Continuous-Time Average-Preserving Opinion Dynamics with Opinion-Dependent Communications

We study a simple continuous-time multi-agent system related to Krause’s model ofopinion dynamics: each agent holds a real value, and this value is continuously attracted by everyother 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 commonvalue, and provide a lower bound on the distance between clusters at a stable equilibrium, under asuitable notion of multi-agent system stability.To better understand the behavior of the system for a large number of agents, we introducea variant involving a continuum of agents. We prove, under some conditions, the existence of asolution to the system dynamics, convergence to clusters, and a non-trivial lower bound on thedistance between clusters. Finally, we establish that the continuum model accurately represents theasymptotic behavior of a system with a finite but large number of agents.