Stationary distribution and cover time of random walks on random digraphs

TL;DR

This paper analyzes the stationary distribution and cover time of simple random walks on random directed graphs, revealing asymptotic behaviors depending on the average degree, with implications for understanding random walk efficiency.

Contribution

It provides new asymptotic results for the stationary distribution and cover time of random walks on D_{n,p} graphs, especially as the average degree grows with n.

Findings

Stationary probability is asymptotic to in-degree over total edges.

Cover time is asymptotic to d log(d/(d-1)) n log n for fixed d.

Cover time approaches n log n as d tends to infinity.

Abstract

We study properties of a simple random walk on the random digraph D_{n,p} when np={d\log n},\; d>1. We prove that whp the stationary probability pi_v of a 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 D_{n,p}. If d=d(n) tends to infinity with n, the stationary distribution is asymptotically uniform whp. Using this result we prove that, for d>1, whp the cover time of D_{n,p} is asymptotic to d\log (d/(d-1))n\log n. If d=d(n) tends to infinity with n, then the cover time is asymptotic to n\log n.