On the 2R conjecture for multi-agent systems

We consider a simple dynamical model of agents distributed on the real line. The agents have a limited vision range and they synchronously update their positions by moving to the average position of the agents that are within their vision range. This dynamical model was initially introduced in the social science literature as a model of opinion dynamics and is known there as the “Krause model”. It gives rise to surprising and partly unexplained dynamics that we describe and discuss in this paper. One of the central observations is the 2R-conjecture: when sufficiently many agents are distributed on the real line and have their position evolve according to the above dynamics, the agents eventually merge into clusters that have inter-cluster distances roughly equal to 2R (R is the vision range of the agents). This observation is supported by extensive numerical evidence and is robust under various modifications of the model. It is easy to see that clusters need to be separated by at least R. On the other hand, the unproved bound 2R that is observed in practice can probably only be obtained by taking into account the specifics of the model's dynamics. In this paper, we study these dynamics and consider a number of issues related to the 2R conjecture that explicitly uses the model's dynamics. In particular, we provide bounds for the vision range that lead all agents to merge into only one cluster, we analyze the relations between agents on finite and infinite intervals, and we introduce a notion of equilibrium stability for which clusters of equal weights need to be separated by at least 2R to be stable. These results, however, do not prove the conjecture. To understand the system behavior for a large agent density, we also consider a version of the model that involves a continuum of agents. We study properties of this continuous model and of its equilibria, and investigate the connections between the discrete and continuous versions.