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

Junhao Peng; Guoai Xu; Renxiang Shao; Lin Chen; H Eugene Stanley

doi:10.1063/1.5028123

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

J Chem Phys. 2018 Jul 14;149(2):024903. doi: 10.1063/1.5028123.

Authors

Junhao Peng¹, Guoai Xu², Renxiang Shao¹, Lin Chen³, H Eugene Stanley³

Affiliations

¹ School of Math and Information Science, Guangzhou University, Guangzhou 510006, China.
² School of Cyberspace Security, Beijing University of Posts and Telecommunications, Beijing 100876, China.
³ Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA.

PMID: 30007392
DOI: 10.1063/1.5028123

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 as Var(FRT) = 2⟨FRT⟩⟨GFPT⟩ - ⟨FRT⟩² - ⟨FRT⟩, where ⟨·⟩ 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.