Quasi-synchronization of bounded confidence opinion dynamics with stochastic asynchronous rule

TL;DR

This paper introduces a generalized bounded confidence opinion model with stochastic asynchronous updates, demonstrating noise-induced quasi-synchronization and unifying synchronization theory for both synchronous and asynchronous opinion dynamics.

Contribution

It develops a generalized model encompassing HK and DW models, proving noise-induced quasi-synchronization in asynchronous systems, and extends the theoretical understanding of opinion dynamics.

Findings

HK dynamics achieve almost sure quasi-synchronization under noise.

Asynchronous models achieve quasi-synchronization in mean, a weaker form.

First theoretical proof of noise-induced synchronization for DW model.

Abstract

Recently the theory of noise-induced synchronization of Hegselmann-Krause (HK) dynamics has been well developed. As a typical opinion dynamics of bounded confidence, the HK model obeys a synchronous updating rule, i.e., \emph{all} agents check and update their opinions at each time point. However, whether asynchronous bounded confidence models, including the famous Deffuant-Weisbuch (DW) model, can be synchronized by noise have not been theoretically proved. In this paper, we propose a generalized bounded confidence model which possesses a stochastic asynchronous rule. The model takes the DW model and the HK model as special cases and can significantly generalize the bounded confidence models to practical application. We discover that the asynchronous model possesses a different noise-based synchronization behavior compared to the synchronous HK model. Generally, the HK dynamics can achieve quasi-synchronization \emph{almost surely} under the drive of noise. For the asynchronous dynamics, we prove that the model can achieve quasi-synchronization \emph{in mean}, which is a new type of quasi-synchronization weaker than the "almost surely" sense. The results unify the theory of noise-induced synchronization of bounded confidence opinion dynamics and hence proves the noise-induced synchronization of DW model theoretically for the first time. Moreover, the results provide a theoretical foundation for developing noise-based control strategy of more complex social opinion systems with stochastic asynchronous rules.