing to the resolution of EP, and stressing the social interaction during the decision making of individuals. SOTC considers a set of individuals, again using the nomenclature of game theory, that have to make a choice between A = def ection and B = cooperation. Assuming that individuals focus on their own self-interest, regardless of the state of the environment, they should always select A, thereby preventing society from the benefit of cooperation. This choice is consistent with the STP. Actually, single individuals make their decisions under the influence of their neighbors, using intuition or fast thinking, which is judged to be other than rational. The single individuals either increase or decrease their social sensitivity according to the self-interest principle, using slow thinking, thereby having the effect of making their fast decision rational. This was the result of the SOTC TLB model. In the previous sections of the chapter we presented the details of the SOTC method for resolving EP. It is perhaps now worthwhile to discuss an alternative technique. One based on a different formulation of the notion of probability, which was introduced to address microscopic physical EP, for example, wave-particle duality. This is

5.4. THE SURE THING PARADOX

177

the quantum notion of probability associated with the square of a wave function, rather than with a classical probability [192]. This view is based on the observation that the adoption of quantum probability has the effect of violating STP [54, 223]. The adoption of quantum probability makes it possible to design analog experiments exploiting the quantum mechanical properties of polarized photons [164, 172]. Consider a polarized photon, the polarization direction of which is defined by the angle θ with respect to the vertical axis |v > and by the angle π/2 − θ with respect to the horizontal axis |h >, see Figure 5.8. This corresponds to the quantum state |Θ >= |v > cosθ + |h > sinθ.

(5.12)

The probability for the photon to go through a vertical polarizer is 2

PA = | ν |Θ | = cos2 θ

(5.13)

and the probability for it to go through a horizontal polarizer is 2

PB = | h |Θ | = sin2 θ.

(5.14)

Naruse et al. [172] use this quantum mechanical property to generate an analogue experiment in which the player M has the probability PA of selecting a slot machine A and the probability PB of selecting a slot machine B. The analog experiment can be made compatible with the widely shared conviction that emotions are an important ingredient of cognition [67, 122, 160] by turning the slot machines into potential partners on a dating site and the players M into a person looking for an adult relationship with one of these partners. This makes it possible to assign to the person A and to the person B a degree of attraction for M , denoted by the symbols ΠA and ΠB , respectively, fitting the normalization condition we have ΠA + ΠB = 1.

(5.15)

It is important to stress that the search for the solution of the problem, M having a relationship in this case, can be done following the

178CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY standard prescriptions of reinforcement learning, which is clearly defined [219]: The learner is not told which action to take, as in most forms of machine learning, but instead must discover which actions yield the highest reward by trying them.

Figure 5.8: The horizontal and vertical axis indicate the eigenfunctions for the process. The angle θ denotes the direction of the vector of interest. In turn, the subject M may have different degrees of attraction for the potential partners. This latter condition goes beyond the conventional reinforcement learning [219], which is, however, essential to establish, with the help of psychological experiment [44], whether or not the adult relations are compatible with determinism. We will discuss this important issue at the end of this section. First of all we stress that to make the player M select either A or B in the single realization of the search process, we adopt the widely

5.4. THE SURE THING PARADOX

179

used algorithms making it possible to draw with equal probability a rational number y on the interval [0, 1]. The searcher M , who is unaware of whether A or B will reward him with a date, has to fix an initial threshold T , 0 < T < 1. (5.16) The values of y < T make the player M select A, thereby establishing the condition PA = T. (5.17) The values of y > T makes M selects B, thereby leading to PB = 1 − T.

(5.18)

Note that we impose boundary conditions that do not allow T to exceed 1, while remaining positive. The standard rules of reinforcement learning are based on modifying the value of the threshold T according to whether M gets a reward or not. If y < T , M has to contact A, with an attraction to him of ΠA . To establish if A will say yes or no to him, we have to select a rational number η from the interval [0, 1]. The values of η ranging from 0 to ΠA represent A saying yes and the values of η ranging from ΠA to 1 represent A saying no. If η < ΠA , M is rewarded with a date and to make it easier for him to achieve this reward again, he makes the following adjustment of the threshold value T (t + 1) = T (t) + Δ,

(5.19)

where Δ 1. If η > ΠA , M is not rewarded with a date and consequently he decides to reduce the level of the threshold T : T (t + 1) = T (t) − Δ.

(5.20)

If y > T then M must also contact the other potential partner B. To assess if B responds with a yes or no, we must again draw a number η from the interval [0, 1]. If η falls within the interval [0, ΠB ] the response is yes, and if the draw is in the interval [ΠB , 1] the

180CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY response is no. In standard reinforcement learning M perceives the potential partners as equivalent. Thus, if B says yes, M increases the probability of contacting B again by lowering the threshold, T (t + 1) = T (t) − Δ.

(5.21)

If B says no, M reduces the probability of contacting B again by raising the threshold, T (t + 1) = T (t) + Δ.

(5.22)

Here again we impose the boundary conditions 0 ≤ T ≤ 1. If M is rewarded with a date at attempt j, we record the outcome with an indicator ξ j = 1. If M is not rewarded with a date at attempt j, we record the outcome with an indicator ξ j = −1. To test whether or not the reinforcement learning is successful we make a large number of realizations (trajectories) N and for any time step t we evaluate the mean value of the indicator function: ξ(t) =

1 N

N

ξi.

(5.23)

i=1

The reinforcement learning can be shown to be in accordance with [172] and [164], and that ξ(t) moves from values close to 0 towards values close to 1, with increasing time, implying that M almost realizes the ideal condition of dating at ever asymptotic time step. If PA PB , M dates A. However, if Δ is not vanishing small, M may keep contacting B and being denied a date. This is the source of fluctuations of ξ(t). These fluctuations have also been found by Naruse et al. [172], as well as by Mihana et al. [164], but they did not interpret the source of these fluctuations. To shed light on the nature of these fluctuations we have to establish whether or not they host crucial events, which may be rare, but are responsible for the transmission of information from one to another complex system [6]. To establish whether or not these fluctuations host crucial events, we adopt the simple method proposed

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181

by Pease et al. [186]. We record the times at which the fluctuations cross a horizontal line and evaluate the time interval between consecutive crossings. Temporal nonsimplicity emerges if the resulting waiting-time PDF ψ(τ ) has the asymptotic IPL shape 1/τ μ , with μ < 3. We see from Figure 5.9 that in the case of the ordinary reinforcement learning μ = 2. This is a result of extraordinary interest for the advocates of reinforcement learning, if we take into account that the brain of healthy individual is known to host crucial events with μ = 2 [7].

Figure 5.9: Waiting-time PDFs calculated using histogram of 10 trajectories, resulting in two IPL indices. The fact that the literature on reinforcement learning does not take cognizance of crucial events may be a consequence of the fact that the crucial events of the SOTC model are a manifestation of criticality, which emerges from the cooperative interaction between the individuals of a society. This makes the emergence of crucial

182CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY events in a process of learning surprising and unexpected. However, it is important to stress the connection between reinforcement learning and the model of Tug of War (TOW) [164, 172]. The TOW model can be used to study the competition between two competing groups and the success of a group is significantly determined by the cooperative behavior of its single members, thereby suggesting that TOW may be a simple model to describe the interaction between two nonsimple networks at criticality. Note the similarity in reasoning to the interpretation of Taylor’s Law. It is important to notice that reinforcement learning can be adapted to take emotions into account. Let us assume that M is passionately attracted by B, even if that attraction for M is significantly weaker than the attraction that A feels for M . The love that M has for B makes him replace Eqs. (5.21) and (5.22) with T (t + 1) = T (t) − Δem .

(5.24)

T (t + 1) = T (t) + Δem ,

(5.25)

Δem = χΔ,

(5.26)

and respectively, where with χ 1. For the numerical calculations done to get the emotional results illustrated in Figure 5.9 we use χ = 10. We see from Figure 5.9 that in this case μ = 2.2. This is another impressive property suggesting a connection with the SOTC model. In fact, the human brain has been shown to generate μ = 2, if it is not involved in a difficult task. Difficult tasks, including meditation, have the effects of making μ increase, significantly departing from the ideal condition μ = 2 [233, 234]. This is a consequence of the lack of attention to the occurrence of crucial events, which are an important ingredient of SOTC [150]. The mathematical details on which the theoretical arguments are based are given in Appendix C. The observation of the waiting-time

5.4. THE SURE THING PARADOX

183

PDF ψ(τ ) must be done with time sequences of length L much larger than Teq . In fact, we see from Figure (5.10) that a sequence with L = 104 < Teq yields exponential waiting-time PDF with a rate compatible with the prediction of Eq. (C.21).

Figure 5.10: Waiting-time PDF for a number of total times L. The lower IPL index is associated with the lower L. Figure 5.11 shows that the numerical work is compatible with the renewal property. The waiting-time PDF for aged times ta = 0, 10, 50, 500 are depicted in the figure. Prior to shuffling the four calculations produce PDFs that superpose. However upon shuffling the inter-event times in the time series the four PDFs separate, with a slope becoming more shallow with increasing aging. In addition to the super-statistics formalism in Appendix C we must also consider the renormalization group transformation of the data devised by Shlesinger and Hughes [?] yielding the waiting-time

184CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY

Figure 5.11: Waiting-time PDF for four aged time series, both shuffled and unshuffled PDF

∞

n 1−a ψ(t) = an bn e−b t . a n=1

(5.27)

With the parameters satisfying the condition b < a < 1 the above series converges and generates the asymptotic IPL behavior for the PDF 1 ψ(t) ∝ μ , (5.28) t where the IPL index is μ=1+

lna . lnb

(5.29)

5.5. CONCLUSIONS FROM SOTC MODEL

185

This prescription is compatible with μ very close to 2.

