Analysis of fluctuations in the first return times of random walks on regular branched networks

TL;DR

This paper derives a formula for the variance of the first return time in finite networks and analyzes its fluctuations specifically on regular branched networks, revealing differences from other network types.

Contribution

It introduces a method to calculate the variance of FRT on finite networks and applies it to regular branched networks, highlighting unique fluctuation behaviors.

Findings

Variance of FRT expressed in terms of mean FRT and GFPT

Distinct fluctuation patterns on Cayley trees compared to other networks

Provides a new analytical approach for FRT variance in finite networks

Abstract

The first return time (FRT) is the time it takes a random walker to first return to its original site, and the global first passage time (GFPT) is the first passage time for a random walker to move from a randomly selected site to a given site. We find that in finite networks the variance of FRT, Var(FRT), can be expressed Var(FRT)~$=2\langle$FRT$ \rangle \langle $GFPT$ \rangle -\langle $FRT$ \rangle^2-\langle $FRT$ \rangle$, where $\langle \cdot \rangle$ is the mean of the random variable. Therefore a method of calculating the variance of FRT on general finite networks is presented. We then calculate Var(FRT) and analyze the fluctuation of FRT on regular branched networks (i.e., Cayley tree) by using Var(FRT) and its variant as the metric. We find that the results differ from those in such other networks as Sierpinski gaskets, Vicsek fractals, T-graphs, pseudofractal scale-free webs, ($u,v$) flowers, and fractal and non-fractal scale-free trees.