Finite Time Bounds for Stochastic Bounded Confidence Dynamics

TL;DR

This paper analyzes the finite-time behavior of stochastic bounded confidence opinion dynamics, focusing on two-agent and bistar graph systems, revealing that opinion differences tend to concentrate around zero under stability conditions.

Contribution

It provides the first finite-time bounds for SBC opinion dynamics, extending understanding beyond asymptotic analysis to practical, short-term behavior in simple network structures.

Findings

Opinion differences concentrate around zero in stable conditions

Finite-time bounds are established for two-agent systems

Results inform understanding of multi-agent opinion evolution

Abstract

In this era of fast and large-scale opinion formation, a mathematical understanding of opinion evolution, a.k.a. opinion dynamics, acquires importance. Linear graph-based dynamics and bounded confidence dynamics are the two popular models for opinion dynamics in social networks. Stochastic bounded confidence (SBC) opinion dynamics was 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 SBC dynamics is quite general and realistic, its analysis is more challenging. This is because SBC dynamics is nonlinear and stochastic, and belongs to the class of Markov processes that have asymptotically zero drift and unbounded jumps. The asymptotic behavior of SBC dynamics was characterized in prior works. However, they do not shed light on its 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 and a bistar graph, which are crucial to the understanding of general 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 SBC dynamics.