Stationary distribution and cover time of random walks on random digraphs

Authors:
Colin Cooper;Alan Frieze
Affiliations:
Department of Informatics, Kings College, University of London, London WC2R 2LS, UK;Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Venue:
Journal of Combinatorial Theory Series B
Year:
2012

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Abstract

We study properties of a simple random walk on the random digraph D"n","p when np=dlogn, d1. We prove that whp the value @p"v 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 D"n","p. If d=d(n)-~ with n, the stationary distribution is asymptotically uniform whp. Using this result we prove that, for d1, whp the cover time of D"n","p is asymptotic to dlog(d/(d-1))nlogn. If d=d(n)-~ with n, then the cover time is asymptotic to nlogn.