Random walks with stochastic resetting in complex networks: a discrete time approach

TL;DR

This paper studies discrete-time random walks with stochastic resets on complex networks, analyzing how resetting affects first hitting times and search efficiency, especially under non-Markovian and heavy-tailed reset distributions.

Contribution

It introduces a framework for analyzing non-Markovian resetting in complex networks, deriving propagator matrices and conditions for ergodicity, and applies these to real network models.

Findings

Existence of non-equilibrium steady states under light-tailed resets

Non-trivial dependence of mean first passage time on reset parameters

Resets can significantly improve search efficiency in large-world networks

Abstract

We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes of the network are target nodes, and we focus on the statistics of first hitting of these nodes. In the non-Markov case of the renewal process, we consider both light- and fat-tailed inter-reset distributions. We derive the propagator matrix in terms of discrete backward recurrence time PDFs and in the light-tailed case we show the existence of a non-equilibrium steady state. In order to tackle the non-Markov scenario, we derive a defective propagator matrix which describes an auxiliary walk characterized by killing the walker as soon as it hits target nodes. This propagator provides the information on the mean first passage statistics to the target nodes. We establish sufficient conditions for ergodicity of the walk under resetting. Furthermore, we discuss a generic resetting mechanism for which the walk is non-ergodic. Finally, we analyze inter-reset time distributions with infinite mean where we focus on the Sibuya case. We apply these results to study the mean first passage times for Markovian and non-Markovian (Sibuya) renewal resetting protocols in realizations of Watts-Strogatz and Barab\'asi-Albert random graphs. We show non trivial behavior of the dependence of the mean first passage time on the proportions of the relocation nodes, target nodes and of the resetting rates. It turns out that, in the large-world case of the Watts-Strogatz graph, the efficiency of a random searcher particularly benefits from the presence of resets.