Lecture notes, a self-study route through mathematical anthropology

These notes turn the explainers, topic notes and models on this site into a sequence you can work through alone. Each unit says what to learn, where to read about it here, and gives one exercise with its answer underneath. Cover the answer with a sheet of paper before you look. Nothing beyond school algebra is assumed; where a unit needs a new idea, such as a permutation or a network density, the reading defines it.

A pencil and a calculator are enough for every exercise. The models run in your browser and are best used after you have done the sum by hand, as a check rather than a shortcut.

Unit 1. Writing kinship down

What to learn: the eight kin-type letters (F, M, B, Z, S, D, H, W), how a string such as MBD is read from Ego outward ("mother's brother's daughter"), the markers e and y for elder and younger, and the difference between a kin type (a genealogical position such as MB) and a kin term (a word in some language, such as English uncle). Learn the diagram symbols at the same time, because you will be drawing from Unit 2 onward.

Read: the notation guide, then how to draw a kinship diagram.

Exercise. (a) Write the strings for "my father's sister's son" and "my mother's mother's brother". (b) Say ZDH in words. (c) Which of these kin types does American English aunt cover: FZ, MZ, FBW, MBW, FFZ?

Answer. (a) FZS and MMB. (b) Sister's daughter's husband. (c) FZ, MZ, FBW and MBW. The first two are aunts by blood, the second two by marriage. FFZ, one generation further up, takes a separate term, great-aunt.

Unit 2. How terminologies sort relatives

What to learn: the six classic terminology types (Hawaiian, Eskimo or Inuit, Sudanese, Iroquois, Crow, Omaha), what each one merges and what it keeps apart, and the cross and parallel distinction on which several of them turn. A parallel cousin is linked to Ego through two siblings of the same sex; a cross cousin through a brother and a sister.

Read: the six kinship systems explained with the kinship terms chart open beside it, then cross and parallel cousins.

Exercise. Sort the four first-cousin daughters FBD, FZD, MBD and MZD into parallel and cross. In an Iroquois-type terminology, which of them does Ego call by the word also used for "sister"?

Answer. Look at the pair of siblings in the middle of each string. FBD runs through F and his brother, MZD through M and her sister: both same-sex pairs, so both are parallel cousins. FZD (F and his sister) and MBD (M and her brother) run through opposite-sex pairs, so they are cross cousins. An Iroquois-type terminology calls the parallel cousins FBD and MZD "sister" and gives the cross cousins a separate term.

Unit 3. Relatedness and ancestors

What to learn: Sewall Wright's path method for the coefficient of relationship r (add (1/2)^n over every path through a shared ancestor, where n is the number of parent-child links in the path), the standard values for siblings, half siblings and cousins, how r differs from the inbreeding coefficient, and why the 2^n ancestor positions of a family tree soon outrun any real population.

Read: the coefficient of relationship, worked examples with the cousin calculator, then pedigree collapse explained with the pedigree collapse calculator.

Exercise. (a) Find r between Ego and the child of Ego's first cousin, assuming the two cousins are linked through a pair of full siblings and in no other way. (b) How many ancestor positions does a family tree have 10 generations back?

Answer. (a) Ego and the cousin share two grandparents. From Ego up to one of them is 2 links; from that grandparent down to the cousin's child is 3 links. One path therefore has 5 links, and there is a second, identical path through the other grandparent:

r = 2 × (1/2)^5 = 2/32 = 1/16

(b) 2^10 = 1,024 positions. Real trees fill them with fewer distinct people as soon as two of the ancestors are themselves related, which is exactly what pedigree collapse measures.

Unit 4. Terminologies as structures

What to learn: formal analysis treats a set of kin terms as a system with its own logic rather than as a list of labels. Three approaches are worth knowing by name: componential analysis (each term as a bundle of features such as generation, sex and lineality), equivalence rules that reduce a term's distant meanings to its closest one, and kin-term algebra in Dwight Read's sense, where the whole terminology is generated by products of a few primitive terms.

Read the topic notes on the formal analysis of kin terms, the generative logic of Crow and Omaha terminologies and the structure of Dravidian terminologies. For a more playful take on the same Dravidian material, using analogies from physics, add relatives, molecules and particles.

Exercise. Work only with English terms, not with genealogy. Compute these products, read as "the X of my Y": (a) brother of father; (b) son of uncle; (c) son of father. What is odd about (c)?

Answer. (a) uncle. (b) cousin. (c) Either brother or, if Ego is male, Ego himself. A product of two terms can have more than one value, and one value can be the speaker. Any algebra of kin terms has to say how it handles cases like this before it can generate anything else.

Unit 5. Sections and marriage classes

What to learn: four-section systems of the Kariera type and eight-subsection systems of the Aranda type as documented by Radcliffe-Brown and later ethnographers; how a rule of marriage plus a rule of descent fixes every child's section; and why these systems are still used in daily life in many Aboriginal communities. Then the harder question of how far any closed model fits real practice.

Read: section systems, Kariera and Aranda, play with the section system simulator, then read the note on closure in Australian Aboriginal kinship.

