title: The largest strongly connected component in the cyclical pedigree model of Wakeley et al.
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(logN). 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(loglogN) generations, while almost all other ancestral lines take at most O(logN) generations.
authors.affiliation: Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany. Electronic address: blath@math.tu-berlin.de.
authors.forename: Jochen
authors.initials: J
authors.lastname: Blath
authors.affiliation: Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany.
authors.forename: Stephan
authors.initials: S
authors.lastname: Kadow
authors.affiliation: Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster, Einsteinstraße 62, 48149 Münster, Germany.
authors.forename: Marcel
authors.initials: M
authors.lastname: Ortgiese
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D005828:Genetics, Population / Q000379:methods*; D006801:Humans; D008957:Models, Genetic; D008962:Models, Theoretical; D010375:Pedigree; D016012:Poisson Distribution; D011156:Population Density; D011336:Probability
keywords: Coalescence in a fixed pedigree; Diploid monoecious Wright–Fisher model; Directed configuration model; Giant component; Random digraph; Wakeley et al.’s cyclical model