Inhomogeneous Hegselmann–Krause models with two types of noise

Abstract

We derive two types of stochastic Hegselmann–Krause opinion formation models with leadership, study their different asymptotic behaviors and the noise effect to the leader’s control. Firstly, we propose a stochastic model with a multiplicative noise, which mimics the randomness from communication uncertainty. Using the Lyapunov functional approach, we provide a sufficient condition leading to the almost surely consensus behavior, which partially explains the mechanism that the group with large size and strong leadership can tolerate strong noise produced by communication uncertainty. The second noise comes from environmental uncertainty, fluctuates the system and makes opinions diverse and inclusive. We derive a stochastic model with additive noise, show that the noise from the environment destroys the leadership. However, the relative fluctuations of the followers’ opinions around the leader’s opinion have a uniformly bounded variance, which means they are still in a same group with the leader’s control. Finally, numerical simulations are performed to confirm theoretical results and explore more findings.