Ring structures and mean first passage time in networks

In this paper we address the problem of the calculation of the mean first passage time 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 > xln 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 graphs 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.