Genealogical mathematics: Proceedings of the MSSB Conference on Genealogical Mathematics February 28-March 3, 1974 at the University of Texas Health Science Center at Houston, Center for Demographic and Population Genetics 9783111654317, 9783111270265

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Genealogical Mathematics

Publications of the Maison des Sciences de l'Homme

Mouton • Paris • The Hague

Genealogical Mathematics Proceedings of the MSSB Conference on Genealogical Mathematics February 28 - March 3, 1974 at the University of Texas Health Science Center at Houston Center for Demographic and Population Genetics

Edited by Paul A . BallonofF

Mouton • Paris • The Hague

© 1974 by Maison des Sciences de l'Homme Printed in France

The participants in the Conference were: John W. Adams Department of Anthropology and Sociology University of South Carolina

Alice Bee Kasakoff Department of Anthropology and Sociology University of South Carolina

John R. Atkins Department of Anthropology University of Washington

F. K. Lehman Department of Anthropology and Linguistics University of Illinois

Paul Ballonoff Center for Demographic and Population Genetics University of Texas at Houston

John R. Lombardl Department of Anthropology Boston University

Thomas E. Duchamp Department of Mathematics University of Illinois

T. Maruyama National Institute of Genetics (Mishima, Japan)

E. Paul Durrenberger Department of Anthropology University of Iowa

Martin Ottenheimer Department of Sociology and Anthropology Kansas State University

Henry Harpending Anthropology Department University of New Mexico T. Robert Harris Department of Sociology and Anthropology Kansas State University Françoise Héritier Laboratoire d'Anthropologie Sociale Centre National de la Recherche Scientifique (Paris)

Dwight W. Read Department of Anthropology University of California at Los Angeles Russell M. Reid Department of Anthropology University of Texas at Austin Klaus Witz Department of Mathematics University of Illinois

Albert Jacquard Institut National d'Études Démographiques (Paris) As director of the Conference and friend of the participants, it was a great pleasure to see these materials collected in one place, and to have participated in the fraternal discussions of the Conference. Paul Ballonoff Houston, Texas March 1974

CONTENTS PREFACE INTRODUCTION:

GENEALOGICAL MATHEMATICS

1

Paul A. Ballonoff CONFERENCE SELF-SUMMARY

19

Klaus Witz PART I GRAFIK:

-

ALGEBRAIC APPROACHES

A MULTIPURPOSE KINSHIP METALANGUAGE

25 27

John R. Atkins MATRIX METHODS IN THE THEORY OF MARRIAGE NETWORKS

53

Thomas E. Duchamp and P. A. Ballonoff AN INTRODUCTION TO A MATHEMATICAL APPROACH TO THE STUDY OF KINSHIP

63

R. J. Greechie and Martin Ottenheimer FROM RELATIONS TO GROUPS: A FORMAL TREATMENT OF THE UNDERLYING LOGIC TO SOME GROUP-THEORETIC MODELS OF KINSHIP

85

T. Robert Harris PROLEGOMENA TO A FORMAL THEORY OF KINSHIP

111

F. K. Lehman and K. Witz KINSHIP ALGEBRA:

A MATHEMATICAL STUDY OF KINSHIP STRUCTURE

135

Dwight W. Read PART II

-

STATISTICAL APPROACHES

STRUCTURAL MODEL OF DEMOGRAPHIC TRANSITION

161 163

Paul A. Ballonoff SYSTÈMES OMAHA DE PARENTE ET D'ALLIANCE. ETUDE EN ORDINATEUR DU FONCTIONNEMENT MATRIMONIAL REEL D'UNE SOCIETE AFRICAINE

197

Françoise Héritier HOW MANY RELATIVES? Alice Bee Kasakoff

215

THE EFFECTS OF EXOGAMY ON DEMOGRAPHIC STABILITY

237

John R. Lombardi RELATIVE AGE AND ASSYMETRICAL CROSS-COUSIN MARRIAGE IN A SOUTH CASTE

257

Russell M. Reid PART III

-

GENETIC APPROACHES

COMPUTATION OF ABNORMAL GENETIC INHERITANCE: ESTIMATION OF PARENTAGE EXCLUSION PROBABILITIES, LEGITIMACY AND ADOPTION RATES

275 277

A. Langaney HOMOGENEISATION OF KINSHIP IN A SMALL POPULATION

285

Th. Leviandier and A. Jacquard IDENTITY OF GENES IN GEOGRAPHICALLY SEPARATED INDIVIDUALS AND THE GENETIC VARIABILITY MAINTAINED IN A POPULATION Takeo Maruyama

291

PREFACE

The Conference on Genealogical Mathematics brought together for three days technical experts from social anthropology, population genetics, demography and mathematics.

