Kinship: The Classificatory Kinship Page

Author: Douglas R. White, University of California, Irvine
see also:Polygyny (Standard_Cross-Cultural Sample)
Sexual Divsion of Labor


"Americans cannot see complexity even in the simplest system" (they are not alone). Simplicity in complexity is one of the mantras of complexity or complex systems. Gregory Bateson taught us schismogenesis but we did not seem to learn from it. Here we look at what social scientists and anthropologists (who ought to know better) thought to be the "simplest system" in human kinship organization. It turns out not to be so.

Why This Page?

As defined by Lewis Henry Morgan, classificatory kinship puts people into "society-wide kinship classes on the basis of abstract relationship rules. These may have to do with genealogical relations locally (e.g., son to father, daughter to mother, daughter to father) but the classes bear no overall relation to genetic closeness. If a total stranger marries into the society, they may simply be placed in the appropriate class opposite to their spouse" (quoting text that I wrote for Wikipedia).

The Failures of Abstract Algebraic Sociological Models

Classificatory kinship was thought in classical sociocultural anthropology to be rigidly prescriptive: A set of relationships with others is defined at birth and fixed with those others for life, adding others newly born or married in. Anthropologists such as A. R. Radcliffe-Brown, following earlier precedents, tried to define group 'sections' of societies with classificatory systems, e.g., 4-section systems, those with six, seven, eight, twelve, etcetera. Mathematical sociologist Harrison White (1963), following the work of Bourbaki algebraist Andre Weil, published in the classic work on kinship by Levi-Strauss (1949), showed by algebraic axioms and theorems how relational classes are connected to the boundary definitions of actual clans or sections. This has remained the dogma of anthropology textbooks to the present date. Mathematical anthropologist John Boyd (1969) tried to prove algebraically the formal equivalence between the logic of categories and the logic of relations, but had to withdraw his proof in an erratta (Boyd 1972). If you want to use algebra to make a demonstration, beware of assuming the consequent. Anthropologists and algebraists have tended to look for closed algebraic systems in kinship, and assuming them, they find them. Little wonder. This was a major point of the White and Reitz (1982:210) axiomatization of regular equivalence, in which there is no supposition of uniqueness for a best-fit regular equivalence algebraic model. Malinowski (1930) would be happy.

Oversimplifation of complex systems has been the bane of simple models of classificatory kinship, as decried by Malinowski (1930) in his critique of Radcliffe-Brown. Thankfully, one does not find reference to 'section systems' in the Wikipedia Indigenous Australian entry. These entries, unlike many other anthropological texts, are edited by Indigenous Australians.

The Detailed Data: Social and Terminological

Among Indigenous Australian groups there are only two detailed ethnographic network studies of the social or genealogical networks of kinship along with the use of classificatory terminology. One is the sutdy on the Groote Eylandt by Frederick Rose (1960) and the other the two-year field study of the Alyawarra, closely related to the Aranda, by Woodrow W. Denham (1975, 1978, 2002, 2003).

There are many studies of Australian kinship terminology and of classificatory kin terminology generally. Read (2000, 2001) has an extremely detailed and rigorous approach to modeling kinship termonilogies using generative algebra, and here the algebraic models work beautifully, impressively so for Australian kinship. The operators are generative products that constitute cognitive and linguistic rules, such as "my father's brother (in English) is called my uncle," Read is able to define precisely the universal characteristics of classificatory kinship terms.

Cognition and the question of how kinship categories are created, however, are not my object in this discussion. Rather, I will be concerned with issues of fit between categories used for Australian kinship terms and marriage practices, in terms of the data collected by Rose and by Denham. My goal will be to clarify issues of simple prescriptive rule-based systems and those of complexity, involving behavioral systems with alternatives, strategic choices, and social emergence (see, for example, Sawyer, 2005). Here we have much to learn about our tendency to oversimplify when it comes to understanding human behavior, especially the behavior of others (see my discussion here of the Natchez controversy), and how to identify complexity in human interaction.

