Diameter and Stationary Distribution of Random $r$-out Digraphs

Let D(n, r) be a random r-out regular directed multigraph on the set of vertices {1, . . . , n}. In this work, we establish that for every r ≥ 2, there exists ηr > 0 such that diam(D(n, r)) = (1 + ηr + o(1)) logr n. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on D(n, r). In particular, we determine the asymptotic behaviour of πmax and πmin, the maximum and the minimum values of the stationary distribution. We show that with high probability πmax = n −1+o(1) and πmin = n −(1+ηr)+o(1). Our proof shows that the vertices with π(v) near to πmin lie at the top of “narrow, slippery towers”; such vertices are also responsible for increasing the diameter from (1+o(1)) logr n to (1+ηr+o(1)) logr n.