Propagation of opinions on a network. Convergence, spreading speed and patterns in the Daley-Kendall model

Abstract: We study a social contagion model describing the propagation of opinions across a network. Our model is a generalization of a model introduced by Daley and Kendall, and consists of a system of coupled differential equations set on the nodes of an infinite graph. The modeling is inspired by SIR-type models from epidemiology. The key idea is that individuals can transmit their opinion to their neighbors (as in epidemiological models), until they perceive that the opinion is already known in their neighborhood. After introducing the model, we analyze the long-time behavior of its solutions. We prove that the solutions converge and we study their asymptotic equilibrium. We also establish estimates on the speed of propagation of the solutions through the network. These results give estimates on the rate of adoption of the opinion and on its velocity in the social network. We also provide simulations that indicate that this model can give rise to patterns: when the individuals take into account individuals that are ''far enough'' in estimating whether or not the opinion is already known, then we observe the emergence of localized ''bubbles'' where the opinion is adopted.