Stochastic Bounded Confidence Opinion Dynamics: How Far Apart Do Opinions Drift?

TL;DR

This paper investigates the finite-time behavior of stochastic bounded confidence opinion dynamics, focusing on a two-agent system, and demonstrates that opinion differences remain close to zero under stability conditions.

Contribution

It provides the first finite-time analysis of stochastic bounded confidence opinion dynamics, extending understanding beyond asymptotic behavior to practical, finite-time scenarios.

Findings

Opinion differences are concentrated around zero in stable conditions

Finite-time behavior aligns with asymptotic stability results

Analysis applies to two-agent systems as a fundamental case

Abstract

In this era of fast and large-scale opinion formation, a mathematical understanding of opinion evolution, a.k.a. opinion dynamics, is especially important. Linear graph-based dynamics and bounded confidence dynamics are the two most popular models for opinion dynamics in social networks. Recently, stochastic bounded confidence opinion dynamics were proposed as a general framework that incorporates both these dynamics as special cases and also captures the inherent stochasticity and noise (errors) in real-life social exchanges. Although these dynamics are quite general and realistic, their analysis is particularly challenging compared to other opinion dynamics models. This is because these dynamics are nonlinear and stochastic, and belong to the class of Markov processes that have asymptotically zero drift and unbounded jumps. The asymptotic behavior of these dynamics was characterized in prior works. However, they do not shed light on their finite-time behavior, which is often of interest in practice. We take a stride in this direction by analyzing the finite time behavior of a two-agent system, which is fundamental to the understanding of multi-agent dynamics. In particular, we show that the opinion difference between the two agents is well concentrated around zero under the conditions that lead to asymptotic stability of the dynamics.