The largest strongly connected component in Wakeley et al's cyclical pedigree model

Abstract

We establish a link between Wakeley et al's (2012) cyclical pedigree model from population genetics and a randomized directed configuration model (DCM) considered by Cooper and Frieze (2004). We then exploit this link in combination with asymptotic results for the in-degree distribution of the corresponding DCM to compute the asymptotic size of the largest strongly connected component $S^N$ (where $N$ is the population size) of the DCM resp. the pedigree. The size of the giant component can be characterized explicitly (amounting to approximately $80 \%$ of the total populations size) and thus contributes to a reduced `pedigree effective population size'. In addition, the second largest strongly connected component is only of size $O(\log N)$. Moreover, we describe the size and structure of the `domain of attraction' of $S^N$. In particular, we show that with high probability for any individual the shortest ancestral line reaches $S^N$ after $O(\log \log N)$ generations, while almost all other ancestral lines take at most $O(\log N)$ generations.