Consensus for clusters of agents with cooperative and antagonistic relationships

In this paper we address the consensus problem in the context of networked agents whose communication graph splits into clusters: interactions between agents in the same cluster are cooperative, while interactions between agents belonging to different clusters are antagonistic. This problem set-up arises in the context of social networks and opinion dynamics, where reaching a consensus means that the opinions of the agents in the same cluster converge to the same decision, that is typically different for the different clusters. Under the assumption that agents belonging to the same cluster have the same amount of trust (/distrust) to be distributed among their cooperators (/adversaries), we propose a modified version of DeGroot's law. By simply constraining how much agents in each group should be conservative about their own opinions, it is possible to achieve a nontrivial solution by means of a distributed algorithm. The result is then particularized to unweighted complete communication graphs, and subsequently extended to a class of nonlinear multi-agent systems.