Monday, April 28, 2008

Common ancestors

In small towns, it is usual that almost everybody is a distant cousin of almost everybody else. What are the kinship statistics of a closed, homogeneous, stable population? The answer depends on the population size, the fertility distribution and mating customs. We will set up some terminology and define a very simplistic population model. The treatment is formally loose as we talk about sets and sequences of sets where we should be dealing with random variables and stochastic processes.

We denote the mother and father of x by m(x) and f(x), respectively. y is an ancestor of x if y = x or y is an ancestor of m(x) or f(x). Given two individuals x and y, their kinship proximity k(x,y) is defined as:

• if y is an ancestor of x or x is an ancestor of y, k(x,y) is the number of generations between x and y,
• else, k(x,y) := min{k(a,x) + k(a,y) : a is a common ancestor of x and y}.

A population X is stratified if it can be decomposed into a disjoint family of subsets or generations {Xi: i integer} such that for every x in Xi, {f(x),m(x)} is included in Xi−1. We define the kinship distribution at generation i as

Ki(n) := P(k(x,y) = n), x, y in Xi, n = 0,1,2,...

where P stands for "probability". Note that the stratification of X implies that k(x,y) is always even when x and y belong to the same generation (because two individuals of the same generation are always at the same distance to their common ancestor), so Ki(n) = 0 if n is odd. X is a stable population if Ki(n) = Kj(n) for every i, j; in this case we denote the kinship distribution simply by K(n). Let us define the fertility distribution at generation i as the following function:

ri(n) := P(x has exactly n children), x is a woman in Xi, n = 0,1,2,...

X is said to be homogeneous if ri(n) = rj(n) for every i, j, in which case we simply write the unique fertility distribution as r(n).

We will study a very simplified model of population dynamics. So, we require that our population X have the following properties:

• X is stratified, stable and homogeneous.
• The median value of r(n) is 2, i.e. the population size does not change over time.
• X is a strict monogamy, i.e. any two individuals have the same mother if and only if they have the same father.
• Siblings do not mate. As for intergenerational incest, it is automatically ruled out by the stratified nature of X.

In a later entry we will explore X kinship distribution by means of computer simulation.