Consensus Convergence with Stochastic Effects

We consider a stochastic, continuous state and time opinion model where each agent’s opinion locally interacts with other agents’ opinions in the system, and there is also exogenous randomness. The interaction tends to create clusters of common opinion. By using linear stability analysis of the associated nonlinear Fokker–Planck equation that governs the empirical density of opinions in the limit of infinitely many agents, we can estimate the number of clusters, the time to cluster formation, and the critical strength of randomness so as to have cluster formation. We also discuss the cluster dynamics after their formation, the width and the effective diffusivity of the clusters. Finally, the long-term behavior of clusters is explored numerically. Extensive numerical simulations confirm our analytical findings.