The Failures of Network Blockmodeling

Analysis of these two bodies of systematic data is extremely challenging. Members of the machine learning team at the MIT Department of Brain and Cognitive Sciences, Kemp, Griffiths and Tenenbaum (2004), devoted a section of their discovery algorithm for latent classes in relational data in the Alyawarra data but failed to realize that their solution is not unique. Peter Bearman (1997), in his empirical analysis of 'Generalized exchange' as a network pattern in Rose's data for the Groote Eylandt makes a similar mistake. He finds evidence of eight groups related in a circular pattern of marriages. He errs in a constructive direction however: he is unable to find how these groups are related to descent, as in the classical algebraic models of Radcliffe-Brown. This is a huge advance because it implies that there are alternative ways in which descent lines can run through the supposed marriage classes, i.e., there is is mobility in the system.

Mobility in the 'system': Reconfiguring our understanding of elementary kinship as a complex system

Are 'Elementary systems' 'complex'?

For Levi-Strauss, an elementary structure is a marriage system in which women of a group are obliged to marry into another that is "explicitly defined" by social institutions. In "complex structures of kinship," such as the Western world, the spouse's group is "indetermined and always open" except for certain kin, including exogamous lineages, nuclear families, families specified by a rule of exogamy or incest prohibitions, or as prohited by law.

What is it that precisely distinguishes marriages in a society like the Groote Eylandt or the Alyawarra from exogamous lineages, in which the spouse's group is "indetermined and always open," that is, 'complex'? Is it that there is a certain set of marriages that is "explicitly defined" so as to prescribe the category into which one must marry? Both the Groote Eylandt or the Alyawarra marry exogamously outside the patrilineage. Some Indigenous Australian groups marry exogamously outside both the patrilineage and the matrilineage. If these were the only marriage rules, they would be defined by Levi-Strauss as "complex" unless they had either a matrimonial moiety system or a classificatory kinship term that prescribes who they marry. An example of the latter might be a kin term glossed genealogically as 'mother's brother's daughter' but really includes a much larger category of relatives.

In the absense of prescribed groups or sections, however, a 'classificatory' term is extensible to a whole range, possibly infinite in extension, of genealogical kin. Many societies, for example, use a counting rule such that if a man traces a genealogical connection to a woman through an even number of female linking relatives, then she is not marriageable (unlike MoMoBrDaDa, MoBrDa or MoDa that have only three or one linking females, excluding the bride). Such a rule is not prescriptive since there are alternatives (and one does not marry a MoDa) but merely proscriptive. So even the classificatory moiety systems that we find for Indigenous Australian groups, including the Alyawarra and the Groote Eylandt, are not necessarily prescriptive in the sense of sections (or subsections), even though they have classificatory kinship terms.

If there were a stronger basis for prescriptive or 'elementary systems' in terms of section systems, that evidence should be found in the data on the Alyawarra or the Groote Eylandt. And, as we shall see: it is not. The possibility of 'elementary systems' remains, but the evidence from Alyawarra and Groote Eylandt is not 'elementary' in the sense of a simple moiety system. The divisions within the moieties (which are very strict) are not signalling something prescriptive but signalling alternatives, as we shall see. Nor is it the case that they are, as Levi-Strauss would have it, 'semi-complex,' where the proscriptions are so embracing as to virtually entail prescriptions. The evidence of Alyawarra, the only really good evidence that we have from Indigenous Australian kinship networks, cognition, and use of terminology, supports the alternate: the kinship system is complex, allowing alternative structural choices that are of consequence within a moiety division. If this is true, the Levi-Strauss misunderstood the fundamental dynamic of the classificatory component of Indigenous Australian system, and the study of kinship requires a new and different theory of complexity.