5.5

Conclusions from SOTC model

The main conclusion drawn from these calculations, concerning resilience, is that criticality is necessary for resilience, but it is not sufficient, as first found by Mahmoodi et al. [152]. For example, the top-down SOTC model generates criticality, but it is not resilient. Therefore information transport from one top-down SOTC model system to another top-down SOTC model system is expected to occur by means of nonsimplicity matching, in spite of the fact that the two systems are not resilient and the information transport may be easily quenched by small groups of committed minorities intent on changing society, or by stray perturbations appropriately timed. The mean field fluctuation around the origin has an IPL with index μ = 1.5; a form of temporal nonsimplicity that ensures the maximum efficiency in the transport of information from one network to another with the same nonsimplicity [146, 254]. This important effect has also been established in the case of a neural network model [148], thereby suggesting that criticality-induced temporal nonsimplicity may be a condition of great importance for the sensitivity of a nonsimple network to its environment. Temporal nonsimplicity must not be confused with critical slowing down, as pointed out and discussed by Beig et al. [36]. Both properties are manifestations of criticality and are characterized by the emergence of an IPL, which makes the survival probability non-integrable. However, critical slowing down is a property of the thermodynamic limit, implying that the number of individuals in the network N is essentially infinite, whereas temporal nonsimplicity is a finite size effect [193]. Temporal nonsimplicity, in the ideal case where the IPL is not truncated and ought to be thought of as a form of perennial outof-equilibrium condition, which indicated the need to extend linear response theory to the non-ergodic condition [106]. An attractive interpretation of the resilient nature of the bottom-

186CHAPTER 5. SELF-ORGANIZED TEMPORAL CRITICALITY up SOTC model is that the ideal condition of full democracy is the most robust form of social organization. According to Helbing and Pournaras [115], in fact, the centralized top-down organizations have various flaws reducing their efficiency and they propose instead a bottom-up pluralistic model inspired by neural processes. We believe that the numerical results of the SOTC model lends support to the conclusion that the bottom-up process of the individual choice is more efficient than the top-down process of global choice. Therefore it seems that the SOTC model of a self-organizing network supports the concluding remarks of Helbing [116]: I am convinced that co-creation, co-evolution, collective intelligence, self-organization and self-governance, considering externalities (i.e., external effects of our actions), will be the success principles of the future. In fact, the spontaneous transition to criticality proposed herein is associated with the emergence of significant resilience and adaptability. This will be made clear in the next chapter devoted to designating temporal nonsimplicity, rather than spatial avalanches, as the signature of criticality and to illustrating the related property of nonsimplicity matching . We think that individual choice is an example of SOTC, which is more interesting than the global choice and for this reason we restrict our attention to study the individual dependence on the network size N . The PDG is frequently used in the field of evolutionary game theory [21, 175]. The latter theory attempts to solve the altruism EP, using the concept of network reciprocity [175]. A game is played multiple times on a network where each person adopts the strategy of the most successful nearest neighbor. Since the clusters of cooperators are richer than the clusters of defectors it is reasonable to assume that the most successful nearest neighbor is a cooperator. However, this attempt at mimicking the action of a collective intelligence failed, in large part, because the social activity of individuals, being subcritical, disrupts the beneficial effects of network

5.5. CONCLUSIONS FROM SOTC MODEL

187

reciprocity [98, 150]. We emphasize that SOTC modeling is an attempt to amend the field of EGT by the limitations which prevented, for instance, the mechanism of network reciprocity from yielding a satisfactory resolution of the AP. The human inclination to cooperate is an outgrowth of biological evolution and was shown to be a consequence of the spontaneous evolution towards criticality in the SOTC model. The time appears ripe to unify the models of biology, sociology and physics made necessary to reach the ambitious goal of achieving a rigorous scientific foundation of this important human characteristic [25, 171]. The spontaneous transition to criticality of SOTC contributes to bypassing the current limitations of the field of evolutionary game theory. SOTC models, as reviewed in this chapter, can be adapted to take into account the top-down processes connected with the non-resilient action of the class of power elites. It is possible to supplement the non-rational decision-making process based on the DMM transition rates with self-righteous biases [108], taking into account the influence of religion or other polarizing influences. We expect that such generalizations of SOTC theory would be misguided and would result in a weakening of societal resilience, spontaneously produced by democracy. However, this is left as a subject for future research.

Chapter 6

Criticality and Crucial Events In this closing chapter we explore the interconnections between criticality, crucial events and temporal nonsimplicity. The purpose is to clarify the underlying reasons for the interpretations presented; interpretations that may have been obscured by the formal discussions of the earlier chapters. We use nonsimplicity matching to emphasize the conditions necessary to efficiently exchange information from one nonsimple network to another. The transfer mechanism can be traced back to the 1957 Introduction to Cybernetics by Ross Ashby. Unlike this earlier work, however, we show that nonsimplicity can be expressed in terms of crucial events, which are generated by the self-regulating processes of the SOTC model. Nonsimple processes, ranging from physical to biological and through sociological, host crucial events that have been shown to drive the information exchange between nonsimple networks. 189

190

CHAPTER 6. CRITICALITY AND CRUCIAL EVENTS

6.1

A little more history

Norbert Wiener observed in a 1948 lecture [266] that the nonsimple networks in the social and life sciences behave differently from, but are consistent with, the physical laws in science. He emphasized that the forces which control social phenomena do not follow from changes in energy, as in the world of physical phenomena, but rather such forces are produced by changes in entropy (information). We refer to this observation honorifically as the Wiener Rule (WR), since he presented it in the form of a conjecture and left it unproven. His lecture appeared the same year he introduced the new science of Cybernetics [265] to the scientific world at large in which his interest in control and communication within and between animals and machines was center stage. Since the start of the millennium the nascent science of networks has been used to model the economic webs of global finance and stock markets; the social meshes of governments and terrorist organizations; the transportation networks of planes and highways; the ecowebs of food networks and species diversity; the physical wicker of the Internet; the bionet of gene regulation, and so on. As these networks in which we are immersed become increasingly nonsimple a number of apparently universal properties begin to emerge. One of those properties is a version of the WR having to do with how efficiently interacting nonsimple networks exchange information with one another. West et al. [254], among many others, noted that the signature of nonsimple networks is given by a 1/f power spectrum Sp (f ), that being an IPL: 1 Sp (f ) ∝ α , (6.1) f with the IPL index α restricted to the interval 0.5 ≤ α ≤ 1.5. In fact, this 1/f − variability appears in a vast array of phenomena including the human brain [132], body movements [71], music [186, 216, 246], physiology [253], genomics [139], and sociology [217]. They [254] used the IPL index as a measure of a network’s nonsimplicity

6.1. A LITTLE MORE HISTORY

191

and reviewed the literature arguing that two nonsimple interacting networks exchange information most efficiently when their indices match. The hypothesis of the complexity matching effect (CME) form of the WR was finally proven by Aquino et al. [13, 14] using ensemble averages and under general conditions by Piccinini et al. [189] using time averages. In less than a decade after the formulation of cybernetics as a research discipline, Ross Ashby, captured in his remarkable introductory book on the subject [17], the difficulty of regulating biological systems. He introduced the notion of requisite variety to emphasize that it is the nonsimplicity of the disturbance that must be regulated against. Of course he used the nomenclature of variety in the disturbance, rather than complexity, or nonsimplicity. In any event, his insightful observation was subsequently generalized to nonsimple networks in a variety of disciplines and led to the conclusion that it is possible to regulate nonsimple networks, if the regulators share the same requisite variety (nonsimplicity) as the networks being regulated. Ashby’s requisite variety is, of course, is in large measure the same as the more recent term CME [254]. The mechanism of the CME, or nonsimplicity matching effect, has been widely observed in phenomena which include such information exchange activities as side-by-side walking [10], ergometer rowing [110], syncopated finger tapping [58], dyadic conversation [2], and interpersonal coordination [88, 156]. These synchronized phenomena are today’s realizations of the regulation of the brain through information exchange, in conformity with the observation of Ashby. For the purposes of this essay it is important to stress that there exists further research directed toward the foundations of social learning [73, 89, ?] that is even more closely connected to Ashby’s challenge of the regulator and regulatee sharing a common level of nonsimplicity. In fact, this social learning research aims at evaluating the transfer of information from the brain of one player to that of another by way of the interaction the two players established through their avatars [92]. This is a virtual reality experiment capturing the social cognition shared by a pair of humans. The results

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are exciting in that the trajectories of pairs of players turn out to be significantly synchronized. But even more importantly than synchronization is the fact that the trajectories of the two avatars have a universal structure based on the shared EEGs of the paired human brains. The SOTC model of the TLB provides a theoretical rational for the universal structure representing the brains of the pairs of interacting individuals, based on the CME. More broadly the theory can be adapted to the communication, or information transfer, between the heart and the brain [191] of a single individual. The key to this understanding is the existence of crucial events. In a system as nonsimple as the human brain [199] there is experimental evidence for the existence of crucial events, which, for our purposes here, can be interpreted as organization rearrangements, or renewal failures. The time intervals between consecutive crucial events τ are described by a waiting-time IPL PDF: ψ(τ ) ∝ 1/τ μ ,

with

1 < μ < 3.

(6.2)

The crucial events generate ergodicity breaking and are widely studied to reveal fundamental biological statistical properties [163]. The transfer of information between interacting networks has been addressed using different theoretical tools, examples of which include: chaos synchronization [202], self-organization [206], and resonance [183]. However, none of these theoretical approaches have to date explained the experimental results that exist for the correlation between the dynamics of two distinct physiological systems [133]. In [152] we relate this correlation to the occurrence of crucial events, which are responsible for the generation of 1/f − variability with an IPL spectrum having an IPL index 3 − μ. Such crucial events also explain the results of a number of psychological experiments including those of Correll [63]. The experimental data imply that activating cognition has the effect of making the IPL index μ < 3 cross the barrier between the Lévy and Gauss basins of attraction, thereby making μ > 3 [105].

6.1. A LITTLE MORE HISTORY

193

The crossing of an attractor basin’s boundary is a manifestation of the significant effect of violating the linear response condition, according to which a perturbation should be sufficiently weak as to not modify a system’s dynamic nonsimplicity [7]. The experimental observation obliged us to go beyond linear response theory (LRT) adopted in earlier works in order to explain the transfer of information from one nonsimple network to another and to formally prove the WR. This information transfer was accomplished through the matching of the IPL index of the crucial events PDF of the regulator with the IPL index of the crucial events PDF of the system being regulated [13, 189]. This is consistent with the general idea of the CME, as mentioned earlier, with the main limitation being that the perturbation intensity is sufficiently small that it is possible to observe the influence of the perturbing system on the perturbed system through ensemble averages, namely by taking an average over many realizations [13, 14], or through time averages, if we know the occurrence time of crucial events [189]. Mahmoodi et al. [152] present a theory with which to understand this universal structure, representing the brain of the two interacting individuals, using the ideas developed herein. In addition, the theory can be adapted to the communication between the heart and the brain [191] of a single individual. The 1/f-variability of the spectrum of a significant dynamic variable is a necessary, but not a sufficient, condition to have maximum information exchange between nonsimple networks. This is where the present theory deviates from the early form of cybernetics, in that the present theory entails the existence of crucial events. Another important property of biological processes is homeodynamics [142], which seems to be in conflict with homeostasis as understood and advocated by Ashby. Lloyd et al [142] invoke the existence of bifurcation points to explain the transition from homeostasis to homeodynamics. This transition, moving away from Ashby’s emphasis on the fundamental role of homeostasis, has been studied by Ikegami and Suzuki [121] and by Oka et al. [179] who coined the term dynamic homeostasis. They used Ashby’s cybernetics to

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deepen the concept of self and to establish if the behavior of the Internet is similar to that of the human brain.

