Consensus Problems on Clustered Networks

In this paper we address consensus in the context of networked agents whose interactions can be modelled by an undirected, signed, weighted, connected and clustered graph. We assume that individuals can be split into three clusters representing the decision classes on a given specific topic. Interactions between agents belonging to the same cluster are cooperative, meaning that the link connecting those agents has a non-negative weight, while interactions between agents belonging to different clusters are antagonistic and therefore a non-positive weight is associated to the link connecting them. We will show that under certain regularity assumptions it is possible to devise a simple modification of DeGroot's algorithm that ensures that the opinions of agents who cooperate converge to consensus, i.e. the opinions of agents belonging to the same class converge to the same decision.