TY - JOUR

T1 - The largest strongly connected component in the cyclical pedigree model of Wakeley et al.

AU - Blath, Jochen

AU - Kadow, Stephan

AU - Ortgiese, Marcel

PY - 2014/12

Y1 - 2014/12

N2 - 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 SNSN (where NN is the population size) of the DCM resp. the pedigree. The size of the giant component can be characterized explicitly (amounting to approximately 80%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)O(logN). Moreover, we describe the size and structure of the ‘domain of attraction’ of SNSN. In particular, we show that with high probability for any individual the shortest ancestral line reaches SNSN after O(loglogN)O(loglogN) generations, while almost all other ancestral lines take at most O(logN)O(logN) generations.

AB - 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 SNSN (where NN is the population size) of the DCM resp. the pedigree. The size of the giant component can be characterized explicitly (amounting to approximately 80%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)O(logN). Moreover, we describe the size and structure of the ‘domain of attraction’ of SNSN. In particular, we show that with high probability for any individual the shortest ancestral line reaches SNSN after O(loglogN)O(loglogN) generations, while almost all other ancestral lines take at most O(logN)O(logN) generations.

UR - https://doi.org/10.1016/j.tpb.2014.10.001

U2 - 10.1016/j.tpb.2014.10.001

DO - 10.1016/j.tpb.2014.10.001

M3 - Article

VL - 98

SP - 28

EP - 37

JO - Theoretical Population Biology

JF - Theoretical Population Biology

SN - 0040-5809

ER -