6.2

Properties of crucial events

The ergodicity hypothesis is of central importance for the interpretation of the theories developed in statistical physics. We mention this because these techniques are used in the social and life sciences as well. Ergodicity assumes that ensemble averages and time averages coincide, thereby sanctioning the use of statistical evaluations based on the observation of the time evolution of a single systems, when an ensemble of infinitely many identical copies of the same system is not available. It was because of the ubiquitous application of this hypothesis in physics that the experimental observation of the fluctuating fluorescence of single nanocrystals [4, 33] generated so much scientific interest. In fact, the alternating light and dark of the fluctuating fluorescence was subsequently proven to be a nonergodic process [49]. When luminescence is activated a cascade of bright (on) and dark (off ) states ensues. The time duration of these states tend to increase with increasing time, a clear indication of non-stationary behavior, giving the misleading impression that the rules generating fluorescence intermittency change over time. That is not the case. The dynamic process generating the intermittent behavior of light and dark is, in fact, renewal. The occurrence of the switch between the two states is a kind of rejuvenation effect, bringing the system to select the time interval of the new state, either on or off, with the same probabilistic prescription as that adopted by the system when the luminescence process is activated 1 . The compat1 For the sake of simplicity and without the loss of generality, we have made the assumption that the on state and the off state ae characterized by the same waiting-time PDF ψ (t) . Actually the literature on blinking quantum dots show that IPL indices are slightly different. This difference is not relevant for the problems discussed in this essay.

6.2. PROPERTIES OF CRUCIAL EVENTS

195

ibility between the renewal and non-stationary conditions is due to the fact that the probability of switching from the on to the off state, or vice-versa, with a time interval independent from the last switch, denoted as τ , has the IPL structure given by Eq.(6.2) with μ < 2. It is well known [86] that in spite of the fact that the occurrence of an event at a time t implies the total rejuvenation of the system, the probability of occurrence of another event after a time interval τ from the previous event, namely at time t + τ , is given by ψ (τ ). The rate of event generation tends to decrease as 1/τ μ−2 . This nonintuitive property implies a subtle recourse to an ideal ensemble experiment, made with many identical systems, all of them with the property that an event occurs at time t. If this ensemble of identical systems were observed, the rate of switching events would decrease in time with the prescription given by Feller [86]. This non-stationary, but renewal, condition of blinking quantum dots generated a search for an efficient procedure to establish with an acceptable level of confidence that their non-stationary behavior is not due to the rules governing their dynamics changing with time. This important property can be assessed through the theoretical property of aging. If the observation of the process under study is done at a time ta after the occurrence of an event, regardless of whether or not further renewal events occur in between, the PDF of times that are necessary to wait to see the occurrence of a new events is different from the PDF that we may record through the observation of many different realizations of the underlying process [5, 32, 104]. To test the theoretical prediction that would make it possible to establish renewal aging though an ensemble observation, we need to find a way to establish renewal aging by observing a single time sequence of data. Figure 6.1 illustrates how to make the renewal aging assessment using a single realization. We move a window of length ta along the time series, locating the left end of the window on the time of occurrence of an event. The window size prevents us from assessing whether or not events occur before the right end

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of the window. We record the time interval between the right end of the window and the occurrence time of the first event that can be perceived. The moving window serves the purpose of mimicking the use of a very large number of identical systems. In fact, if nonstationarity is not due to the dynamic rules changing with time, the exact moment when an event occurs can be selected as the time origin of the observation process. Initiating the observation process at a time interval ta from the occurrence of an event can be done with the events of the time series under study. This is the purpose of the moving window of Figure 6.1.

Figure 6.1: On the top line is depicted a sequence of events denoted as t0 , t1 , ..., t8 above this line is depicted a sequence of processing elements of duration ta . From the aging operation we obtain the second time series τ 1 , τ 2 , ...τ 4 that constitutes the aged experimental time series. Shuffling this last time series allows us to construct the renewal time series. Using the jargon of intermittence studies we refer to the time interval between the occurrence of consecutive events as the laminar region. It is evident that the times we record are portions of the original laminar regions. The histogram must be properly normalized so that the aged waiting-time PDF generated by this observation

6.2. PROPERTIES OF CRUCIAL EVENTS

197

satisfies the normalization condition ∞

dtψ (t) = 1.

(6.3)

0

Since the survival probability Ψ (t) is defined by the integration over the waiting-time PDF: ∞

Ψ (t) =

dt ψ (t ) ,

(6.4)

t

the normalization condition of Eq. (6.3) can be expressed by Ψ (0) = 1.

(6.5)

In the case where the waiting-time PDF ψ (t) used to generate the events of Figure 6.1 is an IPL with an index μ < 3, the events separating a laminar region from the one following, or from the one directly before, are what we call crucial events. In this case the aging experiment illustrated by Figure 6.1, generating only fractions of the original laminar region, has the effect of favoring the longtime laminar regions, because cutting a very large laminar region may also have the effect of leaving very extended the laminar region produced by the delayed observation. The short-time laminar regions are affected much more by the delayed observation. Thus, the resulting aged waiting-time PDF will be characterized by the following property: The weight of the short times will decrease, while the weight of the long times will increase. This is essential to fit the normalization condition of Eq. (6.3). In other words, the tail of the aged PDF will become slower, being equivalent to decreasing the magnitude of the IPL index μ. In the special case 2 < μ < 3, the average value of τ using the hyperbolic waiting-time PDF: ψ (τ ) = (μ − 1)

T μ−1 μ (T + τ )

(6.6)

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is given by τ =

T . μ−2

(6.7)

The waiting-time PDF of age ta is given by ta (ren) ψ ta

(τ ) =

dt R(t )ψ (ta + τ − t )

(6.8)

0

Due to the fact that τ is finite, when ta is very large we can set the rate of event generation to a time-independent value R(t) =

1 . τ

(6.9)

Inserting the hyperbolic PDF given by Eq.(6.6) into Eq. (6.8) and integrating we obtain the following expression (ren)

ψ ta

(τ ) = (μ − 2) T μ−2

1 μ−1

(T + τ )

−

1 μ−1

(T + ta + τ )

(6.10) Sending ta → ∞ results in the infinitely aged waiting-time PDF: (ren)

ψ ta =∞ (τ ) =

(μ − 2) T μ−2 μ−1

(T + τ )

,

(6.11)

which can be derived from Eq. (6.6) by replacing μ with μ − 1. It is interesting to notice that the corresponding infinitely aged survival probability reads (ren)

Ψta =∞ (τ ) =

T T +τ

μ−2

.

(6.12)

On the other hand, the brand new survival probability reads Ψ(ren) (τ ) =

T T +τ

μ−1

.

(6.13)

6.2. PROPERTIES OF CRUCIAL EVENTS

199

To deepen our understanding of why the term survival probability is the appropriate interpretation of the function given by Eq. (6.13), let us fill the times in each laminar region with either the value +1 or the value −1, according to a coin-tossing prescription. This creates a time series ξ (t), having either the value +1 or −1. Let us create a very large number of realizations M of this time series, and select those realizations that have the first laminar region filled with +1’s. Then, let us make an average over all these realizations j so as to create a function Θ(t) defined by M

Θ (t) =

1 ξ (j) (t) M j=1

(6.14)

with M sufficiently large as to adopt a probabilistic approach to evaluating Θ(t). Of course, this function satisfies the property Θ(0) = 1. By increasing the time t the probability of finding crucial events increases. When one crucial event is found, the function Θ(t) falls to zero, because the adoption of the coin tossing procedure to fill the laminar region implies that half of the second laminar regions are filled with +1’s and the other half are filled with −1’s. Thus, we have that ∞

Θ (τ ) =

dt ψ (t ) = Ψ (τ ) .

(6.15)

τ

Let us now address the issue of evaluating the stationary autocorrelation function of this set of virtually infinitely many realizations. It is known that the adoption of renewal theory [100] yields

Φ (τ ) =

ξ (t) ξ (t + τ ) 2

ξ (t)

1 = τ

∞

dt (t − τ ) ψ (t) . τ

(6.16)

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This expression is true because to find the equilibrium autocorrelation function Φ (τ ) we have to move a time window of size τ along a realization and make sure that the moving window is within a laminar region, since overlapping two or more laminar regions sends the autocorrelation function to zero. The adoption of the renewal prescription of Eq. (6.16) yields, after some straight forward algebra along with the definition of τ given by Eq.(6.7): Φ (τ ) =

T T +τ

μ−2 (ren)

= Ψta =∞ (τ ) .

(6.17)

This observation made by Allegrini et al. [5] to study the aging of autocorrelation function of renewal processes with the method called the generalized Onsager principle (GOP). Although the literature on renewal aging is much more extended than this concise review, we limit ourselves to drawing the reader’s attention to the paper of Bianco et al. [38]. The authors of this paper make a significant contribution to the assessment of the renewal (ren) condition by replacing the evaluation of Ψta (τ ) by shuffling the sequence of times generated by the moving window of size ta . The method was successfully used to establish that systems at criticality generate renewal events. It is important to stress that the longtime region of these crucial events is filled with Poisson events. The assessment of the renewal nature of these events would be extremely difficult, and it is to the best of our knowledge an unsolved problem. The use the shuffling illustrated in Figure 6.1 allows us to assess the renewal nature, or the lack of it, in spite of the fact that the analytical form of the aged survival probability is not yet known.

