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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2014
- DOI:
- 10.48550/arXiv.1405.7276
- arXiv:
- arXiv:1405.7276
- Bibcode:
- 2014arXiv1405.7276B
- Keywords:
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- Mathematics - Probability;
- Quantitative Biology - Populations and Evolution;
- 60K35 (Primary);
- 92D10 (Secondary)
- E-Print:
- 21 pages, 2 figures