Continuous-Time Opinion Dynamics With Stochastic Multiplicative Noises

Abstract

Recently, the study of opinion dynamics on complex networks has attracted a great deal of attention. In this brief, we take an interest in how an opinion forms in a social network with stochastic noises. We investigate a stochastic opinion dynamics model, in which individuals obtain information from their neighbors in a social network. We model the social network as a strongly connected graph. Each individual has an initial opinion and iteratively exchanges her opinion based on a stubborn opinion, her neighbors’ opinions and small stochastic noises continually. We first construct a general stochastic opinion evolution problem with multiplicative noises as a stochastic differential equation and then investigate the stochastic stability of this equation. We consider whether an opinion may form as a result of stochastic disturbance and local interactions. We also establish two sufficient conditions which ensure that the average opinion and the error vector system for all individuals are stochastic stable, respectively. Numerical simulations are given to demonstrate the effectiveness of the theoretical results.