Large deviations of random walks on random graphs

TL;DR

This paper analyzes the large deviations in the behavior of unbiased random walks on Erdös-Rényi graphs, revealing how fluctuations and phase transitions relate to localization and entropy through a modified biased walk.

Contribution

It introduces a modified biased random walk framework to explain large deviation phenomena and dynamical phase transitions in random walks on Erdös-Rényi graphs.

Findings

Large deviations of degree sum and entropy are characterized.

Dynamical phase transitions are linked to localization phenomena.

Connections to maximum entropy random walks are established.

Abstract

We study using large deviation theory the fluctuations of time-integrated functionals or observables of the unbiased random walk evolving on Erd\"os-R\'enyi random graphs, and construct a modified, biased random walk that explains how these fluctuations arise in the long-time limit. Two observables are considered: the sum of the degrees visited by the random walk and the sum of their logarithm, related to the trajectory entropy. The modified random walk is used for both quantities to explain how sudden changes in degree fluctuations, akin to dynamical phase transitions, are related to localization transitions. For the second quantity, we also establish links between the large deviations of the trajectory entropy and the maximum entropy random walk.