Their primary reason for attendance of the

Conference was completion of new mathematical developments in the field, and a desire to discuss these developments and project possible future work. Attendance was solicited by three means:

direct invitation to several

workers whose presence was thought particularly desirable; open advertisement; announcement of the Conference by mail to workers known to be interested in the field.

Attendance included persons from all sources.

All

papers were received and circulated one month in advance of the Conference. This allowed the time of the Conference to be spent on discussion of the papers and more general issues.

The present organization of papers re-

sulted from those discussions, while Klaus Witz was decided upon by the Conference to write the Self-Summary. The Conference was held on February 28 to March 3, 1974, under the auspices of the Center for Demographic and Population Genetics, University of Texas, Houston.

Support for the Conference and for publication

of the Proceedings were from several sources.

Initial and primary funding

were by the Mathematical Social Sciences Board

(National Science Founda-

tion).

However, transportation for certain participants was provided by

the Institut National d'Etudes Demographiques, Paris, and by the Maison des Sciences de l'Homme, Paris.

Proceedings were prepared for publication

with the assistance of the M.S.S.B., The Center for Demographic and Population Genetics, and the Maison des Sciences de l'Homme.

We are especially

grateful to Dr. Clemens Heller of the Maison for his interest and assistance in the project, and to R. G. D'Andrade and the M.S.S.B. for assistance creating the Conference in the first place.

INTRODUCTION:

GENEALOGICAL MATHEMATICS

Paul A . B a l l o n o f f

At an increasing rate in the last quarter o f a century, w o r k e r s h a v e laid the foundations for m a t h e m a t i c a l anthropology.

scientific

studies in social

But simultaneously, w e h a v e w i t n e s s e d a convergence o f

interests b y p o p u l a t i o n geneticists and m a t h e m a t i c i a n s o n v e r y problems.

similar

In M a r c h 1974 the Conference o n Genealogical M a t h e m a t i c s

b r o u g h t together practitioners of these several arts for three days o f intense discussion.

The present paper is a summary o f the Conference

itself and of the status of genealogical m a t h e m a t i c s , from the v i e w p o i n t o f the Conference organizer.

The paper titled "Conference

Self-Summary"

by K. W i t z presents a v i e w o f the Conference as seen b y its participants, and this longer paper follows that one in m o r e detail. I.

BACKGROUND

The concept o f "genealogical m a t h e m a t i c s " appears fundamental to w o r k in several fields.

In social anthropology, genealogies as such are a

p r i m a r y objective in field investigations.

D e s c r i p t i o n of societies is

frequently done in terms of kinship systems and terminologies, w h i l e features like m a r r i a g e rules o r patterns are m o s t o f t e n best as k i n - b a s e d , and exemplified o n a genealogical

structure.

described Similarly,

m a n y o f the p o p u l a t i o n statistics o f interest to anthropologists

(and

to social demographers generally) are m o s t easily classified in genealogical terms, or related to genealogically b a s e d concepts. In p o p u l a t i o n genetics we find a similar situation, b u t w i t h different emphasis.

Genealogical concerns result from the biological

fact

of transmission of genes through pairs of individuals to offspring,

and

from the fact that different k i n - b a s e d patterns of transmission m a y result in quite different p r o p e r t i e s in the genetic summary The b a s i c conceptual

statistics.

framework b e i n g involved is first to generate

an "algebra", that is, a discrete relational system and its

structural

2 properties, and second, to study certain statistics of these relational systems.

Generally, the discrete relational systems themselves and the

collection of objects on which they rely or which they generate have direct sociological (ethnographic) interpretation.

These are, for

example, as kinship terminologies (e.g., systems of kin terms), as marriage rules, as concrete marriage networks, as patterns of marriage exchange, as genealogies, as possible histories (e.g., sets of possible genealogies), etc..

The statistical measures of interest on these

structures may be either point valued estimates (e.g., expected values) or distributions (and the distribution may itself be an "expected value" of a set of distributions).

Examples are:

expected frequency of kin

type occurring in a population of finite size, given the terminological system; expected distribution of these kin types; expected average family size needed to maintain a particular marriage rule; average density of kin network given a particular mating and kinship system; average extinction time of a local population; relative endogamy; proportion of females who must marry to maintain the average density of the kin network; expected homozygosity at a typical gene locus; average inbreeding of a population; and distribution of coefficients of relationship of individuals in a population. Therefore, there are two primary areas of concern in each subfield of genealogical mathematics:

structural description, and imputation

of statistics associated with each possible structure.

