Baronchelli, A. & Loreto, V. (2006). Ring structures and mean first passage time in networks. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 73(2), doi: 10.1103/PhysRevE.73.026103
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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⟩×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 Erdös-Renyi random graphs for which the method allows for almost perfect agreement with numerical simulations. Then we extend the approach to the Barabási-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.
|Subjects:||Q Science > QC Physics|
|Divisions:||School of Engineering & Mathematical Sciences > Department of Mathematical Science|
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