Mean encounter times for multiple random walkers on networks

TL;DR

This paper develops a mathematical framework to analyze the collective behavior and encounter times of multiple independent random walkers on networks, providing analytical tools for understanding their dynamics.

Contribution

It introduces a general approach using eigenvalues and eigenvectors of transition matrices to derive analytical expressions for key collective metrics of multiple random walkers.

Findings

Derived formulas for stationary distributions and first-passage times.

Analyzed mean encounter times for various network types and strategies.

Applicable to both local and non-local random walk models.

Abstract

We introduce a general approach for the study of the collective dynamics of non-interacting random walkers on connected networks. We analyze the movement of $R$ independent (Markovian) walkers, each defined by its own transition matrix. By using the eigenvalues and eigenvectors of the $R$ independent transition matrices, we deduce analytical expressions for the collective stationary distribution and the average number of steps needed by the random walkers to start in a particular configuration and reach specific nodes the first time (mean first-passage times), as well as global times that characterize the global activity. We apply these results to the study of mean first-encounter times for local and non-local random walk strategies on different types of networks, with both synchronous and asynchronous motion.