Exact mean first-passage time on generalized Vicsek fractal

TL;DR

This paper analytically derives the exact mean first-passage time for random walks on Vicsek fractals and their generalizations, revealing consistent scaling behavior regardless of the seed used.

Contribution

It introduces an efficient, mapping-based method to precisely compute mean first-passage times on generalized Vicsek fractals, surpassing previous spectral techniques.

Findings

Exact mean first-passage time solutions for Vicsek fractals

Method applicable to all generalized Vicsek fractals

Scaling relation remains unchanged in large size limit

Abstract

Fractal phenomena may be widely observed in a great number of complex systems. In this paper, we revisit the well-known Vicsek fractal, and study some of its structural properties for purpose of understanding how the underlying topology influences its dynamic behaviors. For instance, we analytically determine the exact solution to mean first-passage time for random walks on Vicsek fractal in a more light mapping-based manner than previous other methods, including typical spectral technique. More importantly, our method can be quite efficient to precisely calculate the solutions to mean first-passage time on all generalized versions of Vicsek fractal generated based on an arbitrary allowed seed, while other previous methods suitable for typical Vicsek fractal will become prohibitively complicated and even fail. Lastly, this analytic results suggest that the scaling relation between mean first-passage time and vertex number in generalized versions of Vicsek fractal keeps unchanged in the large graph size limit no matter what seed is selected.