A novel homogenous protocol for multi-agent clustering over directed graphs

Whereas mechanisms of consensus in networks have been thoroughly studied, many multi-agent systems, arising in nature and industry, exhibit disagreement, as exemplified by cleavage of opinions in social groups (instead of an agreement on some value) or uniform spread of mobile agents over some area (instead of their rendezvous at some point). A special case of disagreement, where the solutions split into several synchronous groups, is referred to as clustering (referred also to as “clusterization” or “cluster consensus”). Although maintaining disagreement may seem “easier” than to establish consensus, only a few models, providing multi-agent clustering, are currently known. Most of them either assume either a special interaction topology or some structural heterogeneity in the network, caused e.g. by presence of dedicated “informed” leaders or separation of couplings into attractive and repulsive. In this paper, we offer a novel homogenous distributed protocol that can provide both consensus and clustering of identical agents, interacting over general directed graph, and establish criteria for their convergence. These protocols resemble usual distributed consensus algorithms, where agents are coupled via non-monotone output maps. Besides having the manifold of consensus equilibria, such a system has an infinite number of “cluster” equilibria, corresponding to the splitting of agents into several groups (clusters).