Controllability Gramian-based measures of graph product networks

This paper addresses the controllability measures of composite networks based on graph products, in which the graph product networks are either Kronecker product networks or Kronecker sum networks. The primary objective pursued here is to reveal the controllability Gramian-based measure links between the graph product composite network and its subnetworks. First, the analytical expression of the average controllability for the Kronecker product network is derived by using the corresponding controllability measures of its subnetworks. An upper bound of volumetric control energy is further obtained for the Kronecker product network. Then, the upper bounds are derived for the controllability measures of the Kronecker sum network, which provide a valuable reference for designing controllable networks in practical applications. Finally, the controllability of the Kronecker product network is compared with that of the Kronecker sum network, and it is found that when the subnetworks satisfy certain conditions, the Kronecker product network has better average controllability than the Kronecker sum network.