A connection between a system of random walks and rumor transmission

We establish a relationship between the phenomenon of rumor transmission on a population and a probabilistic model of interacting particles on the complete graph. More precisely, we consider variations of the Maki-Thompson epidemic model and the “frog model” of random walks, which were introduced in the scientific literature independently and in different contexts. We analyze the Markov chains which describe these models, and show a coupling between them. Our connection shows how the propagation of a rumor in a closed homogeneously mixing population can be described by a system of random walks on the complete graph. Additionally, we discuss further applications of the random walk model which are relevant to the modeling of different biological dynamics.