Almost Sure Exponential Stability of Altafini Model with Multiplicative Noise for Opinion Dynamics

Abstract

In this paper, we investigate the stability analysis of stochastic opinion dynamics in which the agents are interfered by multiplicative noise. We propose a stochastic Altafini model of opinion evolution in signed social networks with multiplicative noise and give the analysis of almost sure exponential stability (ASES). Compared with the deterministic opinion dynamic and stochastic opinion dynamics in additive noise environments, our results concern the effects of the agents’ free will, i.e., multiplicative noise, in signed social networks. We establish a sufficient condition with stochastic multiplicative noise which can guarantee the stochastic system is almost sure exponential stability (ASES) when the parameters of the strength of noise is limited by the conditions. Numerical simulations are given to illustrate the above theoretical results.