Stationary distribution and cover time of random walks on random digraphs

Abstract

We study properties of a simple random walk on the random digraph Dn,p when np=dlogn, d>1.We prove that whp the value πv of the stationary distribution at vertex v is asymptotic to deg−(v)/m where deg−(v) is the in-degree of v and m=n(n−1)p is the expected number of edges of Dn,p. If d=d(n)→∞ with n, the stationary distribution is asymptotically uniform whp.Using this result we prove that, for d>1, whp the cover time of Dn,p is asymptotic to dlog(d/(d−1))nlogn. If d=d(n)→∞ with n, then the cover time is asymptotic to nlogn.