To avoid misunderstanding, it is not that the ideal type of elementary systems cannot be defined as a strictly prescriptive system, but that (1) moiety systems fulfill this type only in the very loosest manner, and would easily allow choice among lineages or other units, while (2) the strictest example of prescription, the Australian kinship imagined by Radcliffe-Brown and others in terms of section systems, lacks instantiation in any single well documented ethnographic case, and (3) there is no evidence for Australian kinship that the use of classificatory kinship systems, which is a very real phenomena (Read 2001), has any intrinsic connection to prescriptive marriage as opposed to strategic alternatives, variability, flexibility, and complexity.

Redefining Complexity in Kinship Systems

Bearman (1997) finds networks of generalized exchange among groups of Groote Eylandt kin, but not elementary structure in terms of a rule as to whom you marry. He found no rule, that is, given the memberships of father and mother in adjacent groups in the directed cycle of wife taking and wife giving groups, as to which of the eight marriage groups a son or daughter would belong to or where they would find their mates.

The analysis of the Alyawarra data by Denham, Atkins and McDaniel (1979:20) was thought to resolve into six male descent lines that married in a perfect direct cycle. The cycle, however, is one where, because the average ages of marriage for brothers and sisters differ by 28 years (14 for the daughter, 42 for the brother). "The age difference between ego and his wife is fourteen years, between ego and his mother is twenty-eight years, and between ego and his father is forty-two years." The generational time for women averages 14 years while that for men averages 42. Tracing a chain of brothers-in-law through the wife (WiBrWiBrWiBr...) never forms a closed loop, but forms a helix connected by marriages with a second helix in the opposite patrimoiety. Their algebraic model was that these helices jointly constituded a double helix and thus a newly discovered form of elementary system in which marriage was prescriptive.

Based on these findings from Denham's field data and his collaborative analysis with Atkins and McDaniel, Tjon Sie Fat (1981, 1983, 1990) worked out a brilliant algebraic classifcation of new elementary systems composed of double helices with variable differences in their generational spans for males and for females.

Reanalysis by Denham and White (2005), however, showed something far more interesting. While the Alyawarra double helix model assumed that a man's grandson in the paternal line with marry a women in the same prescriptive classification as his DaDaDaDaDaDa, there is no evidence for this in the Alyawarra data. They contain no married woman for whom we know the descendants of her MoMoMoMoMoMo. Further, there were no living Alyawarra who could recall a MoMoMoMoMoMo even though on average she would have lived only two generations back (96 years) from the male perspective. In general, Indigenous Australians are known to avoid the reckoning of kin much more than two generations back. Thus there was no evidentiary basis for the algebraic generational-closure algorithm that Denham, Atkins and McDaniel used to construct the a closed model of the double helix. This leaves open the possibility of many genealogically helices of women that connect to each other through marriage, and that can be classified into two matrimoieties but denies that there is any prescriptive marriage into a given classificatory category on the basis of kin terms. That is, like intermarrying lineages, the linkage among helices is complex.

More revealing, however, what was Denham and White did discover, and not simply as a possibility but an actuality. There were at least three different ways of viewing the algebraic structure of marriages, each of while could morph into another. Instead of closure into six classificatory male lines, the classificatory lines could be realigned in different ways, depending on alternative patterns of permitted marriages in successive generations. This finding, moreover, was consistent with what White and Reitz (1982:210) had found about algebraic systems that lack a closure principle. That is, the various alternative structures do not have a point of intersection -- unique form -- that could be said to underlie them all. 'No one ring doth rule them all,' meaning no unique 'best structure' that comprehends them all. The system, instead, is flexible and composed of alternative possibilities, in short, 'complex' in the term favored by Levi-Strauss.

So it seems we have found a new sort of evolvable complexity in social structure, where the cognitive rules do not extend infinitely into the past, behavior is discretionary, and representations of 'general' models of what is only local structure can be modified on the fly, guided by how people vote in their choices of whom to marry, which alters consistency with one structure over another that is more consistent with the kin classification into which their spouses fit.

