Ancestries in random $d$-DAGs

Abstract

We consider a random recursive DAG $G_n$ on the vertex set $[n]$ where every vertex $i\geq 2$ has out-degree $d$, with the targets chosen uniformly at random among the earlier $i-1$ vertices. For this model, we propose a novel way to investigate the descendants of $n$ (which have recently been studied in a paper by Janson) through what we call ancestry processes. The ancestor process $a_i(n)$ of a vertex $i$ is defined as the number of ancestors of $i$ in $G_n$, and is closely related to the evolutions of multi-draw P\'olya urns. Results on the descendants can then be obtained via asymptotic results on functionals of the ancestry processes, generally leading to technical integral expressions. This method yields the answer to two questions posed by Janson, the first on the size of the joint descendants of vertices $n$ and $n+1$, and the other on the location of the earliest non-descendant. We further prove limit theorems for the ancestry processes $a_i(n)$ depending on $i$, determine the location of the earliest source node, and provide an alternative proof of a first-moment result contained in Janson's work.