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

摘要

We establish a link between Wakeley et al's (2012) cyclical pedigree modelfrom population genetics and a randomized directed configuration model (DCM)considered by Cooper and Frieze (2004). We then exploit this link incombination with asymptotic results for the in-degree distribution of thecorresponding DCM to compute the asymptotic size of the largest stronglyconnected 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 thuscontributes to a reduced `pedigree effective population size'. In addition, thesecond 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 theshortest ancestral line reaches $S^N$ after $O(\log \log N)$ generations, whilealmost all other ancestral lines take at most $O(\log N)$ generations.