Cluster synchronization and controllability of complex multi-agent networks

This paper discloses the similarities between the condition for realizing cluster synchronization and that for uncontrollability in diffusively coupled multi-agent networks, both of which are built upon the characteristics of the networks´ topologies. We first generalize the notions of equitable partitions and almost equitable partitions to make them applicable to directed, weighted graphs. Consequently, we are enabled to characterize the controllable subspace of a given diffusively coupled multi-agent system using graph theoretic ideas. After comparing the condition to realize cluster synchronization and the condition for the network to be controllable, we conclude that those diffusively coupled multi-agent networks that are not controllable usually realize cluster synchronization asymptotically. Simulation results are provided to illustrate the theoretical results.