Abstract
We derive an approximate but explicit formula for the Mean First Passage Time of a random walker between a source and a target node of a directed and weighted network. The formula does not require any matrix inversion, and it takes as only input the transition probabilities into the target node. It is derived from the calculation of the average resolvent of a deformed ensemble of random sub-stochastic matrices H=\langle H\rangle +\delta HH=⟨H⟩+δH, with \langle H\rangle⟨H⟩ rank-11 and non-negative. The accuracy of the formula depends on the spectral gap of the reduced transition matrix, and it is tested numerically on several instances of (weighted) networks away from the high sparsity regime, with an excellent agreement.
Highlights
The formula does not require any matrix inversion, and takes it as input only the local transition weights into the target node
Its accuracy depends on the existence of a “large" spectral gap between the Perron-Frobenius eigenvalue and the blob of all other eigenvalues of the reduced transition matrix T (j) – obtained from the full transition matrix of the walker by erasing the target node’s row and column
We have shown that – for a variety of “not too sparse" networks – this condition is not hard to materialize, and leads to an excellent agreement of our approximate formula with numerical simulations as well as the exact formula (3)
Summary
The exploration of a complex network by a walker that hops randomly from one node to another according to a given probabilistic rule has received much attention in recent years [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20], with many applications (see [21] for an excellent review), including the self-organization and generation of networks [22,23,24]. The computation of the MFPT involves a cumbersome inversion of a reduced matrix of transition probabilities. For this reason, any analytical treatment of the MFPT has in general proven difficult, with a number of attempts made to derive exact expressions – often valid when transition matrices have special symmetries – as well as approximate and mean field results (see Sec. 1.1 for details). We address these issues by proposing an approximate but explicit formula for the MFPT of a walker on directed and weighted networks. Our formula does not require matrix inversions, it depends only on the local information about the target node, and sheds light on the interplay between structural and spectral properties of the underlying network. The two Appendices are devoted to technical calculations and examples
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