Research areas in mathematical and computational anthropology

Mathematical anthropology is less a single field than a habit of work: state a social rule precisely, find the structure the rule implies, and then check that structure against what people actually do. The habit has produced several distinct lines of research, some over a century old and some that only became practical once computers could handle whole genealogies or thousands of simulated agents.

This page sketches how the field grew, then takes the main lines one at a time. Each section ends with two to four references that are good places to start, and points to the explainers and topic notes on this site that cover the same ground. If you want a sequenced route through the basics first, use the lecture notes.

How the field grew

Morgan's book collected kinship vocabularies from many languages and asked what their differences meant; Kroeber, nearly four decades later, argued that the vocabularies could be compared feature by feature. The decisive mathematical step came out of a wartime meeting in New York between Claude Lévi-Strauss and the mathematician André Weil. Weil's appendix to Les Structures élémentaires de la parenté (Paris, 1949), which dealt with the system W. Lloyd Warner had described under the name Murngin (the Yolngu peoples of northeast Arnhem Land), showed that marriage rules of this kind can be written as permutations of marriage classes, so that questions about who may marry whom become questions about groups. The book reached English readers in 1969 as The Elementary Structures of Kinship (Boston: Beacon Press).

The 1960s turned this into a program. White's book worked out which structures satisfy a set of axioms for prescriptive marriage, Philippe Courrège (1965) gave the same models an axiomatic form, and John Paul Boyd's "The algebra of group kinship" (1969) carried the program into the Journal of Mathematical Psychology. In parallel, componential analysis and Lounsbury's formal accounts treated terminologies as semantic systems with rules. Lorrain and White (1971) then asked what a role is in any network, kin or not, and François Lorrain's Réseaux sociaux et classifications sociales (1975) pursued the algebra of social structure in book length.

From the 1980s the lines diverged and multiplied: Read's kin-term algebras, Hage and Harary's graph models of Oceanic exchange and communication, White and Jorion's genealogical networks, and, alongside them, the separate growth of cultural evolution theory and agent-based simulation. Much of the newer formal work on Dravidian, Crow-Omaha and Aboriginal Australian systems appeared in specialist venues such as the online journal Mathematical Anthropology and Cultural Theory.

Kinship algebra and terminology analysis

This line asks two related questions. The first concerns marriage-class systems: sections, subsections and exchanging clans whose rules of marriage and descent fix the class of every child. Written as permutations, these rules generate a group, and the properties of the group (its size, whether the order of steps matters) say how the system behaves over generations. The explainer on Weil's algebra is the gentle entry point, and the Kariera and Aranda explainer shows the systems themselves. Our notes take the algebra further: prescriptive systems as permutations, Abelian kinship systems and non-Abelian kinship systems.

The second question concerns terminologies. Why does a language put certain kin types under one word, and can the whole set of terms be derived from a few rules? Answers range from componential analysis and equivalence rules to Read's view of a terminology as an algebraic structure generated by products of terms, in which the American consanguineal terms form what mathematicians call an inverse semigroup. Dravidian and Crow-Omaha systems are the classic test cases; see the notes on Dravidian terminology, Crow and Omaha generative logic, formal analysis of kin terms and the physics-flavored relatives, molecules and particles.

Key references:

• White, Harrison C. (1963). An Anatomy of Kinship: Mathematical Models for Structures of Cumulated Roles. Englewood Cliffs, NJ: Prentice-Hall.
• Lounsbury, Floyd G. (1964). "A formal account of the Crow- and Omaha-type kinship terminologies." In Ward H. Goodenough (ed.), Explorations in Cultural Anthropology: Essays in Honor of George Peter Murdock, pp. 351-393. New York: McGraw-Hill.
• Read, Dwight W. (1984). "An algebraic account of the American kinship terminology." Current Anthropology 25(4): 417-449. doi:10.1086/203160
• Kronenfeld, David B. (2009). Fanti Kinship and the Analysis of Kinship Terminologies. Urbana: University of Illinois Press.

