The largest strongly connected component in the cyclical pedigree model of Wakeley et al.

摘要

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(logN) generations. (C) 2014 Elsevier Inc. All rights reserved.