A new and testable theory of complexity in kinship

What White (2004) proposed (also in White and Johansen 2004) was a project carried out with the collaboration of French social anthropologists (Hamberger, Houseman, Daillant, White and Barry 2004) and implemented by anthropologists and historians (e.g., Grange 2006): a longitudinal behavioral census within each ethnographic and historical study of kinship systems on which comprehensive network data have been collected. Part of this project, funded by the Agence Nationale de Recherche (ANR 2006-2008) is aimed at testing what White and Houseman (2002) proposed about complexity: namely that marriage preferences will shape the histogram of rank-frequency among types of genealogial kin taken as mates (-- away from the exponential distribution expected at random--) toward a power-law distribution. What they predicted is that Western type social class systems will show power laws on affinal relinking (marriage cycles among families) while societies in which marriage is often with blood relatives will show power laws on consanguineal relinking frequencies. The methods of network analysis introduced by White and Jorion (1994, 1996) and elaborated in White and Johansen (2004) formed the initial basis for the network analysis, but now have been superceded by new software and developments in data coding that are part of the ANR project.

Readings amd Abstracts

Abstract: A field experiment conducted in Central Australia in 1971-72 explored differences between what Aborigines actually did and what they said they did when anthropologists interviewed them. Fieldwork entailed observing behavior and recording it in numerically coded forms; analysis entails extracting patterns computationally that would not appear in traditional ethnographic data. This paper focuses on discrepancies between expected and observed with regard to descent, marriage and kinship. First it examines field methods and the resulting dataset, then it reviews a wide range of analytical methods that have been used to interpret the data. The alternative analytical methods reviewed here serve to test "competing hypotheses" about the nature and operation of Alyawarra descent, marriage and kinship. At the same time, however, the cumulative result of using these diverse methods has been increasingly complex and subtle understandings of previously unknown aspects of Central Australian social organization. The fact that the data continue to repay increasingly sophisticated analyses thirty years after they were recorded attests to the success of the field experiment.


  • The Radcliffe-Brown normative model functions as a kind of "cognitive core" for the Alyawarra system of descent, marriage and kinship. It is not incorrect, but it is seriously incomplete and inadequate.
  • Examination of alternate models of Alyawarra social structure cannot be uniquely resolved into a single model but a nested model with a unique simplest structure embedded in models that are more complex. Each layer of models conforms to actual marriages that are in 98% agreement with the RB section memberships.
  • The elegance of the Atkins' double helix model is marred by the fact that patterns of marriage among the deceased are quickly forgotten and no longer cast their shadow as a constraint on future behaviors. Thus the "kinship system" can evolve dynamically across a class of network models influenced stochastically by age distributions at marriage in accordance with Tjon Sie Fat's algebraic model, and to incorporate non-Alyawarra lineages in ways that are incompatible with both the RB and the Atkins models, thereby violating the Axiom of Algebraic Closure. An open format model that does not require the assumption of Generational Closure is one of the nested models that provide a good fit to the ethnographic data.
  • The problem posed by the widespread extra-normative application of Omaha terms led us to discover recurring patterns in Alyawarra behavior that give the people a great deal of discretionary control over marriage by applying Omaha terms non-reciprocally in violation of the Axiom of Universal Reciprocity.
  • Not only was the field experiment successful, but our findings require some fundamental rethinking of models for Australian kinship. Abstract. Ring cohesion, as a theory relevant to social cohesion, offers itself in the analysis of matrimonial relinking as an outgrowth of a structural approach: "Structural studies are, in the social sciences, the indirect outcome of modern developments in mathematics which have given increasing importance to the qualitative point of view in contradistinction to the quantitative point of view of traditional mathematics. It has become possible, therefore, in fields such as mathematical logic, set theory, group theory, and topology, to develop a rigorous approach to problems which do not admit of a metrical solution. The outstanding achievements in this connection - which offer themselves as springboards not yet utilized by social scientist - is to be found in J. von Neumann and O. Morgenstern, Theory of Games and Economic Behaviour; N. Wiener, Cybernetics; and C. Shannon and W. Weaver, The Mathematical Theory of Communication". [the abstract enfolds a quote from Levi-Strauss, Structural Anthropology, 1963, Chapter XV, Social Structure, section on "Structure and Measure", p. 283]

    KEY-WORDS - Kinship network, Family relinking, Social cohesion, Structural endogamy.

