Mean first-passage time for random walks on undirected networks

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.