Tripartite and Sign Consensus for Clustering Balanced Social Networks

In this paper we address two forms of consensus for multi-agent systems with undirected, signed, weighted, and connected communication graphs, under the assumption that the agents can be partitioned into three clusters, representing the decision classes on a given specific topic, for instance, the in favour, abstained and opponent agents. We will show that under some assumptions on the coop-erative/antagonistic relationships between the agents, simple modifications of DeGroot's algorithm allow to achieve tripartite consensus (if the opinions of agents belonging to the same class all converge to the same decision) or sign consensus (if the opinions of the agents in the three clusters converge to positive, zero and negative values, respectively).