Clustering in diffusively coupled networks

This paper shows how different mechanisms may lead to clustering behavior in connected networks consisting of diffusively coupled agents. In contrast to the widely studied synchronization processes, in which the states of all the coupled agents converge to the same value asymptotically, in the cluster synchronization problem studied in this paper, we require all the interconnected agents to evolve into several clusters and each agent only to synchronize within its cluster. The first mechanism is that agents have different self-dynamics, and those agents having the same self-dynamics may evolve into the same cluster. When the agents’ self-dynamics are identical, we present two other mechanisms under which cluster synchronization might be achieved. One is the presence of delays and the other is the existence of both positive and negative couplings between the agents. Some sufficient and/or necessary conditions are constructed to guarantee nn-cluster synchronization. Simulation results are presented to illustrate the effectiveness of the theoretical analysis.