The number of descendants in a random directed acyclic graph

02/24/2023
by   Svante Janson, et al.
0

We consider a well known model of random directed acyclic graphs of order n, obtained by recursively adding vertices, where each new vertex has a fixed outdegree d≥2 and the endpoints of the d edges from it are chosen uniformly at random among previously existing vertices. Our main results concern the number X of vertices that are descendants of n. We show that X/√(n) converges in distribution; the limit distribution is, up to a constant factor, given by the dth root of a Gamma distributed variable. Γ(d/(d-1)). When d=2, the limit distribution can also be described as a chi distribution χ(4). We also show convergence of moments, and find thus the asymptotics of the mean and higher moments.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset