Modeling and Analysis of Competitive Propagation with Conversion

In this paper we propose a class of propagation models for multiple competing products over a social network. We consider two propagation models: the social conversion mechanism and the self conversion mechanism, corresponding respectively to the endogenous and exogenous factors. A novel concept, the product-conversion graph, is proposed to characterize the interplay among competing products. According to the chronological order of social and self conversions, we develop two Markov-chain models and, based on the independence approximation, we approximate them with two respective difference equations systems. Theoretical analysis on these two approximation models reveal that the systems’ asymptotic behaviors vary with the structure of the product-conversion graph. In addition to the theoretical work, the accuracy of the independence approximation and the asymptotic behavior of the Markov-chain model are investigated via numerical analysis; we focus on the case where social conversion occurs before self conversion. Finally, by modifying the model, we propose and analyze a two-player uncooperative initial seeding game. The uniqueness of a Nash equilibrium is proved and, at the Nash equilibrium, the nodes’ eigenvector centralities are important in determining the amount of initial seeding resources spent on them.