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A process of rumor scotching on finite populations

Rumor spreading is a ubiquitous phenomenon in social and technological networks. Traditional models consider that the rumor is propagated by pairwise interactions between spreaders and ignorants. Spreaders can become stiflers only after contacting spreaders or stiflers. Here we propose a model that considers the traditional assumptions, but stiflers are active and try to scotch the rumor to the spreaders. An analytical treatment based on the theory of convergence of density dependent Markov chains is developed to analyze how the final proportion of ignorants behaves asymptotically in a finite homogeneously mixing population. We perform Monte Carlo simulations in random graphs and scale-free networks and verify that the results obtained for homogeneously mixing populations can be approximated for random graphs, but are not suitable for scale-free networks. Furthermore, regarding the process on a heterogeneous mixing population, we obtain a set of differential equations that describes the time evolution of the probability that an individual is in each state. Our model can be applied to study systems in which informed agents try to stop the rumor propagation. In addition, our results can be considered to develop optimal information dissemination strategies and approaches to control rumor propagation.