Effect of the interdependence between subnets on the structural controllability of complex networks

Controllability is a hot issue in complex networks. In this paper, without changing the number and the positions of the controllers, the effect of interdependence (viz., inter-coupling relations) between two directed subnets that are controllable or uncontrollable on the structural controllability of complex networks is investigated. According to the Kalman rank criterion and PBH rank criterion, some sufficient conditions and necessary conditions for the structural controllability or uncontrollability of interdependent networks are presented. The results show that (1) if the uncontrollable and controllable nodes in one subnet are inter-coupled with the controllable and uncontrollable nodes in the other subnet, respectively, and the directions are from controllable nodes to the uncontrollable nodes, then the interdependent network may be controllable; (2) the interdependent network can be obtained structurally controllable if there are simply directed paths covering all the uncontrollable nodes of the two subnets, separately; and (3) the interdependent networks are structurally controllable no matter what kind of inter-couplings between two controllable subnets. All of these results are adaptive to undirected subnets. The final example verifies the effectiveness and correctness of the results provided in this paper.