Determining mean first-passage time on a class of treelike regular fractals

TL;DR

This paper develops explicit methods to calculate mean first-passage times on a family of treelike fractals, providing exact formulas and scaling behaviors, and demonstrating their applicability to various fractal networks.

Contribution

The paper introduces new techniques for deriving exact mean first-passage times on regular treelike fractals, including formulas for partial and entire MFPTs, and extends these methods to other fractal networks.

Findings

Exact formulas for PMFPT and EMFPT on fractals

Scaling laws of MFPT with network size

Applicability to Vicsek and scale-free fractals

Abstract

Relatively general techniques for computing mean first-passage time (MFPT) of random walks on networks with a specific property are very useful, since a universal method for calculating MFPT on general graphs is not available because of their complexity and diversity. In this paper, we present techniques for explicitly determining the partial mean first-passage time (PMFPT), i.e., the average of MFPTs to a given target averaged over all possible starting positions, and the entire mean first-passage time (EMFPT), which is the average of MFPTs over all pairs of nodes on regular treelike fractals. We describe the processes with a family of regular fractals with treelike structure. The proposed fractals include the $T$ fractal and the Peano basin fractal as their special cases. We provide a formula for MFPT between two directly connected nodes in general trees on the basis of which we derive an exact expression for PMFPT to the central node in the fractals. Moreover, we give a technique for calculating EMFPT, which is based on the relationship between characteristic polynomials of the fractals at different generations and avoids the computation of eigenvalues of the characteristic polynomials. Making use of the proposed methods, we obtain analytically the closed-form solutions to PMFPT and EMFPT on the fractals and show how they scale with the number of nodes. In addition, to exhibit the generality of our methods, we also apply them to the Vicsek fractals and the iterative scale-free fractal tree and recover the results previously obtained.