Stochastic Stability of Opinion Dynamics with Stubbornness

In this paper we investigate the stochastic stability of opinion dynamics model in which each individual's current opinion is influenced by others. We assume that the social network is strongly connected topology and that each individual's opinion continually is subjected to small stochastic multiplicative noises and a stubborn opinion. We first propose a general stochastic opinion dynamics model with a stubborn opinion. We then consider the stochastic stability of the system. According to the mathematical concepts, it turns out that there are at least three different types of stochastic stability: stability in probability, moment stability and almost sure stability. We focus on the stochastic stability in probability in this paper. The time evolution of opinion dynamics is described by the stochastic differential equation with multiplicative noise. We establish clearly the condition under which the average of opinions can reach stochastic stability in probability and we develop sufficient condition that the error system for all opinions is stochastic stability in probability. Besides theoretical analysis, we demonstrate our results through numerical computations and simulations as well.