Ring structures and mean first passage time in networks

Andrea Baronchelli, Vittorio Loreto

Published 2005-01-27, updated 2006-02-08Version 2

In this paper we address the problem of the calculation of the mean first passage time (MFPT) on generic graphs. We focus in particular on the mean first passage time on a node 's' for a random walker starting from a generic, unknown, node 'x'. We introduce an approximate scheme of calculation which maps the original process in a Markov process in the space of the so-called rings, described by a transition matrix of size O(ln N / ln<k> X ln N / ln<k>), where N is the size of the graph and <k> the average degree in the graph. In this way one has a drastic reduction of degrees of freedom with respect to the size N of the transition matrix of the original process, corresponding to an extremely-low computational cost. We first apply the method to the Erdos-Renyi random graph for which the method allows for almost perfect agreement with numerical simulations. Then we extend the approach to the Barabasi-Albert graph, as an example of scale-free graph, for which one obtains excellent results. Finally we test the method with two real world graphs, Internet and a network of the brain, for which we obtain accurate results.