6.3

Importance of crucial events

Can the phenomenon of criticality be unambiguously defined in a social model? This is a difficult question to answer, because even in social models that are members of the Ising universality class [257], such as the DMM, allowance has to be made for the fact that

6.3. IMPORTANCE OF CRUCIAL EVENTS

201

such networks have a small number of elements, much smaller than the virtually infinite Avogadro number of elements in a physical network. This size dependence has the effect of breaking the singularity condition of ordinary thermodynamic systems. To address this condition our research group [236, 273] defined the occurrence of criticality through the observation of temporal nonsimplicity. In the case of a phase transition occurring in a model of the Ising universality class, with a finite number of interacting elements, at criticality the mean field ξ(t) fluctuates around a vanishing value. Moreover, the time interval between consecutive crossings of the origin is described by a markedly non-exponential waiting-time PDF ψ (τ ) [236]. However, in the sub-critical regime the interval between consecutive crossings of the origin is exponential and in the supercritical regime the interval between consecutive crossings of the nonvanishing mean field is again exponential. Consequently, temporal nonsimplicity emerges at criticality and the proper functioning of the network requires that the IPL PDF of time intervals, between consecutive crucial events, to be exponentially truncated [36, 145, 242]. The adoption of temporal nonsimplicity as the signature of the occurrence of criticality prompted Zare and Grigolini [273] to note that this may be a more important indicator than the observation of avalanches, where the number of elements participating in the process is also an IPL PDF. This assumption was confirmed by Mafahim et al. [148], who found that two networks in critical states signaled by temporal nonsimplicity exchange information with an efficiency substantially larger than in the corresponding state of criticality signaled by avalanches with IPL PDFs. The reason for the close connection between maximal efficiency of information transport and temporal nonsimplicity is based on the theory illustrated in [146, 236, 242]. Criticality generates non-Poisson renewal events characterized by the IPL indexes and the exchange of information is based on the occurrence of the non-Poisson renewal events of one network influencing the occurrence of the non-Poisson renewal events of the other network, this being entailed by the CME, first articulated by West et al. [254, 257].

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We conjectured elsewhere [150, 151] that the SOTC model spontaneously generates temporal nonsimplicity [236]. The present chapter is devoted to establishing that this conjecture is correct and to prove it we adopt a numerical approach, applied to the individual choice rule in the bottom-up SOTC model of the preceding chapter.

6.3.1

Making crucial events visible

To prove the existence of crucial events by making them visible in the SOTC model we monitor the times at which the fluctuations ζ (t) re-crosses zero. When this was done we found that the three waiting-time PDFs coincide, those being for the mean field ξ (t), the mean echo response strength K(t), and the mean global payoff Π (t) . For simplicity, in Figure 6.2 is illustrated the waiting-time PDF for the variability of the mean field around its average value. The fact that fluctuations in K(t), ξ (t), and Π (t) around their average values yield indistinguishable results is an incontrovertible consequence of the fact that all three properties are driven by the non-Poisson renewal events with the same statistical properties, that is, by crucial events. It is known that in nonsimple dynamic networks of finite size the IPL PDFs are exponentially truncated [36]. As a consequence, the non-Poisson nature of the crucial events is established by analyzing the intermediate time region rather than the entire time domain for which there is data for the process. Therefore, to estimate with accuracy the IPL index generated by the SOTC model of the previous chapter we focus on the time region between t ≈ 2 and t ≈ 200, as illustrated in Figure 6.2. We find that the waiting-time PDF is IPL at intermediate times with an index having the numerical value μ ≈ 1.3, rather than the traditional value μ = 1.5, generated by DMM at criticality [36]. It is interesting to note that the length of the time region characterized by μ ≈ 1.3 depends on the size of the network N . Figure 6.3 shows that for N = 400 the IPL region of the PDF is more extended than in the N = 100 case. We also see in the latter figure that for

6.3. IMPORTANCE OF CRUCIAL EVENTS

203

Figure 6.2: The waiting-time PDF of the time intervals between consecutive average value crossings of the function ζ (t) defined for the mean field. We adopted for the calculation the parameter values: T = 1.5, χ = 4, N = 100 on a two dimensional lattice with periodic boundary conditions. Reproduced from [152] with permission.

N = 25 the short time region is characterize by a very large value of μ and by a pronounced exponential shoulder, both conditions are indicative of non-crucial events being generated on those time scales. Although the fluctuation intensity is very large, much larger than for N = 100 and N = 400, the extended IPL region is lost and with it the efficiency of the process of information transport is also lost, as we subsequently show. An appreciation of the importance of the SOTC model can be gained by briefly reviewing some of the research work previously done [?] on the random growth of surfaces, which can be interpreted

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Figure 6.3: The waiting-time PDFs for the time intervals between consecutive origin crossings of the function ς (t) defined for different network sizes N ; top N = 400, middle N = 100 and bottom N = 25 on left. We adopted for the calculations the parameter values: T = 1.5, χ = 4. Reproduced from [151] with perrmission. as a form of SOC [?], and suggests that the Laplace transform of the survival probability ∞

Ψ (t) ≡

dt ψ (t )

(6.18)

t

has the following form, Ψ (u) =

1 α

u + λ (u + Δ)

1−α

(6.19)

6.3. IMPORTANCE OF CRUCIAL EVENTS

205

where α = μ − 1 < 1. Here λ is a parameter measuring the interaction between elements having a decay rate Δ ∝ λ that determines the exponential truncation of the survival probability Ψ (t). In the case where λ 1 an extended time interval exists, 1/λ t 1/Δ, where the waiting-time PDF has the IPL form 1 (6.20) t1+α with α ≈ 0.30. This structure is lost for N = 25, when temporal nonsimplicity is no longer visible. The most important reason for the Laplace transform of the survival probability given by Eq. (6.19) is that when an extended IPL emerges from it, the process is distinctly non-ergodic. The spectrum of fluctuations in the non-ergodic case cannot be derived from the Wiener-Khintchine theorem, resting on the stationary assumption, as it does. It is necessary to take into account the fact that the IPL index μ is less than two, being μ = 1.3 in this case, so that the average time interval between consecutive events diverges, thereby making the process driven by the crucial events non-stationary. This anomalous condition yields the power spectrum for a time series of length L [145]: 1 1 Sp (ω) = 2−μ β (6.21) L ω with the IPL index β = 3 − μ. (6.22) ψ (t) ∼

In the case where the process yields a slow, but stationary autocorrelation function, we would have β < 1 [145]. Evaluating the power spectrum in this case becomes computationally challenging because, as shown by Eq.(6.21), the noise intensity decreases with increasing length L of the time series. Nevertheless, the results depicted in Figure 6.4, which yield a slope of β ≈ 1.66, thereby affording satisfactory support to the claim that the origin crossings of ζ (t) are renewal, non-stationary events. In summary, the SOTC model spontaneously generates the crucial events of criticality-induced temporal nonsimplicity.

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Figure 6.4: Spectrum of the fluctuations of K(t) for the parameter values T = 1.5, χ = 4, N = 100. Reproduced from [151] with perrmission.

6.4

Dynamic nonsimplicity matching

It cannot be overly stressed that the synchronization between two nonsimple dynamic networks is separate and distinct from chaos synchronization; they are not the same. The matching of the dynamics considered here is instead due to the non-Poisson, renewal events (crucial events) of the driving network exerting a profound influence on the crucial events of the responding network, see [13, 189] for a more complete discussion. Piccinini et al. [189] point out that the crucial events are generated by criticality, that is, where the network undergoes a phase transition. In the coupled networks considered here such phase transitions are not the result of changing an externally controlled parameter to a critical value, but rather they

6.4. DYNAMIC NONSIMPLICITY MATCHING

207

are a consequence of an internal, dynamically driven, spontaneous process. In Figure 6.5 we illustrate the remarkable synchronicity between identical SOTC nonsimple networks, A and B, of the same size, with N = 100 [152]. We select a random subgroup SA of network A, consisting of 5% of the units of A, and we assign to each of them the strategy of a unit of B, also randomly selected. We follow the same prescription with a subgroup SB consisting of 5% of units of B following the strategy adopted in randomly selecting units of A. We see in Figure 6.5 that a remarkable synchronization between the coupled SOTC model networks is realized, even with this weak coupling between networks. To establish the sensitivity of this kind of synchronization to network parameters we apply the same procedure to two SOTC model networks A and B, with the size of the networks ranging from N = 25 to 900. We examine the time-averaged cross-correlation function C(τ ): 1 L−τ

L−τ 0

C(τ ) = 1 L

dt ξ A (t) − ξ A

L 0

dt ξ A (t) − ξ A

2

ξ B (t + τ ) − ξ B

L 0

dt ξ A (t ) − ξ A

,

(6.23)

2

and present the numerical results in Figure 6.6. It is evident from the curves in the figure that the magnitude of the cross-correlation does not vary monotonically with the size of the network, but in fact manifests a maxima for an intermediate size network. To appreciate the significance of this result, we introduce another important result recently obtained in the field of evolutionary game theory [214]. This latter paper stresses the connection between the emergence of cooperation and that of memory. Our SOTC model is based on the memory of the profitability of the last two trades, which determines the shift from the previous value of the echo response parameter to its present value. This way of determining the level of influence an element has on its nearest neighbors may be related to their model. Figure 6.6 apparently confirms this relation

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Figure 6.5: The mean fields ξ A (t) and ξ B (t) of two identical SOTC model networks connected to each other as described in the text is an illustration of synchronization. The self-organization is realized through the individual choice mechanism. We adopted the values: T = 1.5, χ = 4, and N = 100 in the caculations. Reproduced from [151] with perrmission.

insofar as it establishes that the cooperation-induced efficiency increases with decreasing size of the interacting networks. However, Figure 6.6 shows that there exists an intermediate size, N = 100, at which the efficiency of information transported from one SOTC network to another is maximum. The heuristic interpretation of this effect is that temporal nonsimplicity is a property of the finite size of the network, with the standard deviation Δζ being proportional to 1/N ν , as shown in Figure 5.5. There is a dependence on the number of units that is even more significant than in the case of ordinary

6.4. DYNAMIC NONSIMPLICITY MATCHING

209

Figure 6.6: Cross-correlation functions between identical selforganizing networks given by Eq.(6.23), calculated for six sizes of networks for curves from top to bottom: M = 100, 225, 400, 49, 900 and 25, with T = 1.5 and χ = 4. Reproduced from [151] with perrmission. criticality [36], thereby clarifying why the communication efficiency increases with decreasing network size. The signature of nonsimplicity is therefore not related to the number of units involved in an avalanche, as previously assumed in most cases and is the basis for SOC, but is even more importantly the signature of temporal nonsimplicity. Criticality is revealed in the case of self-organization, as well as, in the case of criticality generated by the fine tuning of the echo response parameter K. In the case considered here, as we have shown earlier, with the help of Figures 5.4 and 5.6 the fluctuating field may be K(t) itself, which,

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as we have seen in the case of individual choice fluctuates around K ≈ 1.8, when N = 100. Figure 6.3 shows that the domain over which μ = 1.3 decreases with increasing N and at N = 25 a Poisson shoulder emerges, implying that temporal nonsimplicity is lost altogether. Therefore, this loss of temporal nonsimplicity explains the interesting result that an optimal network size exists, at which the efficiency of information transport becomes maximal. In other words, the intensity of the nonsimple fluctuations transmitting information increases with decreasing N , but an excessively small network size annihilates their temporal nonsimplicity altogether. It is important to stress that the location of the peak efficiency at N = 100 depends on the other parameters defining the EGM. The weakening of cooperation with the increase of the number of players is a subject of interest [111, 215], thereby generating the issue of establishing if there exists a universal, robust optimal size of the number of interacting units [174], which is independent of the other parameters characterizing the network dynamics. We cannot rule out that a more refined treatment of the dependence on the parameters of the EGM may lead to an optimal value of N much smaller than 100 in Figure 6.6. However, the emergence of a waitingtime IPL EGM seems to prevent us from accounting for the results of the experimental investigation of Nosenzo et al. [174], setting N = 2 as the optimal size for cooperation emergence.

