  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.
