Analytic expression for the mean time to absorption for a random walker on the Sierpinski gasket

Abstract

The exact analytic expression for the mean time to absorption (or mean walk length) for a particle performing a random walk on a finite Sierpinski gasket with a trap at one vertex is found to be T((n))=[3(n)5(n+1)+4(5(n))-3(n)]/(3(n+1)+1) where n denotes the generation index of the gasket, and the mean is over a set of starting points of the walk distributed uniformly over all the other sites of the gasket. In terms of the number N(n) of sites on the gasket and the spectral dimension d of the gasket, the precise asymptotic behavior for large N(n) is T((n))-->1/3(2N(n))(2/d)-N1.464. This serves as a partial check on our result, as it is (a) intermediate between the known results T-N2 (d=1) and T-N ln N (d=2) for random walks on d-dimensional Euclidean lattices and (b) consistent with the known result for the asymptotic behavior of the mean number of distinct sites visited in a random walk on a fractal lattice.

Random walks on deterministic scale-free networks: Exact results

“…For a generic graph, given the corresponding adjacency matrix A and the coordination matrix Z, the numerical calculation of the mean time to absorption can be performed by exploiting a differential equation where the normalized discrete Laplacian ∆ = AZ −1 − I appears [10,11,25,26]. More precisely, for the topological structures analyzed here, the Laplacian ∆ g is a V g ×V g matrix which depends on the generation g and we have Therefore, the mean time to absorption averaged over all possible starting sites i = 1 reads as:…”

Mean first-passage time for random walks on generalized deterministic recursive trees

“…Recent papers have considered the exact determination of the mean first-passage time for some self-similar network models, like the Sierpinski fractals [8,9], pseudofractal web [10], Apollonian networks [11,33], Koch networks [12], etc., including some trees as the iterative fractal scale-free network [13] or the T-graph [14,15], The approach used considers the topology of the networks and employs decimation techniques or counting methods which usually require long and complex calculations. Here we provide a technique to compute the MFPT which is based on the relationship between the MFPT and the eigenvalues of the Laplacian matrix of the networks, but avoids their explicit computation.…”

Close or connected: Distance and connectivity effects on transport in networks

“…An important issue in the field is to quantify the impact of topological properties of a network on its transport properties. As a paradigm of transport process, random walks on complex networks have been intensely studied [4][5][6][7][8][9], and the mean first-passage time (MFPT) [10] to a target nodewhich quantifies the time needed for a random walker to find a target on the network -has been widely used as an indicator of transport efficiency [11][12][13][14][15][16][17][18].…”