Stationary distribution and cover time of random walks on random digraphs

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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(n1)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/(d1))nlogn. If d=d(n) with n, then the cover time is asymptotic to nlogn.

Keywords

Random digraphs
Random walk
Cover time

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Supported in part by NSF grant CCF0502793.