Exact mean first-passage time on the T-graph

Abstract

We consider a simple random walk on the T-fractal and we calculate the exact mean time taug to first reach the central node i0. The mean is performed over the set of possible walks from a given origin and over the set of starting points uniformly distributed throughout the sites of the graph, except i0. By means of analytic techniques based on decimation procedures, we find the explicit expression for taug as a function of the generation g and of the volume V of the underlying fractal. Our results agree with the asymptotic ones already known for diffusion on the T-fractal and, more generally, they are consistent with the standard laws describing diffusion on low-dimensional structures.

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].…”