The Maki-Thompson Model with Spontaneous Stifling on Symmetric Networks

Abstract

We investigate rumor spreading in a generalized Maki-Thompson model with spontaneous stifling, evolving on quasi-transitive networks. Individuals are either ignorants, spreaders, or stiflers; spreaders stop by contact with other spreaders or stiflers or after an independent random waiting time sampled from a given distribution, modeling a spontaneous loss of interest. The topology of the underlying population network is incorporated by modeling it as a broad class of symmetric networks, whose vertices are partitioned into finitely many orbit types. This yields a unified framework for homogeneous and heterogeneous networks. For sequences of finite quasi-transitive graphs, and for infinite quasi-transitive graphs with subexponential growth, we establish a Functional Law of Large Numbers and a Functional Central Limit Theorem for the densities of each vertex type for the three states. The mean-field limit is described by a system of nonlinear integral equations, while fluctuations are asymptotically Gaussian and governed by a system of stochastic integral equations with explicit covariance. Our results show how the topology and the law of spontaneous stifling jointly shape the speed and variability of rumor outbreaks. As a special case, our model reduces to the classical Maki-Thompson model when spontaneous stifling is absent.