Distribution of repetitions of ancestors in genealogical trees

TL;DR

This paper derives the probability distribution of ancestor repetitions in genealogical trees for simple neutral models, revealing a power-law distribution for ancestor weights in large populations.

Contribution

It provides an exact calculation of the stationary distribution of ancestor repetitions in genealogical trees using a renormalization group approach.

Findings

Distribution P_∞(w) is a power law for small w.

The shape of P_∞(w) is affected by model extensions.

The model offers insights into ancestor repetition patterns over generations.

Abstract

We calculate the probability distribution of repetitions of ancestors in a genealogical tree for simple neutral models of a closed population with sexual reproduction and non-overlapping generations. Each ancestor at generation g in the past has a weight w which is (up to a normalization) the number of times this ancestor appears in the genealogical tree of an individual at present. The distribution P_g(w) of these weights reaches a stationary shape P_\infty(w) for large g, i.e. for a large number of generations back in the past. For small w, P_\infty(w) is a power law with a non-trivial exponent which can be computed exactly using a standard procedure of the renormalization group approach. Some extensions of the model are discussed and the effect of these variants on the shape of P_\infty(w) are analysed.