Cluster-based Controllability in Structured Networks

This paper investigates cluster-based controllability in structured networks represented by directed graphs. A structural condition for identifying clusters in a given network is presented first, followed by definitions of leader and follower clusters. By distinguishing intra- and inter-cluster edges, the concept of structural controllability is extended to the cluster level. The notion of Cluster Structural Controllability (CSC) is introduced, and the analysis shows that CSC depends solely on the nonzero structure of inter-cluster edges. While intra-cluster edges ensure consensus within each cluster, they do not affect controllability; only inter-cluster edges determine whether the consensus values of follower clusters can be independently controlled. Furthermore, an inter-cluster controllability matrix is defined to describe the relationship between leader and follower cluster consensus values. It is theoretically proven that the network satisfies CSC if and only if the inter-cluster controllability matrix has full row rank.