Determining global mean-first-passage time of random walks on Vicsek fractals using eigenvalues of Laplacian matrices

TL;DR

This paper derives an exact formula for the average first-passage time of random walks on Vicsek fractals using Laplacian eigenvalues, revealing power-law growth with system size and bounds for general trees.

Contribution

It provides the first explicit solution for GMFPT on Vicsek fractals and establishes bounds for GMFPT on general treelike networks, linking spectral properties to dynamical processes.

Findings

GMFPT scales approximately as a power-law with system size

Upper and lower bounds for GMFPT are established for trees

Leading behavior of bounds varies with network structure

Abstract

The family of Vicsek fractals is one of the most important and frequently-studied regular fractal classes, and it is of considerable interest to understand the dynamical processes on this treelike fractal family. In this paper, we investigate discrete random walks on the Vicsek fractals, with the aim to obtain the exact solutions to the global mean first-passage time (GMFPT), defined as the average of first-passage time (FPT) between two nodes over the whole family of fractals. Based on the known connections between FPTs, effective resistance, and the eigenvalues of graph Laplacian, we determine implicitly the GMFPT of the Vicsek fractals, which is corroborated by numerical results. The obtained closed-form solution shows that the GMFPT approximately grows as a power-law function with system size (number of all nodes), with the exponent lies between 1 and 2. We then provide both the upper bound and lower bound for GMFPT of general trees, and show that leading behavior of the upper bound is the square of system size and the dominating scaling of the lower bound varies linearly with system size. We also show that the upper bound can be achieved in linear chains and the lower bound can be reached in star graphs. This study provides a comprehensive understanding of random walks on the Vicsek fractals and general treelike networks.