Return times of random walk on generalized random graphs

TL;DR

This paper analytically derives the return time distributions of random walks on complex random graphs with arbitrary degree distributions, providing insights into their stochastic dynamics and phase transitions.

Contribution

It introduces a novel analytical approach to determine return time distributions on complex networks, extending beyond simple lattice models.

Findings

Derived explicit return time distributions for random walks on complex networks.

Validated analytical results with numerical simulations.

Revealed differences in walk dynamics across various network types.

Abstract

Random walks are used for modeling various dynamics in, for example, physical, biological, and social contexts. Furthermore, their characteristics provide us with useful information on the phase transition and critical phenomena of even broader classes of related stochastic models. Abundant results are obtained for random walk on simple graphs such as the regular lattices and the Cayley trees. However, random walks and related processes on more complex networks, which are often more relevant in the real world, are still open issues, possibly yielding different characteristics. In this paper, we investigate the return times of random walks on random graphs with arbitrary vertex degree distributions. We analytically derive the distributions of the return times. The results are applied to some types of networks and compared with numerical data.