Mean first-passage time for random walks on undirected networks
School of Computer Science, Fudan University,
2 Shanghai Key Lab of Intelligent Information Processing, Fudan University, Shanghai 200433, P.R. China
3 Department of Mathematics, College of Science, Shanghai University, Shanghai 200444, P.R. China
4 Department of Electronic Engineering, City University of Hong Kong, Hong Kong, P.R. China
Published online: 23 November 2011
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) ~ d−γ, the scaling of the lower bound is N1−1/γ. 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.
© EDP Sciences, Società Italiana di Fisica and Springer-Verlag, 2011