On Krause's consensus formation model with state-dependent connectivity

We study a model of opinion dynamics introduced by Krause: each agent has an opinion represented by areal number, and updates its opinion by averaging all agent opinions that differ from its own by less than 1. Wegive a simple proof of convergence into clusters of agents, with all agents in the same cluster holding the sameopinion. We then introduce a particular notion of equilibrium stability and provide lower bounds on the inter-clusterdistances at a stable equilibrium. To better understand the behavior of the system when the number of agents islarge, we also introduce and study a variant involving a continuum of agents, obtaining partial convergence resultsand lower bounds on inter-cluster distances, under some mild assumptions.