Exercise. Take a four-section system written with abstract labels A, B, C and D. Sections A and B intermarry; so do C and D. A man's children belong to C if he is A, to A if he is C, to D if he is B, and to B if he is D. Ego is a man of section A. Find the sections of (a) his mother, (b) his MBD, (c) his FZD, and compare (b) and (c) with the section he marries into.

Answer. (a) Ego is A, and only a C man has A children, so his father is C. C men marry D women, so his mother is D. (b) A mother's brother shares her section, D, and a D man's children are B. So MBD is B. (c) His father is C, so FZ is also C. A C woman marries a D man, whose children are B. So FZD is B as well. Both cross cousins sit in B, the very section an A man marries into. The two rules produce bilateral cross-cousin marriage without ever stating it.

Unit 6. Groups and permutations

What to learn: the idea in André Weil's 1949 appendix to Lévi-Strauss, which was to treat the marriage rule and the descent rule as permutations of the classes, compose them, and ask which group they generate; what a group is, in plain words; and the later split between Abelian systems (where the order of steps never matters) and non-Abelian ones (where it can).

Read: André Weil and the first kinship algebra, then the notes on prescriptive systems as permutations, Abelian kinship systems and non-Abelian kinship systems.

Exercise. Use the system from Unit 5. Let w send a man's section to his wife's section, and c send a man's section to his children's section. (a) Write w and c out in full. (b) What is c∘c? (c) Do w and c commute, that is, is w∘c the same as c∘w? Read w∘c as "apply c first, then w".

Answer. (a) w: A to B, B to A, C to D, D to C. c: A to C, C to A, B to D, D to B. (b) c∘c sends every section back to itself (A to C to A, and so on): a man and his son's son share a section, the pattern of alternating generations. (c) Yes. w∘c sends A to C and then to D, B to D and then to C, C to A and then to B, D to B and then to A. c∘w sends A to B and then to D, B to A and then to C, C to D and then to B, D to C and then to A. The results match. With the identity, the four maps form the Klein four-group, the textbook Abelian case. In words, w∘c gives a man's son's wife's section and c∘w gives his wife's brother's children's section; that they coincide says the son takes his wife from the section of his own mother's brother's daughters.

Unit 7. Kin networks and social networks

What to learn: two ways of turning a genealogy into a graph (people as nodes, or couples as nodes in a P-graph, where each person is a line from the parents' couple to his or her own), why marriages between relatives appear as cycles, and then the standard measures of social network analysis: density, degree, betweenness, clustering and path length, and the evidence about natural group sizes.

Read: the note on information flows in kinship networks, teaching kinship with a spreadsheet, network density and degree centrality with the network density calculator, then small-world networks and what the evidence says about Dunbar's number.

Exercise. (a) Ego marries his FBD. In a P-graph, how many couple nodes and how many person lines make up the cycle this marriage closes? (b) A hypothetical hamlet of 6 households has 7 undirected exchange ties. What is its density?

Answer. (a) Four of each. The nodes are the grandparents' couple, Ego's parents' couple, the FB's couple and Ego's new couple. The lines are F (grandparents to Ego's parents), FB (grandparents to his own couple), Ego (his parents to the new couple) and the bride (her parents, the FB's couple, to the new couple). (b) Density is the share of possible ties that exist:

density = 2L / (n(n - 1)) = (2 × 7) / (6 × 5) = 14/30 ≈ 0.47

Unit 8. Evolving culture, and testing claims

What to learn: transmission biases (conformist, prestige and content biases) and the simple recursions that model them; agent-based models and the idea of "growing" a pattern from individual rules; Galton's problem, the reminder that neighboring or related societies are not independent data points; and cultural consensus analysis, which estimates shared knowledge and each person's agreement with it.

Read: transmission biases in cultural evolution, agent-based models in anthropology and the note on a herder and farmer agent-based model; then Galton's problem explained, cultural consensus analysis and the statistics starter.

Exercise. In the standard conformist model, a variant held by a share p of the population changes each generation by

Δp = D × p × (1 - p) × (2p - 1)

where D, between 0 and 1, is the strength of the bias. Start at p = 0.6 with D = 0.5 and find p after one and two generations. What happens if you start at exactly p = 0.5?

Answer. Generation 1: Δp = 0.5 × 0.6 × 0.4 × 0.2 = 0.024, so p = 0.624. Generation 2: Δp = 0.5 × 0.624 × 0.376 × 0.248 ≈ 0.029, so p ≈ 0.653. The majority variant gains a little more each round. At p = 0.5 the factor (2p - 1) is zero and nothing changes, but the balance is fragile: nudge p to 0.51 and the bias carries it upward, nudge it to 0.49 and it slides the other way. Set the recursion up in two spreadsheet columns and copy it down 30 rows to see the whole curve.

After unit 8

The research areas page lists the key books and papers behind each unit, grouped by line of work, and the contents page has every explainer and note on the site. A good test of what stuck: draw a three-generation diagram of a hypothetical family, write the kin type of every person from one Ego, compute r for three of them, and say which of them an Iroquois-type terminology would call "sister".