Mean first-encounter times of simultaneous random walkers with resetting on networks

TL;DR

This paper derives exact formulas for the average time until two random walkers on networks meet, considering resetting behaviors, and demonstrates how resetting can significantly reduce encounter times across various network types.

Contribution

It provides a spectral analytical framework for calculating mean first-encounter times with resetting on networks, extending understanding of random walk dynamics.

Findings

Resetting reduces encounter times in various network models.

Spectral methods effectively analyze resetting effects.

Results applicable to human mobility and epidemic modeling.

Abstract

We investigate the dynamics of simultaneous random walkers with resetting on networks and derive exact analytical expressions for the mean first-encounter times of Markovian random walkers. Specifically, we consider two cases for the simultaneous dynamics of two random walkers on networks: when only one walker resets to the initial node, and when both walkers return to their initial positions. In both cases, the encounter times are expressed in terms of the eigenvalues and eigenvectors of the transition matrix of the normal random walk, providing a spectral interpretation of the impact of resetting. We validate our approach through examples on rings, Cayley trees, and random networks generated using the Erd\H{o}s-R\'enyi, Watts-Strogatz, and Barab\'asi-Albert algorithms, where resetting significantly reduces encounter times. The proposed framework can be extended to other types of random walk dynamics, transport processes, or multiple-walker scenarios, with potential applications in human mobility, epidemic spreading, and search strategies in complex systems.