Mean first-passage time for random walks on undirected networks

Details Mean first-passage time for random walks on undirected networks Mean first-passage time for random walks on undirected networks

2011-11-07

arxiv.org/abs/1111.1500v1

Eur. Phys. J. B 84, 691-697 (2011)

Physics

Condensed Matter

Statistical Mechanics

7 pages, no figures; definitive version published in European Physical Journal B

Scientific paper

10.1140/epjb/e2011-20834-1

In this paper, by using two different techniques we derive an explicit formula for the mean first-passage time (MFPT) between any pair of nodes on a general undirected network, which is expressed in terms of eigenvalues and eigenvectors of an associated matrix similar to the transition matrix. We then apply the formula to derive a lower bound for the MFPT to arrive at a given node with the starting point chosen from the stationary distribution over the set of nodes. We show that for a correlated scale-free network of size $N$ with a degree distribution $P(d)\sim d^{-\gamma}$, the scaling of the lower bound is $N^{1-1/\gamma}$. Also, we provide a simple derivation for an eigentime identity. Our work leads to a comprehensive understanding of recent results about random walks on complex networks, especially on scale-free networks.

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