Characterizing Energy-Related Controllability of Composite Complex Networks via Graph Product

This article characterizes the energy-related controllability of composite complex networks. We consider a class of composite networks constructed from simple factor networks via Cartesian product. The considered factor networks are leader-follower signed networks with neighbor-based Laplacian dynamics, adopting positive and negative edges to capture cooperative and competitive interactions among network units. Different from most existing works focusing on classical controllability, this article investigates the controllability of composite networks from energy-related perspectives. Specifically, controllability Gramian-based metrics, including average controllability and volumetric control energy, are characterized based on the Cartesian graph product, which reveals how the energy-related controllability of a composite network can be inferred from the spectral properties of the local factor systems. These results are then extended to layered control networks, a special, yet widely used, network structure in many man-made systems. Since structural balance is a key topological property of signed networks, a necessary and sufficient condition to verify the structural balance of composite signed networks is developed, which is applicable to generalized graph product.