The role of

genealogical mathematics is thus philosophically similar to that of quantum mechanics in theoretical physics. The present purpose is not at all a comprehensive study in the history of genetics or of anthropology, but certain key papers and ideas should be mentioned, since the topic is not entirely new.

Past efforts

can be roughly grouped into three areas of interest, with some overlap between the histories of genetics and anthropology.

These areas are

formal logic, kinship algebras and group theory, and the probability formulation of genetic identity by descent. While the anthropologists of the kinship persuasion typically

3

reference Greenberg (1949) as the earliest application of "formal logic" (in the sense of Carnap (1958)) in fact in biology applications occurred more than a decade before.

Woodger (1937) laid out in logical form a

system he termed "logistic", and used this to create a general definition of relationships by descent.

His objective, however, was biologi-

cal, not social, as was quickly recognized by Cotterman (1940). Cotterman exploited the "logistic" technique to create a system of genealogical notation useful in calculations of a variety of genetic statistics. Although (as will be discussed further below) Cotterman's work is usually referenced only for its probability content, its significance is therefore much broader.

Along with several papers by Etherington

(1939, 1941, and others) Cotterman placed on solid footing the study of algebraic foundations of genetics as a topic apart from genetics itself.

Such a separation of topics is often unnoticed by geneticists

(or mathematicians.') but has found flower most recently in Jacquard (1970), Lyubich (1971) and Bertrand (1966).

The reader curious about

this topic should see these articles. The limitations of the "logistic" technique and therefore the reason for subsequent domination of the field by followers of Etherington can be clearly seen in Woodger (1952).

This work significantly improved on

earlier notational style, but was only able to derive the simplest results on genetic statistics.

The historic lesson, therefore, is that

"logical" forms, as such, can only create or define the needed structures, but that other techniques must be used for statistical purposes.

In the

case of Etherington, the needed technique was essentially to find eigenvalues ("roots of the train equation") corresponding to particular algebraic forms.

For Cotterman, the technique was to use the algebra to

define possible cases, and to devise a "bayesian" probability method for inferences about these cases. It is common in genetics to begin the accepted history of "inbreeding" with reference to Sewall Wright (1921).

This paper is in fact

important as the earliest correct study of the effect on homozygosity

k of a population rigorously following particular kin-based mating patterns. However, anthropologists had long been aware of regular structures which appeared in natural human populations.

The earliest self-consciously

mathematical treatment of these appears to have been Macfarlane (1882). This article used a simple relational scheme (concatonation of symbols "c" for "child" and "p" for "parent") to enumerate a case of relationship up to degree 5, where the degree of a relationship was simply the number of c or p symbols in the chain.

While not used explicitly, a very similar

scheme was used by Haldane and Jayakar (1962) to enumerate close degree kin types and study their inbreeding effects for genetic purposes. But the work of Macfarlane was on the one hand ignored anthropologically even in the study of "kinship algebra" in the last twenty years, and on the other hand fell a degree short of the necessary steps to connect marriage-based concepts to terminological kinship structures. The next mathematical step appears to have been by A. Weil (1948).

In

an appendix to Levi Strauss' Elementary Structures of Kinship, Weil constructed an elementary mathematical structure, called a permutation, which represented the transmission of clan labels through marriage rules. This idea was of sufficient interest to provoke a literature on the topic.

Kemeny et al. (1965) developed it as an example in an elementary

mathematics text; Harrison White (1963) enumerated cases possible under different assumptions on commutativity of "matrix operators" representing descent and marriage.

P. Courrege (1965) presented a paper more

carefully constructed from algebraic foundations, rather than from "axioms" in natural language, as in White's test.

Liu (1969) published

a group theoretical treatment of systems of the type considered by White, Courrege and Weil, but did not appear to believe there was a direct connection between the approaches. Thus the group theoretical treatment of kinship has been neither uniform nor entirely well received, but has undoubtedly been productive. For example, F. Lorrain (1974b) created a method for studying generalized networks of the type discussed in Chapter 1 of White (1963).

T. R.

Harris' paper for the Genealogical Mathematics Conference was an explicit

5 attempt to re-do White's "axioms" more rigorously, and to show that groups are their natural result. The reader should note that this discussion has deliberately omitted a number of popularly referenced papers.