    Abstract. The paper deals with matrimonial rings, a particular kind of cycles in kinship networks which result when spouses are linked to each other by ties of consanguinity or affinity. By taking a network-analytic perspective, the paper endeavours to put this classical issue of structural kinship theory on a general basis, such as to allow conclusions which go beyond isolated discussions of particular ring types (like cross-cousin marriage, sister exchange, and so forth). The paper provides a definition and formal analysis of matrimonial rings, a method of enumerating all isomorphism classes of matrimonial rings within given genealogical bounds, a series of network-analytic tools - such as the census graph - to analyse ring structures in empirical kinship networks, and techniques to effectuate these analyses with the computer program PAJEK. A program package containing the required macros can be downloaded from the web. The working of the method is illustrated at the example of kinship networks from four different parts of the world (South-America, Africa, Australia and Europe).

    KEY WORDS - Matrimonial rings, Kinship, Social network analysis, Graph theory, Enumeration theory, Social anthropology

    Abstract: We examine data on and models of small world properties and parameters of social networks. Our focus, on tie-strength, multilevel networks and searchability in strong-tie social networks, allows us to extend some of the questions and findings of recent research and the fit of small world models to sociological and anthropological data on human communities. We offer a ***navigability of strong ties*** hypothesis about network topologies tested with data from kinship systems, and potentially applicable to corporate cultures and business networks.

    Abstract. A representational language for genealogical networks is developed that provides a better means of visualizing and analyzing some of the basic organizing principles of kinship. In contrast to the conventional genealogical diagram, with circles for females and triangles for males, a marriage link between pairs of opposite sex, and branches from that union to children, the p-graph uses a simpler graph-theoretic construction in which a node represents either a married couple or an unmarried individual, and arcs are directed from the parental couple to the nodes corresponding to their children. When a child is married, there will be two parental nodes, one for the wife's parents and one for the husband's. Thus the arcs (directed links from parents to children) in this graph need only be distinguished as to one of two types: those for parent(s) to sons and to daugthters, respectively. This representation makes for a compact graph in which different kinds of marriage (e.g., types of consanguineal marriages or relinkings to affines) will show up as cycles in the underlying undirected graph. Graph theoretical analysis of p-graphs can then easily order marriages through time and identify marriage patterns as changing configurations of marriage cycles through time, without distortion of the genealogical network itself.

    Abstract. Confusions between substantive and relational concepts of kinship as a social network have led to a number of problems that are clarified by a temporally ordered relational theory of network structure. The ordered-network approach gives rise to a novel means of graphing the social field of kinship relations, while allowing kinship to be locally defined in culturally relative terms. Its utility is exemplified in applications to kinships among US Presidents, Old Testament Canaanites, and native Australians of Groote Eylandt. The formal concepts treated in the mapping of kinship networks are: kinship axioms, parental graph structure, core, circuits of consanguineally and affinally linked kin, sides and divides, homeomorphic mappings, homomorphisms as potentially simplifying mappings of kinship, elementary structure, and order-structure. Representational theorems are proven about homeomorphisms, cores and circuits, and the ambiguity of elementary structures. The last set of theorems leads to clarifying and redefining some of the basic concepts of elementary, semi-complex and complex structures of kinship in terms of properties of generationally ordered networks. The conclusions of the formal argument are 'post-structural' in the narrow sense of demonstrating the need for specifying contingent historical processes in the structural analysis of kinship as a social field. The open-ended approach to change, one that is implied by the study of ordered structures that unfold in a temporal succession, connects to issues of population variability, selection, and evolutionary processes. The kinship structures that are mapped in this approach are not intended as any sort of complete representations of kinship 'systems', but merely as scaffoldings that help to bring into view kinship as a social field, providing a baseline for other mappings (which may be superimposed) of social processes such as communicative fields, exchange processes, transmission of learned behaviors, social rights and inheritance, political and religious succession, and the like.