6.5

Summary and closing observations

The ubiquity of EP was clearly suggested by the discussion of paradox in multiple disciplines in Chapter 2. Some kinds of paradox was seen to have long pedigrees, as documented by an impressive literature in which scholars of various backgrounds have contributed solutions, with varying degrees of success. Others have only recently been identified. Regardless of its source, new or old, what has been made abundantly clear is the intimate connection between paradox and nonsimplicity, that being that nonsimplicity entails paradox.

6.5. SUMMARY AND CLOSING OBSERVATIONS

211

Statistics provided a preliminary explanation of paradox in Chapter 3, where we detailed the divergence from simplicity to complexity, from the Machine Age to the Information Age. We argued that the Industrial Revolution of the nineteenth and early twentieth centuries did not accept the existence of paradox, since at least one pole of such paradox would potentially be in conflict with the remarkable social engine that was transforming society. Moreover, the Information Revolution of the late twentieth and early twenty-first centuries shed these perceptual bonds by accepting all manner of variability and subsequently even embraced paradox, albeit reluctantly. In Chapter 4 we introduced a network-based model of decision making that was used as the basis for an eventual understanding of EP in Chapter 5. A number of calculations that revealed the interpretive context of the model was given in the text. The DMM was used to model the intuitive component of decision making and is based on an individual imitating the opinions of others. It shares a number of the mathematical properties of the physics models such as scaling, criticality and universality, but relies solely on sociologically based assumptions. This model provides an infrastructure for how decisions can be made fast and without rationality. The complementary modeling strategy discussed in this chapter addresses rationality in decision making using a version of evolutionary game theory. The evolutionary PDG incorporates the dynamics of game theory into the decision making process and allows individuals to modify their behavior over time in response to how successful the behavior of others have been in being rewarded for their decisions. The manifesto on computational social science by Conte et al. [61] stimulated much of the social application of our research work, in large part due to the listing of scaling and criticality as two crucial aspects necessary to understanding modern society. We have emphasized a number of times that in our analysis criticality was not forced upon the networks by setting the echo response parameter K to a critical value, in marked contrast to our earlier work [150, 236, 273]. The critical value of K is instead reached spontaneously, without artificially enhancing altruism, by assuming that

212

CHAPTER 6. CRITICALITY AND CRUCIAL EVENTS

each person selects the value of Kr (t) that assigns to themselves the maximal benefit at that time. Chapter 5 synthesizes the two modeling strategies forming a new kind of two-level model. This composite model of the two networks, one a DMM and the other an evolutionary PDG, couples them together in a unique way. The two-level model enables the internal dynamics to spontaneously converge on a critical state that simultaneously satisfies the self-interest of each member of society, while also achieving maximum benefit for society as a whole. This is the selforganized temporal criticality (SOTC) model [151, 152, 262], which is self-adjusting and does not require the external fine tuning of a parameter to reach criticality. Note that the SOTC condition of criticality is reached regardless of whether we adopt the individual bottom-up choice rules, or the global top-down choice rules [151]. The global choice rule implies the existence of a leader and consequently of an intelligence driving the social system. However, the stability of the critical state generated by the bottom-up SOTC model of society far exceeds that of the topdown SOTC model of society. Such results suggest an interpretation for why power elites always seem to fail in the short term, whereas democracies appear to be more stable. The fact that criticality is also generated spontaneously by adopting the individual choice rule is a compelling indication that the SOTC model can simultaneously be interpreted as a spontaneous transition to the condition of swarm intelligence. In this final chapter we explored the interconnections between criticality, crucial events and temporal nonsimplicity. The purpose being to clarify the underlying reasons for the interpretations that may have been obscured by the general discussions of the earlier chapters. The phenomenon of nonsimplicity matching is used to emphasize the conditions necessary to efficiently exchange information from one nonsimple network to another. The information transfer mechanism can be expressed in terms of crucial events, which are generated by the processes of the SOTC model. Nonsimple processes, ranging from physical to biological and through sociologi-

6.5. SUMMARY AND CLOSING OBSERVATIONS

213

cal, host crucial events that have been shown to drive the information transport between nonsimple networks.

Appendix A

Master Equations A.1

The Decision Making Model

As demonstrated by Grinstein et al. [107], any discrete system, defined by means of local interactions, with symmetric transitions between states and randomness originating from the presence of a thermal bath, or internal causes, belongs to the universality class of kinetic Ising models. One such system is the dynamical decision making model (DMM) [38, 235, 236] and is the one we have implemented herein. Positioning N such individuals at the nodes of a two-dimensional lattice yields a system of N coupled two-state master equations [38, 235], which, under the assumption of nearest neighbor interactions, contains time-dependent transition rates: dp(i) (t) = Gi (t)p(i) (t), p(i) (t) = dt

(i)

p1 (t) (i)

p2 (t)

,

(A.1)

where p(i) (t) is the probability of the element i(= 1, 2, ..., N ) in the network at the time t being in one of two states si (t) = ±1, and is (i) (i) normalized such that p1 (t) + p2 (t) = 1 for every i. The matrix of 215

216

APPENDIX A. MASTER EQUATIONS

time-dependent coupling rates for individual i is given by (i)

Gi (t) =

(i)

−g12 (t) g21 (t) (i) (i) −g21 (t) g12 (t)

;

(A.2)

where the individual rates are K (i) (i) M1 (t) − M2 (t) , M (i) K (i) (i) (i) M1 (t) − M2 (t) , g21 (t) = g exp M (i) (i)

g12 (t) = g exp −

(A.3)

and K is the strength of the interaction between elements on the network. This parameter is also called the echo response parameter, since its magnitude denotes the strength of the imperfect imitation response between individuals on the lattice. The symbol M (i) denotes the total number of elements linked to the element i and (i) Mn (t) is the number of those elements in the state n = 1, 2 at time (i) (i) t. Of course, M (i) = M1 + M2 at all times. In the case where each element in the network is coupled to all the other elements, we have all-to-all (ATA) coupling, such that M (i) = N , and the time (i) dependence of the total number of elements in states n = 1, 2, Mn = Nn (t) implies that the transition rates become erratic functions of time [235].

A.1.1

All to All coupling

In the all-to-all (ATA) coupling case,when the total number of elements within the network becomes infinite (N → ∞), the fluctuation frequencies collapse into probabilities according to the law of large numbers. In physics this replacement goes by the name of the mean-field approximation, in which case the transition rates in the master equations given by Eq.(A.3) are the same for each individual i and are written as

A.1. THE DECISION MAKING MODEL

217

(i)

g12 (t) ≡ g12 (t) = g exp [−K{p1 (t) − p2 (t)}] , (i)

g21 (t) ≡ g21 (t) = g exp [K{p1 (t) − p2 (t)}] .

(A.4)

The formal manipulation of the two-state master equation is facilitated if we introduce a new variable defined as the difference in the probabilities P (t) ≡ p1 (t) − p2 (t).

(A.5)

Subtracting the lower from the upper equation in the two-state master equation, after some algebra, yields the highly nonlinear rate equation for the difference variable dP (A.6) = −(g12 + g21 )P + (g12 − g21 ), dt where the nonlinearity enters through the transition rate dependence on the difference variable g12 = g exp [−KP ] ; g21 = g exp [KP ] .

(A.7)

Consequently, the nonlinear rate equation can be written ∂V (P ) dP =− , dt ∂P

(A.8)

and the network dynamics are determined by the potential function V (P ), which is a symmetric double-well potential with the explicit form

V (P ) =

2g K +1 P sinh (KP ) − cosh (KP ) K K

(A.9)

Note that the network is not a mechanical system and yet the global dynamics are described by a double-well potential given by Eq.(A.9) and depicted in Figure 1.5.

218

APPENDIX A. MASTER EQUATIONS It is now convenient to define the mean field variable ξ(K, t) =

1 N1 (K, t) − N2 (K, t) = N N

N

si (K, t)

(A.10)

i=1

where si (K, t) is the state of element i for a given K at time t. The fluctuations of the mean field as depicted in Figure 4.4b characterize the entire network, capturing the cooperation between elements at any moment of time for increasing K. It is interesting that at the critical value of the control parameter the ATA version of the DMM undergoes a phase transition. Note that the amplitude of ξ (K, t) depends on the value of the imitation parameter K. When K = 0, all elements in the network are independent Poisson processes; therefore an average taken at any moment of time over all of them yields zero. Once the value of the coupling becomes non-zero, K > 0 and single elements become less and less independent, resulting in nonzero averages. The quantity, KC , is the critical value of the control parameter K, at which point a phase transition to a global majority state occurs. If the number of elements is large, but finite, we consider the mean-field approximation to be nearly valid, and we replace Eq.(A.10) with the stochastic quantity ξ (K, t) = P (K, t) + f (K, t),

(A.11)

where f (K, t) is a small amplitude random fluctuation. Inserting Eq.(A.11) into Eq.(A.8) and performing some straight-forward algebra, we obtain the stochastic differential equation [38, 236] to lowest order in the strength of the fluctuations: dξ (K, t) ∂V (K, ξ) =− + ε(K, t) dt ∂ξ

(A.12)

The additive fluctuations ε (K, t), have amplitudes √ that are computationally determined to be on the order of 1/ N , with statistics that depend on K.

