Polarization in coopetitive networks of heterogeneous nonlinear agents

The mechanisms of regular cooperative behavior in multi-agent networks, such as consensus and synchronization, have been thoroughly studied. However, many natural and engineered networks do not synchronize, exhibiting persistent disagreement or clustering. One of the reasons for this “irregular” behavior is competition among some pairs of agents. Whereas cooperative interactions are usually represented by attractive couplings, bringing the trajectories closer, competition between two agents is naturally modeled by a repulsive coupling, leading to disagreement among the agents and preventing their trajectories from convergence. Such couplings may e.g. describe interactions of antagonistic individuals in social groups, competing economic agents and repelling particles. Networks where agents can both cooperate and compete are said to be coopetitive. To study the dynamics of general coopetitive networks, in particular, mechanisms of agents' splitting into several clusters, remains a challenging problem. A simple yet insightful model of polarization under coopetitive interactions was proposed. These papers address consensus-type dynamics over signed graphs, where arcs of positive and negative weights correspond, respectively, to cooperative and competitive couplings between the agents. If the graph is structurally balanced, these protocols lead to either consensus or “bipartite consensus” (polarization): the agents split into two competing “camps”, and the values (opinions) of agents from different camps agree in modulus but differ in signs. The results are limited to single integrator agents, interacting over static signed graphs. In this paper, we extend these results to time-varying graphs and nonlinear heterogeneous agents that satisfy a relaxed passivity condition.