Genealogical network analysis

Instead of starting from rules, this line starts from recorded genealogies and treats them as networks. In a P-graph, introduced by White and Jorion, couples (and unmarried individuals) are the nodes and each person is a line running from the parents' couple to his or her own. The payoff is that every marriage between people who were already related closes a cycle, so patterns such as repeated intermarriage between the same families, often called relinking, can be counted rather than guessed. Later work enumerates these matrimonial rings systematically and measures structural endogamy, the part of a population tied together by such cycles. Our note on information flows in kinship networks covers the underlying idea, and the note on closure in Australian Aboriginal kinship shows what happens when tidy models meet large recorded genealogies.

Key references:

• White, Douglas R., and Paul Jorion (1992). "Representing and computing kinship: a new approach." Current Anthropology 33(4): 454-463. doi:10.1086/204097
• White, Douglas R. (1997). "Structural endogamy and the network 'graphe de parenté'." Mathématiques, informatique et sciences humaines 137: 101-125.
• Hamberger, Klaus, Michael Houseman, Isabelle Daillant, Douglas R. White and Laurent Barry (2004). "Matrimonial ring structures." Mathématiques et sciences humaines 168. doi:10.4000/msh.2965
• Hamberger, Klaus, Michael Houseman and Douglas R. White (2011). "Kinship network analysis." In John Scott and Peter J. Carrington (eds), The SAGE Handbook of Social Network Analysis. London: SAGE.

Social network analysis

Anthropologists were among the first to use networks as an analytic idea. John Barnes used it in 1954 to describe the web of kinship, friendship and acquaintance in a Norwegian island parish, and the Manchester school, gathered in J. Clyde Mitchell's 1969 volume, brought the personal network into the study of Central African towns. The formal side then grew mostly in sociology and mathematics: density and centrality, structural equivalence, blockmodels, and the small-world model of Watts and Strogatz, in which a few random shortcuts in a clustered network make paths short. The density and centrality explainer, the network density calculator, the small-world explainer and the piece on Dunbar's number cover the basics.

Key references:

• Barnes, J. A. (1954). "Class and committees in a Norwegian island parish." Human Relations 7(1): 39-58. doi:10.1177/001872675400700102
• Mitchell, J. Clyde, ed. (1969). Social Networks in Urban Situations: Analyses of Personal Relationships in Central African Towns. Manchester: Manchester University Press.
• Lorrain, François, and Harrison C. White (1971). "Structural equivalence of individuals in social networks." Journal of Mathematical Sociology 1(1): 49-80. doi:10.1080/0022250X.1971.9989788
• Wasserman, Stanley, and Katherine Faust (1994). Social Network Analysis: Methods and Applications. Cambridge: Cambridge University Press.

Cultural evolution and phylogenetics

Cultural evolution treats beliefs, skills and customs as variants that spread through a population by learning, and asks how the way people learn (from parents, peers or prestigious models; by copying the majority or the successful) shapes what spreads. Two book-length mathematical treatments, by Cavalli-Sforza and Feldman and by Boyd and Richerson, set the terms of the field. A second strand borrows phylogenetic methods from biology: if societies descend from common ancestors, as their languages suggest, then a family tree of languages can serve as a scaffold for testing how traits such as descent rules or residence changed. This also answers Galton's problem, the objection that related societies are not independent cases. Studies in this vein have linked the spread of cattle to the loss of matrilineal descent in Africa and argued that early Austronesian societies were matrilocal. Start with transmission biases and Galton's problem.

Key references:

• Cavalli-Sforza, L. L., and Marcus W. Feldman (1981). Cultural Transmission and Evolution: A Quantitative Approach. Princeton: Princeton University Press.
• Boyd, Robert, and Peter J. Richerson (1985). Culture and the Evolutionary Process. Chicago: University of Chicago Press.
• Mace, Ruth, and Mark Pagel (1994). "The comparative method in anthropology." Current Anthropology 35(5): 549-564. doi:10.1086/204317
• Jordan, Fiona M., Russell D. Gray, Simon J. Greenhill and Ruth Mace (2009). "Matrilocal residence is ancestral in Austronesian societies." Proceedings of the Royal Society B 276: 1957-1964. doi:10.1098/rspb.2009.0088

Agent-based models

An agent-based model builds a population of individual decision makers in software, gives each one simple rules, and lets their interactions produce aggregate patterns. Joshua Epstein and Robert Axtell's Sugarscape showed in 1996 how group formation, cultural transmission, combat and trade could emerge from local rules on a grid of resources. The best-known anthropological case is the model of Long House Valley in northeastern Arizona, home to the Kayenta Anasazi (the paper's term; Ancestral Puebloan is more common now), which reproduced the main features of the valley's population history, its growth, shifting settlements and decline, from rules for farming households and a reconstruction of the valley's past environment. Epstein later framed the approach as generative social science: if you can grow a pattern from plausible rules, you have a candidate explanation for it. See the explainer on agent-based models and the note on a herder and farmer model.