A.2. SUM AND SEM

A.2

219

SUM and SEM

In Section 4.4 a simple two-level network is introduced, one level of the composite is a DMM network and the other level is an evolutionary PDG network. To establish the distinct interactions between the two levels in the cases of the selfish model (SEM) and success model (SUM), we adopt a natural extension of the DMM [257], requiring the introduction of a second strength parameter K2 and K → K1 , as discussed in Section 5.2.3. The transition rate from the cooperator (1) state to the defector (2) state for individual i at time t, in this coupled model, is given by: ⎤ (i) (i) Θ (t) − Θ (t) K 1 (i) (i) 1 2 ⎦, = g exp ⎣− (i) M1 (t) − M2 (t) − K2 (i) (i) M Θ1 (t) + Θ2 (t) (A.13) and the transition rate from the defector to the cooperator state is given by exchanging indices: (i) g12 (t)

(i)

⎡

⎡

g21 (t) = g exp ⎣

K1 (i) (i) M1 (t) − M2 (t) + M (i)

(i) Θ1 (t) K2 (i) Θ1 (t)

− +

(i) Θ2 (t) (i) Θ2 (t)

⎤

⎦.

(A.14) The meaning of this prescription is easily established. When K2 = 0 the composite model is the same as in the DMM, including the onset of criticality for the imitation parameter value of K1c ≈ 1.65 as depicted in Figure A.1. This figure illustrates the second-order phase transition generated by the DMM, namely, either the SEM or SUM when K2 = 0, corresponding to the condition where no bias exists for the choice of either cooperation or defection. In the initial condition the N units of the network are randomly assigned to either the cooperation or the defection state with the same probability, making zero the mean field of the network where either +1 or −1, according to whether

220

APPENDIX A. MASTER EQUATIONS

Figure A.1: Mean field of the lower level when K2 = 0 for a total length of time, or equivalently, a number of steps L = 106 , for a lattice size of N = 32 × 32 and a total of ten realizations. the individual is in the cooperation or in the defection state, respectively. Of course, due to the fact that we are using a finite number of individuals, N = 1024, the mean field fluctuates around the vanishing mean value. As we increase K1 the intensity of the fluctuations tends to decrease. This is a finite-size effect discussed in detail in the applications of DMM [36, 220]. In the ATA case, where each individual is coupled to all the others, the DMM yields analytical results for the mean field. These results are interpreted as the space coordinate of a particle in a nonlinear over-damped potential, driven by random fluctuations generated by the finite size effect. At criticality, the potential is quartic,

A.2. SUM AND SEM

221

it exerts a weaker containment on diffusion and increases the width of its equilibrium distribution. The coupling between the two states has the payoff matrix: 1+λ 1+τ

−s −λ

,

and in the numerical calculations s = 3 and τ = 2. Note that the cooperator and the defector payoffs are determined by the states of their nearest neighbors. Thus, we have in the contributions to the transition rates (i)

(i)

Θ1 (t) = (1 + λ)

(i)

M1 (t) M (t) −s 2 M M

(A.15)

and (i)

(i)

Θ2 (t) = (1 + τ )

(i)

M1 (t) M (t) −s 2 M M

(A.16)

(i)

where M1 (t) is the number of neighbors in the cooperative state (i) at time t, M2 (t) is the number of neighbors in the defector state at time t and M = 4. It is important to state that we also evaluate the benefit for the community by averaging over all possible pairs of interacting individuals, according to the prescription: Bij = 2,

(A.17)

if both individuals of the pair (i, j) are cooperators; Bij = 1 + τ − s,

(A.18)

if one individual of the pair (i, j) is a cooperator and the other is a defector; Bij = 0, if both individuals of the pair (i, j) are defectors.

(A.19)

222

APPENDIX A. MASTER EQUATIONS

Notice that the choice between the cooperation and defection state is done with λ ≥ 0, while the payoff benefits for the society are evaluated, as shown by Eqs. (A.17), (A.18) and (A.19), by setting λ = 0. The calculation is done in this way because the psychological reward λ, is an incentive to cooperate that does not directly increase the social payoff, even if it has the eventual effect of increasing society’s wealth by stimulating global cooperation. In summary, the average societal benefit is given by Θ=

1 4N

Bij

(A.20)

{i,j}

denoting the sum over all possible pairs (i, j). To define the SEM we set λ = 0 and establish the interaction between the two network levels by assigning a positive value to the coupling coefficient K2 . In this case, a cooperator is encouraged to adopt the cooperator state for a longer time, if the payoff associated with the choice of the cooperator condition is larger than that associated with the defector state. The SUM is established by setting both λ = 0 and K2 = 0. The influence of the DMM on the PDG level is established computationally by randomly selecting a fraction r of the total number of steps L to allow each individual to adopt the strategy of the most successful nearest neighbor. We note that small values of r in SUM play the same role as small values of imitation parameter K2 in SEM. The influence of morality on SEM is studied by setting λ > 0, while keeping λ = 0 for the evaluation of the social benefit, as stated earlier.

A.2.1

Criticality-induced network reciprocity

In Figure A.2 we show the social benefit as a function of K1 when K2 = 0, namely, the strategy choice is only determined by unbiased imitation. The social benefit in the supercritical regime is obviously maximal when all individuals select the cooperation strategy and it vanishes when they all select the defection strategy. Much

A.2. SUM AND SEM

223

more interesting is the social benefit for K < K1c . We see that there is an increase of social benefit with increasing K1 . This is a consequence of the fact that imitation generates clusters of cooperators and clusters of defectors and the increasing social benefit is due to the increasing size of the clusters of cooperators.

Figure A.2: Social benefit as a function of K1 when K2 = 0 and the same parameter values used in Figure A.1. We notice that the social benefit increases linearly with the imitation parameter value throughout the subcritical domain. From [149] with permission. The results depicted in Figure A.3 support our claim that the social benefit increases in a way that is qualitatively very similar to the increase of the number of individuals that have four neighbors in the same state, either cooperators or defectors. A cluster of individuals belonging to the same state increases as a function of K1 with

224

APPENDIX A. MASTER EQUATIONS

a prescription qualitatively similar to the increase of the number of individuals with four neighbors in the same state.

Figure A.3: The triangles represent the calculations for the social benefit.The black squares are the calculations for the percentage of individuals having four neighbors in the same state. From [149] with permission. We now adopt the SUM, namely, we perturb the imitation-induced strategy choice, making the individuals attend to the success of their nearest neighbors. We run the model for L time steps, and randomly select 1% of the time L to update their strategy, adopting the one with their most successful nearest neighbor. As stated earlier, we use a two-dimensional regular network, where each person has 4 neighbors. The results depicted in Figure 4.12 are truly impressive. The adoption of the strategy of the most successful nearest neighbor has

A.2. SUM AND SEM

225

a very modest effect in the subcritical region. By way of contrast, at criticality, the effects of this choice become macroscopic and the imitation-induced phase transition generates the two branches, one with a majority of cooperators and the other with a majority of defectors. These branches are generated with equal probability and the SUM selects the branch with a majority of cooperators.

A.2.2

Morality stimulus on SEM at criticality

The parameter λ is only of psychological benefit implying no direct payoff benefit for society. We interpret this parameter to be the strength of the influence that the morality network has on the SEM, but without explicit modeling of the morality network dynamics. Here we find the imitation-induced intelligence has a twofold effect. The collective intelligence is shown to lead to the abrupt extinction of cooperators, if λ = 0, and to the abrupt extinction of defectors when λ = λc . With the values of the parameters adopted here, λc = 2.5. In Figure 4.13 is illustrated the effect of a strategic choice when λ = 0, shows that in the sub-critical regime a very large value of K2 is required for the extinction of cooperation. As we approach criticality, namely as the network becomes more and more intelligent, even a very weak interest for the personal payoff benefit leads to the extinction of cooperators, which, in fact, is shown to occur for K2 ≈ 0.60 when K1 = 1.6. The swarm intelligence of the network makes the model very sensitive to the influence of morality, as is clearly shown by Figure A.4. We see, in fact, that as a consequence of the imitation-induced criticality, the average social benefit undergoes a kind of first-order transition at λc = 2.5, with a discontinuous jump from no benefit to a very large benefit, indeed, when K1 = 1.65 and K2 ≈ 1.20. This indicates that the criticality-induced intelligence wisely turns the psychological reward for the choice of cooperation into a significant social benefit. However, too large a value of λ may have the same deleterious

226

APPENDIX A. MASTER EQUATIONS

Figure A.4: The average benefit of the SEM as a function of λ, which clearly has a critical value. The curves are determined by the averaging of ten realizations for each value of the imitation strength. From [149] with permission. effects as too small a value. The corresponding cumulative probability is illustrated in Figure A.5, which shows that the PDF of the time intervals between consecutive renewal events is an IPL, with the index μ = 1.5 when λ = λc = 2.5 and it is an exponential function for both λ < λc and λ > λc . This is a clear indication that the collective intelligence generated by criticality [36, 146, 148, 220] is lost if the moral incentive to altruism is either too weak, or too strong.

A.2. SUM AND SEM

227

Figure A.5: The survival probability Ψ (τ ) of the time interval between consecutive regressions to the origin. It is a surrogate for the survival probability for changing states. From [149] with permission.

