A genealogical model for the ancestor paradox

Naive reasoning suggests that the ascending family tree of any individual is a binary tree, which would imply that this individual has $2^n$ ancestors $n$ generations back. Such an exponential growth is however not compatible with the size of the world population at that time. This apparent paradox is explained by the presence of inbred unions, which effectively reduces the number of distinct ancestors.

We propose a stochastic model to study this phenomenon, in the form of an oriented acyclic random graph. We present conditions under which the family tree "collapses", i.e. has a diamond shape rather than the pyramid shape of the binary tree. These criteria on the random graph parameters are then translated into conditions on the studied population, in particular on the proportion of unions between cousins in each generation.