Key references:

• Epstein, Joshua M., and Robert Axtell (1996). Growing Artificial Societies: Social Science from the Bottom Up. Washington, DC: Brookings Institution Press; Cambridge, MA: MIT Press.
• Axtell, Robert L., Joshua M. Epstein, Jeffrey S. Dean, George J. Gumerman, Alan C. Swedlund and others (2002). "Population growth and collapse in a multiagent model of the Kayenta Anasazi in Long House Valley." PNAS 99 (suppl. 3): 7275-7279. Full text at PubMed Central
• Epstein, Joshua M. (2006). Generative Social Science: Studies in Agent-Based Computational Modeling. Princeton: Princeton University Press.

Cognitive and consensus approaches

Cognitive anthropology began with the same kinship data. Componential analysis tried to find the features a speaker must know to use each term correctly, and Wallace and Atkins asked whether those features are psychologically real or only an analyst's convenience. A later and very practical result is cultural consensus theory. Given a set of people's answers to the same questions, it estimates both the culturally shared answer key and how closely each person's answers track it, without knowing the key in advance. That gives a direct check on whether a domain of knowledge is shared at all. Read the cultural consensus explainer and the statistics starter.

Key references:

• Goodenough, Ward H. (1956). "Componential analysis and the study of meaning." Language 32(1): 195-216. doi:10.2307/410665
• Wallace, Anthony F. C., and John Atkins (1960). "The meaning of kinship terms." American Anthropologist 62(1): 58-80. doi:10.1525/aa.1960.62.1.02a00040
• Romney, A. Kimball, Susan C. Weller and William H. Batchelder (1986). "Culture as consensus: a theory of culture and informant accuracy." American Anthropologist 88(2): 313-338. doi:10.1525/aa.1986.88.2.02a00020
• D'Andrade, Roy (1995). The Development of Cognitive Anthropology. Cambridge: Cambridge University Press.

Spatial cognition

How people represent space is a natural meeting point for formal models and fieldwork. Thomas Gladwin's study of navigators on Puluwat Atoll in the Caroline Islands described how they sail between islands without instruments, using star courses, ocean swells and reference islands held in memory. Edwin Hutchins compared ship navigation with Micronesian navigation and argued that the computation is spread across people, tools and procedures rather than sitting in one head. Stephen Levinson and colleagues showed that languages differ in the frames of reference they use (for example "to my left" against "to the north"), and that speakers' nonverbal problem solving tends to follow the frames of their language. Graph theory enters too: Hage and Harary's Island Networks models communication between islands as networks and asks which positions become centers.

Key references:

• Gladwin, Thomas (1970). East Is a Big Bird: Navigation and Logic on Puluwat Atoll. Cambridge, MA: Harvard University Press.
• Hutchins, Edwin (1995). Cognition in the Wild. Cambridge, MA: MIT Press.
• Levinson, Stephen C. (2003). Space in Language and Cognition: Explorations in Cognitive Diversity. Cambridge: Cambridge University Press.
• Hage, Per, and Frank Harary (1996). Island Networks: Communication, Kinship, and Classification Structures in Oceania. Cambridge: Cambridge University Press.

Where the lines meet

The divisions above are for finding your way, and much of the interesting work crosses them. Genealogical network analysis feeds straight into the debate about marriage-class models, as the closure note shows. Phylogenetic methods take up comparative questions first asked about kinship terminologies. And an agent-based model is a natural way to ask whether rules written down by the algebraic line can work in a population of realistic size, with uneven numbers of sons and daughters. A useful first exercise: pick one rule from the kinship algebra section, such as the four-section system in Unit 5 of the lecture notes, and work out by hand what happens to it in a hypothetical community of 30 people where one section has no marriageable women in a given generation.