Appendix B

Importance of Deception Every child makes up stories, plays games and sometimes mixes fantasy with reality. When it is done in a clever manner and without apparent malice adults tend to reward the behavior, but when it is done to cover a mistake or to disguise some spiteful act adults usually punish, or at least scold the child for such behavior, all the while explaining why what they have done is unacceptable. Linking truthtelling to the child’s intent often leads to confusion as to what is the appropriate thing to do, resulting in ambiguity. A clear distinction between the universal acceptability of the truth and the blanket condemnation of a lie is often lost in socially conflicting messages. One lesson a child learns is that a lie told to protect someone from a truth that might cause another pain, or even discomfort, is not only commendable, but telling the truth in that situation is selfish and insensitive. Another lesson learned with equal clarity is that it is always better to tell the truth than it is to lie. These paradoxical lessons learned as a child often led to adults with inconsistent behavior, such as the woman who tells her cubby husband 229

230

APPENDIX B. IMPORTANCE OF DECEPTION

that he must diet, while explaining to her overweight girlfriend that she should only diet if she is comfortable with it. Another is the man who is a trader on the floor of the stock market and a member of the socialist party. How we balance truth and non-truth is our lives is one kind of EP not addressed in this essay. A non-truth has a spectrum of social nuances, which is some semblance of order might be: lie, untruth, falsehood, fabrication, prevarication, deception, half-truth, equivocation, story, myth, white lie, and fib. This ordering is idiosyncratic, but most people would agree that a lie is significantly less true than a fib, even though neither is the truth. Of course, if you come in second in a competition, does it matter how small the margin of victory actually was? For most people the answer would be yes and the margin of acceptability is determined how they have resolved the truth paradox for themselves. Here we draw attention to the use of deception as a tool for achieving a particular outcome. Standing in front of a team and convincing them that they are the best at what they do is not considered to be lieing, but it is certainly not telling the truth. However, ignoring the rational and activating the irrational in a pep-talk can excite the team with the prospect of being the best at football, basketball or computer programing. Shifting the focus from what is real to what is possible makes the truth time dependent, with the challenge being willingness to believe in the win. In fact, most continuing winners maintain that winning is a state of mind and conscious intent is necessary for success. A fairly universal characteristic of Westerners is an attraction to competition, whether it is nature or nurture, there is no denying that most social interactions, from keeping up with the neighbors, to having the largest wardrobe, to driving the fastest car, all have an element of competition. Even those that are less extravagant still might enjoy a day at the races (horses or cars), an evening of cards (bridge or poker), or routing for your favorite sports team in front of the TV. In large part competitiveness is determined by what is compatible with an individual’s temperament, talents and temerity.

231 One activity that blends a person’s ability to deceive and the desire to win is draw poker. In three-card draw poker the ante is the minimal cost to play each hand. After each person puts their ante into the ’pot’ at the center of the table, they are dealt five cards face down and the play is for the content of the pot. At this point an individual can decide not to play and lays down the hand, or elect to discard up to three cards, which will be replaced by the dealer, after the wager. Before deciding the number of new cards desired, the first person to the dealer’s left initiates a round of betting. That player can wager up to some predetermined maximum amount, or check. The next person in the sequence can match the bet, or increase the wager up to the same predetermined value, or decide to throw the hand away, thereby forfeiting the contents of the pot. Each of the players is given the same option, matching all the prior bets, raising the last bet even more, or throwing the hand away. All the money wagered is put into the ever increasing pot. There are some variations in the rules of betting that need not concern us here. The point is that after this round of wagering, the new cards are dealt and people determine whether or not they have improved the value of their hands. A second round of betting is then initiated. It should be mentioned that in the book that launched the mathematics of game theory, Theory of Games and Economic Behavior [245], the authors von Neumann and Morgenstern emphasized the importance of bluffing as a psychological strategy for winning. A player bluffs when they wager to give the impression that the value of the cards they hold are higher than they objectively are under the rules of the game. This can be done in a number of ways. For example, when a betting round comes to a particular player, three options are available that will give the impression the player has a good hand: 1) meet all bets and raises made prior to their turn and then raise the last bet; 2) if no prior bet is made the player can make a wager; 3) the player can check (or meet any bets previously made, but not raise), then if a subsequent player makes a wager on

232

APPENDIX B. IMPORTANCE OF DECEPTION

the same round of play, the player raises their bet. The bare rules do not capture the true dynamics of a round of betting and the open discussion that accompanies it. One may become quite skilled at poker by learning the probability of each hand and betting accordingly. However to truly master the game requires a knowledge of how to read one’s opponents, that is, understand their personal reactions to being dealt winning cards, or losing ones. Most importantly, to make a living at poker demands mastering the psychology of bluffing, otherwise when facing professionals, the cards alone will rarely determine the winner. Or so I have been told.

Appendix C

Analytic Arguments and STP Much of the formal manipulation regarding the sure thing paradox (STP) in Section 5.4 was postponed to facilitates discussed. Those delayed mathematical details are presented in this appendix.

C.1

Limits of T

Let us define pA and pB , as the respective probabilities of getting a response of yes from A, as well as, from B and qA and qB as the respective probabilities of getting a response of no from A, as well as, from B. We can explicitly construct these probabilities as pA = T W,

(C.1)

pB = (1 − T )(1 − W ),

(C.2)

qA = T (1 − W )

(C.3)

qB = (1 − T )W.

(C.4)

and

233

234

APPENDIX C. ANALYTIC ARGUMENTS AND STP

Note that these probabilities are so normalized that they sum to one: pA + pB + qA + qB = 1. (C.5) Thus, the probability for M to obtain a yes from either A or B is: pA + pB = (2W − 1) + 1 − W.

(C.6)

An efficient reinforcement algorithm is expected to make T move from the initial value that we set equal to 1/2 to the optimal value corresponding to the largest success, pA + pB = 1. Thus, we see that the optimal value of T is given by Topt =

W 2W − 1

(C.7)

Let us make the assumption that A likes M more than B likes M , namely W > 0.5. We see that in this case, if W < 1, then Topt > 1, thereby implying that in that case the reinforcement learning algorithm must be supplemented by boundary constraints. This suggests that we first study the case W = 1.

C.2

The special case W=1

In this case A always maintains yes, whereas B always maintains no. The time evolution of T (t) is determined only by the no’s of B. Consequently, dT (t) = (1 − T (t))Δ, (C.8) dt which we solve subject to the initial condition T (0) = 0.5,

(C.9)

mirroring the total lack of information of M when the play starts. The solution to the rate equation is therefore given by T (t) = 1 − 0.5 exp[−Δt].

(C.10)

C.2. THE SPECIAL CASE W=1

235

As shown by Figure C.1, numerical simulation yields a result indistinguishable from the exact solution of Eq. (C.10).

Figure C.1: Comparison between numerical simulation and the theoretical prediction of Eq. (C.10). Let us focus our attention on the mean field given by Eq.(5.23). When N = 1, we obtain for the mean field the difference in the probabilities ξ (t) = p(t) − q(t),

(C.11)

p(t) = pA (t) + pB (t)

(C.12)

where the total yes probability is

and the total no probability is q(t) = qA (t) + qB (t)

(C.13)

236

APPENDIX C. ANALYTIC ARGUMENTS AND STP

Using Eqs. (C.1) −(C.4) with W = 1, we obtain p (t) = T (t)

(C.14)

q (t) = 1 − T (t)

(C.15)

and Thus, inserting these last two expressions into the equation for the mean field yields ξ (t) = 2T (t) − 1,

(C.16)

which, inserting the solution to the rate equation becomes ξ (t) = 1 − e−Δt

(C.17)

Note that using the definition of the mean field we obtain 2

ξ (t)

= p(t) + q(t) = 1.

(C.18)

Thus, the standard deviation Δξ, defined by Δξ =

2

ξ (t)

− ξ (t)

2

(C.19)

can be written Δξ = e−Δt/2

C.2.1

2 − e−Δt .

(C.20)

Violation of the renewal condition

Eq. (C.8) is obtained by assuming that M does not make its threshold T (t) change when the girl A says yes. Thus, the sequence of yes n times is the probability that the random number y ends in the interval [0, tm ] n consecutive times. Therefore, the survival probability is obtained through n

Ψ(tm , n) = T (tm )n = (1 − (1 − T (tm ))) ≈ e−n(1−T (tm )) , (C.21)

C.2. THE SPECIAL CASE W=1

237

where tm = n1 + .... + nm .

(C.22)

This is a kind of diffusional time with a statistical distribution that cannot be evaluated using any of the central limit theorems, since the positive numbers ni are derived from many different Poisson probability prescriptions. Using Eq. (C.10) we write Eq. (C.21) Ψ(tm , n) = e−nΛ(tm ) ,

(C.23)

where the time-dependent rate is Λ(tm ) ≡ 0.5e−Δtm .

(C.24)

This double exponential form of the survival probability was first constructed by Emil Gumbel in his study of extreme values. This allows us to define the global survival probability Λmax

Ψ (n) =

dΛΠ (Λ) exp[−Λn]

(C.25)

Λmin

According to the formalism of superstatistics [6, 34, ?], the PDF for the decay rate Π (Λ) has the form Π (Λ) =

Aμ−1 Λμ−2 −ΛA e Γ (μ − 1)

(C.26)

The numerical results show that Eq. (C.25) should yield Ψ(n) ∝

1 , n

(C.27)

corresponding to the crucial IPL index μ = 2.

(C.28)

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Index 1/f-variability, 195 Acemoglu, 153 adaptability, 188 Adrian, 68 aging experiment, 199 all to all coupling ATA, 218 Alllegrini, 202 altruism paradox top-down process, 166 altruistic gene, 39 ambiguous figure, 18 Anderson 1977 Noble Prize, 64 antinomy, 4 Aquinis, 77 Ariely, 150 Aristotilean logic, 15 Aristotle, 4 arms race, 55 Army United States, 54 Asbiyyah effect, 138 Ashby, 193, 195 Asimov, 86

attention flicker, 24 Axelrod, 129, 147 Bak, 146 beetle new species, 47 bell-shaped curve height, 66 bell-shaped distribution, 69 belonging paradox, 40 Bernoulli, 127 Bianco, 202 Black Swan, 76 bluffing, 234 Bohr, 2, 6, 11, 142 both/and, 11, 41 thinking, 16 bottom-up influence flow, 163 SOTC model, 187, 204 spontaneous, 22 bottom-up results, 165 Box, 98 Braathen, 51 brain two-level model, 152 264

INDEX Brown, 8 bursty behavior, 82 business organizations, 42 butterfly effect, 13

