Cluster transitions in a multi-agent clustering model

Clustering is a phenomenon that may emerge in multi-agent systems through self-organization: groups arise consisting of agents with similar dynamic behavior. It is observed in fields ranging from the exact sciences to social and life sciences; consider e.g. swarm behavior of animals or social insects, dynamics of opinion formation, or the synchronization (which corresponds to cluster formation in the phase space) of coupled oscillators modeling brain or heart cells. We consider the formation of clusters in a multi-agent system with each agent belonging to a multi-dimensional space. We mainly focus on cluster transitions with varying coupling strength. While for the one-dimensional case the transition generically involves at most two clusters, this is not necessarily true for higher dimensions. For a system with three agents, all-to-all coupling, and equal weights, we investigate necessary and sufficient conditions guaranteeing the occurrence of only two possible cluster structures: a single cluster containing all three agents, and a configuration with three clusters; for no value of the coupling strength is there an intermediate stage consisting of two clusters. We illustrate the results with a numeric example on opinion formation.