These in general are more con-

cerned with problems of cognition or with linguistic properties of the "genealogical" structures or their statistical implications.

While I

will not here argue the point in depth, there is a natural division between what may be gained from treating cultural systems as semantic systems (where the first objective is description of particular cultures), and from first studying more abstract mathematical systems and then interpreting their properties in terms of ethnographic observation.

Essential-

ly, it is the difference in physics between the study of physics proper, that is, heat, size, mass, etc. of particular objects and systems, and the study of foundations of physics, which is the study of operators on Hilbert space, etc., which are only related to physics proper when interpreted in a physical context.

While genealogical mathematics cannot yet

be called "foundations of anthropology", it has most certainly moved well beyond direct description of culture. The third historically developed area of concern derives directly from population genetics.

In population genetics, "identity by descent" refers

to the condition where two genes, in the same individual or in different individuals, are identical by having been transmitted by descent from a single ancestral gene.

(Two genes may be identical such as by independent

mutation, without being identical by descent.) Development of this concept into a useful tool for population genetics is classically and correctly attributed to G. Malecot (1948) , but in fact there were earlier uses of the idea.

Most explicit was made by Cotterman

(1940) in an unpublished doctoral dissertation.

However, Haldane and

Moshinsky (1939) used a formula quite similar to the inbreeding formula given by Malecot.

While both these articles are occasionally referenced,

recognition of the probability conception certainly existed earlier, since Wahlund (1928) explicitly uses it in studying linkage dissociation rates in migration between population isolates.

6 For the present purpose, study of identity by descent may be considered, along with the work of Etherington, as an early self-conscious use of genealogical mathematical techniques to find a statistic of importance, and to do so in general rather than context dependent manner. II.

SUMMARY OF CONFERENCE PAPERS

As an alphabetic and historic privilege, we begin the summary of Conference papers with that of John Atkins.

Atkins introduces a muta-

tional system he calls GRAFIK, for "general relational algebra for investigating kinship". His concern is with the "grammar" of this algebra, e.g., the mode of composition of atomistic algebraic phrases into statements describing properties of kinship systems.

His basic universe is interpreted

as all persons who may appear in an ethnographic context (e.g., living or dead not significant —

a dead uncle is still "uncle" in this usage).

Atkins creates a sex partition of the basic universe, and defines three basic single symbol relations: M (mate of). and 1:

P (parent of), 1 (child of) and

He then notes these basic mathematical properties of P

asymmetric (antisymmetric and irreflexive); nontransitive,

many-many; and of M:

irreflexive, symmetric, intransitive and many-

many. The most important results involve studies of expressions of the form /R/ where R is a chain of basic relations with carefully specified properties and /R/ is the set of persons (a,b), a / b, a,beU, for whom aRb is true.

Strings of basic symbols are then composed to form differ-

ent possible kin expressions, and an extensive example is developed for English kin terms.

The typical form is /

for all i,j,k,

Let any member of the defined set Y^ be called a non-regres-

sive ordered (n+1)tuple. If iH-1 = 3, triplets such as (a,a,a) or (a,b,b) or (a,b,c) would qualify as non-regressive, but the triplet (a,b,a) would not qualify. The concept of non-regressive (rH-l)tuples is crucial to the following definition. 5.4

Df.

/ R ^ . ,.Rn/ = f (x, 2 )|a(y 0 ,y 1 ) ...,y n ) 6 Y n :

n (x,z) = u(y„,y ) & V \ > 0 n q=l

(y y ) g R J}. "q-l'-'q q

Let this be called an n-factor

eversive, or elementwise non-regressive, relational composition.

('Unre-

stricted geneaproduct'.) This eversive mode of relational composition—the

geneaproduct—pro-

hibits genealogical tracings that involve "doubling back," as when siblings are characterized as co-grandchildren (implying a tracing that goes twice through a common parent), or when someone is described as a child of a parent of himself.

E.g., the ordinary composition IP holds

for self-pairs as well as for siblings (i.e., I c qp), but the eversivelycomposed relation /IP/ holds for siblings only—diverse pairs whose members share at least one parent. Use of the geneaproduct--that is, eversive compositions marked by the /..../ enclosure—is vital to efficiency in the construction of Egocentric geneaclasses. Note that: 5.41

/RS...T/ E RS...T, with equality if n=l.

5.42

/RST/ E /R/ST// E R/ST/ E RST.

5.43

/RST/ E //RS/T/ E /RS/T