265

control, 74 control paradox, 53 correlation, 116 Correll, 194 Couzin, 99 crisis, 116, 119 Carver, 81 critical phase catastrophy theory, 13 extended, 157 central limit theorem, 26, 63, 68 critical phenomena, 143 chess master, 45 critical point, 118 citations critical points, 157 IPL, 79 criticality, 107, 135, 144, 156, 203 Civil War, 34 extended, 157 Clausius, 7 imitation-induced, 227 CME, 194 phase transition, 27, 97 crucial events, 194 self-organized temporal cognition, 24, 88 SOTC, 28, 141, 212 tor-down, 22 two categories, 145 Cohen, 80 crticality collective behavior, 64 spontaneously reached, 88 collective intelligence, 135 crucial event commitment flicker, 103, 119, 162 proof , 204 committed minority, 119, 121, 175 crucial events, 162, 191 complexity, 12 bottom-up altruism, 167 chaos, 12 Curfs, 18 psychpathology, 14 Currie, Madame, 72 complexity matching effect Cybernetics, 192 CME, 193 cybernetics, 14, 65 Conferate Army, 34 conflict, 51 Dali, 76 conflict intervention, 52 Darwin, 37, 79, 132 consensus, 87, 103, 136 dcision making consistency, 44 consistency, 44 Conte, 212 decay contradiction, 76 inverse power-law, 118

266 deceit, 45 deception, 231 decision making, 100 dynamic, 16 decision making model DMM, 27, 97 deep truth, 74, 142 deliberation, 151, 155 devil’s advocate, 100, 124 distribution heavy tailed, 78 inverse power law, 77, 82 IPL, 86 normal, 69 of grades, 75 Pareto, 77, 81 DMM, 106, 150 overview, 103 draw poker, 233 dynamics bottom-up, 146 echo response hypothesis, 106 echoes, 105 Edison, 49 efficiency cooperation-induced, 210 EGM, 127, 129, 151 Einstein, 10, 33 1922 Nobel Prize, 10 either/or, 11, 39, 41 thinking, 16 Ekeland, 12 elite, 162 elites

INDEX fall of, 174 empirical paradox EP, 31 Eniac, 14 entropy, 7, 14 statistical, 8 EP, 142, 147 empirical paradox, 16 ergodicity, 196 Escher, 18 Euclid, 5 evenness, 85 evolutionary game model EGM, 101 evolutionary game theory, 27, 88, 212 EGT, 98 explore, 133 extreme variability, 76 face-vase, 18 failure, 47 of theory, 32 Failure Award, 49 Farson, 48, 49 feedback, 14 Feller, 197 Feynman, 67 finite size system, 158 Fisher, 14 flock, 119 fluctuations, 67 intensity, 170 random, 71

INDEX fluorescence, 196 fMRI, 21 forces information-based, 82 fractal, 83 gadfly, 123, 124 Galileo, 50 Galton, 79 game theory, 56 evolutionary, 155 strategy, 128 Gauss, 68 Ghandi, 72 Gintis, 127, 134 Godel incompleteness theorem, 4 gravity, 6 Grigolini, 127, 134, 203 Grinstein, 217 groupthink, 99, 123, 148 habituation, 82 Haidt, 163 Hamilton, 129, 147 Heisenberg, 11 Helbing, 174, 188 heroes, 33 Hilbert, 5 Hitler, 72 Hobbs, 36 holf, 148 homeodynamics, 42, 195 homeostasis, 195 human nature, 36

267 Hume, 163 hypothesis echo response, 105 Ikegami, 195 imbalance, 78 social, 80 imitation hypothesis, 104 imitation mechanism, 100 importance, 202 individual choice, 188 Industrial Revolution, 10, 26, 63, 211 information, 14 exchange, 82, 161 Information Age, 11, 26, 63, 81, 211 information exchange, 193 maximally effucuent, 203 information forces, 192 information transmission, 173 innovation, 55 innovation paradox, 48 intelligence collective, 99 imitation-induced, 137 swarm, 145 interdependence, 52 internal control, 153 Internet, 81 intuition, 24, 88, 151, 152 inverse power law, 80 invisible hand, 39 Ising model, 107, 118, 144 Ising models, 217

268 Ising universality class, 100 Jackson President, 35 Janis, 99 Jordan, 76 Kahneman, 148, 151, 156 Kant, 163 Kao, 99 Kepler three laws, 12 Keyes, 48, 49 kin selection, 132 Kipling, 45 Kleinschmidt, 20 knowable unknown, 71 laminar region, 198 Laplace, 68 lattice two-dimensional, 111 Law Taylor’s, 184 law frequency of error, 68 Taylor’s, 83 laws Newton’s, 7 leadership, 41, 44, 88 learning paradox, 40 Lewis, 40, 177 Lewis M, 42 LFE, 68, 70 life

INDEX secret of, 8 Lighthill, 12 linear logical thinking, 147 linear logic breakdown, 3 linear responce theory violation, 194 Lloyd, 195 logic, 142 Lorenz, 13 Machiavelli, 72 Machine Age, 26, 63, 73, 211 Mackay, 113 macrodynamics, 64 Mafahim, 203 Mahmoodi, 127, 187, 195 Mandelbrot, 83 Martin, 79 Martinello, 146 Mary Edwards Walker, 33 master equations, 217 coupled two-state, 217 two state, 153 May, 127, 128, 151 Mayer, 52 mean benefit, 161 mean imitation strength, 161 Medal of Honor, 33 Menciu, 163 Michelangelo, 72 microdynamics, 64 Milgram, 101 Mill, 80

INDEX mob, 142 morality stimulus, 227 Morgenstern, 127, 233 Morin, 13 multilevel selection, 39, 132 nanocrystal, 196 natural law, 67 Natural Philosophers, 68 natural selection, 37 near-critical state, 159 network scale free, 81 social, 3 network dynamics EGM, 129 network reciprocity, 188, 224 New York, 34 Newton, 73 noise, 106 non-Poisson events, 203 non-stationary crucial events, 207 non-stationary behavior, 196 nonsimple phenomena, 15 nonsimple networks, 162 nonsimple phenomena, 16 nonsimplicity social, 142 the world, 72 normal curve, 73 Nosenzo, 211 Nowak, 127, 128, 130, 151, 174

269 O’Boyle, 77 O’Neil, 76 Oka, 195 Onsager, 113, 144 Onsager Principle generalized, 202 optical illusion, 18 organizing paradox, 41 Oswego New York, 34 oxymoron, 143 paradox, 1 altruism, 33 capability, 2, 78 conflict, 50 control, 52 empirical, 32 empirical, EP, 6 entailment, 143 innovation, 47 IPL, 79 logical, 3 organization, 40 para bellum, 55 physical, 6 St. Petersburg, 127 strategic, 43 survival, 45 visual, 18 wave-particle duality, 11 Pareto, 26, 63, 77, 79 distribution, 80 Pareto distribution inverse power law, 77

270 Pareto’s law, 86 Parker, Bonnie, 72 Paulus, 100 Pavlov strategy, 174 payoff, 56 payoff matrix, 128 PDG, 58, 127, 129, 156 Pease, 182 perceptual states, 22 performing paradox, 41 phase transitions, 65 Picasso, 76 Planck, 10 1918 Noble Prize, 10 Plato, 163 Poincare, 13 Poisson, 85, 108 poker player, 45 potential DDM, 158 double well, 22, 25 Pournaras, 174 power curve, 46, 83 predator, 119 predictability, 72 preemptive strike, 58 principle of superposition, 73 prisoner’s dilemma game PDG, 88 psychological reward, 224 quantum hypothesis, 10 Quetelet, 79 Quine, 4

INDEX rainbow, 10 Rand, 130, 151 randomness, 67 rapid transition process RTP, 170 reaction to loss, 50 reciprocity, 131 regulation CME, 193 nonsimplicity, 193 reinforcement learning, 183 rejuvenation, 197 renewal process, 196 renewal theory, 201 renormalization group, 145 requisite variety, 193 resilience, 162, 174, 188 rule global choice, 172 individual choice, 164 scale-free, 118 scaling, 108 self-interest, 39, 143 self-organization, 87, 142, 157, 172 self-organized criticality SOC, 146 SoC, 87 self-organized temporal criticality SOTC, 87, 147 self-referential, 5 selfish agents, 174 selfish gene, 39 selfishness, 36, 88

INDEX selfishness model SEM, 133 selflessness, 36, 88 SEM, 136, 221 Shannon, 14 Sherman, 34 Sigmund, 174 Sinatra, 76 small world problem, 101 small world theory, 102 Smith A, 39 Smith WK, 40, 41, 177 SOC, 146, 206 social science computational, 87 sociobiology, 37, 45 sociology modeling, 98 Sornette, 87 SOTC, 141 bottom-up model, 162 TLB model, 204 top-down, 173 top-down model, 163 SOTC model, 28, 212 crucial events, 204 spatial selection, 132 speciation, 46 stability, 44 standardized variable, 70 Stanley, 107, 118 statistics, 65, 67 LFE, 69 normal, 66 understanding, 71

271 stochastic oscillators two-state, 157 stochastic resonance, 24 strategic paradox, 43 strategies, 129 Strogatz, 101 success, 47 success model SUM, 133 SUM, 136, 221 survival probability, 206 Suzuki, 195 swarm intelligence, 162 synchronicity, 209 Syracuse Medical College, 34 Taleb, 76 Taoism, 16 Tarde, 104 Taylor LR, 46 Taylor RAJ, 46 Taylor’s law, 85 temporal complexity, 147 temporal nonsimplicity, 191 criticality-induced, 207 Tennessee, 34 thinking, 43 complex, 13 crticality, 86 logical, 4 nonlinear, 45 thinking fast, 160 thinking slow, 156 Thomas, 35

272 time, 14 time reversibility, 7 tipping point, 13, 121, 142 TLB network, 166 top-down SOTC model, 187 TLB, 172 transition, 144 transition rate, 152 transitions, 114 Turalska, 100, 130 Tversky, 150 two brain model, 148 two-level brain TLB, 160 two-level model, 212 uncertainty, 80 Union Army, 34 United States Army, 54 universality, 108 unknowable unknown, 71 variability, 46, 67 normal curve, 74 Vilone, 153 von Neumann, 127, 233 wager, 233 waiting-time PDF IPL, 165 Wang, 134 war, 58 Watts, 101 wench-witch

INDEX visual paradox, 24 West, 47, 100, 130, 192, 203 Wiener, 14, 65 Wiener Rule WR, 192 Wiener-Khintchine theoren, 207 Wilson, 145 Wilson DS, 38 Wilson EO, 38 wisdom of the crowd, 98 wisdom of the crowd, 99 Wittmann, 100 world map mental, 66 world view Pareto, 81 World War II, 14 World Wide Web, 81 WR, 195 WWW, 81 young love, 53 Zapperi, 146 Zare, 203 zealot, 119 zealots, 121 Zelaya, 53