An approximation for return time distributions of random walks on sparse networks

TL;DR

This paper introduces an approximation method combining message-passing and mean-field techniques to accurately estimate the return time distribution of random walks on large, sparse undirected networks, highlighting the influence of global network structure.

Contribution

The paper presents a novel approximation approach for return time distributions that effectively captures both short- and long-term behaviors in large sparse networks.

Findings

Excellent agreement between approximation and true distributions

Global structure influences return times mainly through total edges

Local network structure significantly affects random walk properties

Abstract

We propose an approximation for the first return time distribution of random walks on undirected networks. We combine a message-passing solution with a mean-field approximation, to account for the short- and long-term behaviours respectively. We test this approximation on several classes of large graphs and find excellent agreement between our approximations and the true distributions. While the statistical properties of a random walk will depend on the structure of the network, the observed agreement between our approximations and numerical calculations implies that while local structure is clearly very influential, global structure is only important in a relatively superficial way, namely through the total number of edges.