    Other References

    Bearman, Peter S. 1997. Generalized Exchange. American Journal of Sociology 102(5):1383-1415.

    Boyd, John P. 1969. The algebra of group kinship. Journal of Mathematical Psychology 6:139-167.
    ---. 1972. Erratum Journal of Mathematical Psychology 9:339.

    Denham, W. 1975. Population properties of physical groups among the Alyawarra tribe of Central Australia. Archaeology and Physical Anthropology, Oceania 10 (2): 114-151.
    ---. 1978. Guide to contents, structure and operation. Vol. 1 of The Alyawarra ethnographic data base. HRAFlex Books OI5-001. New Haven, CT: Human Relations Area Files Press.
    ---. 2002. Group compositions in band societies database.
    ---. 2003. Alyawarra ethnographic archive.

    Denham, W., J. Atkins and C. McDaniel. 1979. Aranda and Alyawarra Kinship: A Quantitative Argument for a Double Helix Model. American Ethnologist 6(1):1-24.

    Grange, Cyril. 2006. 'Matrimonial Networks of the French Jewish Upper Class in Paris - 19th century - 1950.' Social and Historical Dynamics: Satellite Workshop of the European Conference on Complex Systems 2006 (ECCS '06), Oxford, University of Oxford.

    Kemp, C., Griffiths, T. L. & Tenenbaum, J. B. 2004. Discovering latent classes in relational data. AI Memo 2004-019 (pdf) - see Part 4 on blocking Alyawarra kin terms

    Levi-Strauss, Claude. 1949. Les Structures elementaires de la parente, The Elementary Structures of Kinship, ed. *Rodney Needham, trans. J. H. Bell, J. R. von Sturmer, and Rodney Needham, 1969.

    Malinowski, Bronislaw. 1930. Must Kinship Studies Be Dehumanised by Mock Algebra? Man 30:256-257.

    Read, Dwight. 2000/2001. Formal Analysis of Kinship Terminologies and its Relationship to what constitutes Kinship. Mathematical Anthropology and Culture Theory 1(1):1-46. Anthropological Theory 1:239-256.

    Rose, Frederick G.G. 1960. Classification of Kin, Age Structure and Marriage among the Groote Eylandt Aborigines : A Study in Method and a Theory of Australian Kinship. Berlin: Akademie Verlag.

    Sawyer, R. Keith. 2005. Social Emergence: Societies as Complex Systems. Cambridge: Cambridge University Press.

    Tjon Sie Fat, Franklin E. 1981. More complex formulae of generalized exchange. Current Anthropology 22 (4): 377-99.
    ---. 1983. Age metrics and twisted cylinders: Predictions from a structural model. American Ethnologist 10:583604.
    ---. 1990. Representing Kinship: Simple Models of Elementary Structures. Standort: FB f.Ethnologie: Gesellschaft 673.

    White Douglas R., and Ulla Johansen. 2004. Network Analysis and Ethnographic Problems: Process Models of a Turkish Nomad Clan. Lanham, Maryland: Lexington Press (Rowman).

    White, Douglas R., and Karl P. Reitz. 1983 Graph and Semigroup Homomorphisms Social Networks 5:193-234.

    White, Harrison C. 1963 An Anatomy of Kinship: Mathematical Models for Structures of Cumulated Roles. Englewood Cliffs NJ: Prentice-Hall.