The Meeting Time of Multiple Random Walks

Abstract: This article rigorously analyzes the meeting time between pursuers and evaders performing random walks on digraphs. There exist several bounds on the expected meeting time between random walkers on graphs in the literature, however, closed-form expressions are limited in scope. By utilizing the notion that multiple random walks on a common graph can be understood as a single random walk on the Kronecker product graph, we are able to provide the first analytic expression for the meeting time in terms of the transition matrices of the random walkers when modeled by either discrete-time Markov chains or continuous-time Markov chains. We further extend the results to the case of multiple pursuers and multiple evaders performing independent random walks. We present various sufficient conditions for pairs (or tuples) of transition matrices that satisfy certain conditions on the absorbing classes for which finite meeting times are guaranteed to exist.