Deducing complete selection rule set for driver nodes to guarantee networkʼ structural controllability

Structural controllability is critical for operating and controlling large-scale complex networks. In real applications, for a given network, it is always desirable to have more selections for driver nodes which make the network structurally controllable. Different from the works in complex network field where structural controllability is often used to explore the emergence properties of complex networks at a macro level, in this paper, we investigate it for control design purpose at the application level and focus on describing and obtaining the solution space for all selections of driver nodes to guarantee structural controllability. In accord with practical applications, we define the complete selection rule set as the solution space which is composed of a series of selection rules expressed by intuitive algebraic forms. It explicitly indicates which nodes must be controlled and how many nodes need to be controlled in a node set and thus is particularly helpful for freely selecting driver nodes. Based on two algebraic criteria of structural controllability, we separately develop an input-connectivity algorithm and a relevancy algorithm to deduce selection rules for driver nodes. In order to reduce the computational complexity, we propose a pretreatment algorithm to reduce the scale of networkʼ structural matrix efficiently, and a rearrangement algorithm to partition the matrix into several smaller ones. A general procedure is proposed to get the complete selection rule set for driver nodes which guarantee networkʼ structural controllability. Simulation tests with efficiency analysis of the proposed algorithms are given and the result of applying the proposed procedure to some real networks is also shown, and these all indicate the validity of the proposed procedure.