Continuous-time average-preserving opinion dynamics with opinion-dependent communications

TL;DR

This paper analyzes a continuous-time opinion dynamics model where agents' opinions evolve based on opinion-dependent communication, proving convergence to clusters and establishing the continuum limit for large populations.

Contribution

It introduces a novel continuous-time opinion model with opinion-dependent interactions and demonstrates convergence, cluster formation, and the continuum approximation.

Findings

Agents' opinions converge to clusters with shared values.

A lower bound on inter-cluster distances at equilibrium is established.

The continuum model accurately predicts large-agent system behavior.

Abstract

We study a simple continuous-time multi-agent system related to Krause's model of opinion dynamics: each agent holds a real value, and this value is continuously attracted by every other value differing from it by less than 1, with an intensity proportional to the difference. We prove convergence to a set of clusters, with the agents in each cluster sharing a common value, and provide a lower bound on the distance between clusters at a stable equilibrium, under a suitable notion of multi-agent system stability. To better understand the behavior of the system for a large number of agents, we introduce a variant involving a continuum of agents. We prove, under some conditions, the existence of a solution to the system dynamics, convergence to clusters, and a non-trivial lower bound on the distance between clusters. Finally, we establish that the continuum model accurately represents the asymptotic behavior of a system with a finite but large number of agents.