MODELING AND ANALYSIS OF COMPETITIVE PROPAGATION WITH CONVERSION ∗

Abstract. In this paper we model a class of propagation processes for multiple competing products over a social network and we characterize the dynamical properties resulting from varying individual behaviors. We consider four types of product-adoption processes for any individual: self adoption, social adoption, self conversion and social conversion. Based on these four product-adoption processes, we build a discrete-time Markov-chain model and approximate it by a difference equations system. We refer to the resulting difference equations system as the network competitive propagation model. Unlike previous analyses of interacting memes, our model predicts the long-term coexistence of mutually-exclusive competing products in a single-layer network. The accuracy of the network competitive propagation model is evaluated via theoretical and numerical analysis. The theoretical analysis reveals the existence of a unique consensus equilibrium, in which the state probability distributions of all the individuals are identical. This consensus equilibrium is globally asymptotically stable and depends only on the self conversion parameters; in this sense we say that the self-conversion process dominates the system’s asymptotic behavior. Without self conversion, our model still exhibits a consensus property, but the consensus value varies with the graph topology and initial conditions. Finally, our simulation work reveals the special clustering property of the Markov-chain model with no self conversion and relates our model to real data