Random walks on complex trees

TL;DR

This paper investigates how random walks behave on complex trees, revealing significant differences from looped networks in discovery rates, displacement, and passage times, with implications for understanding diffusive processes on tree structures.

Contribution

It provides a detailed analysis of random walk dynamics on complex trees, highlighting unique properties such as logarithmic MFPT dependence and the impact of topological distances.

Findings

Vertex discovery rate slows down on trees

Mean first passage time shows logarithmic degree dependence

Distance dominates source-target topological effects

Abstract

We study the properties of random walks on complex trees. We observe that the absence of loops reflects in physical observables showing large differences with respect to their looped counterparts. First, both the vertex discovery rate and the mean topological displacement from the origin present a considerable slowing down in the tree case. Second, the mean first passage time (MFPT) displays a logarithmic degree dependence, in contrast to the inverse degree shape exhibited in looped networks. This deviation can be ascribed to the dominance of source-target topological distance in trees. To show this, we study the distance dependence of a symmetrized MFPT and derive its logarithmic profile, obtaining good agreement with simulation results. These unique properties shed light on the recently reported anomalies observed in diffusive dynamical systems on trees.