Energy-Related Controllability of Signed Complex Networks With Laplacian Dynamics

This article investigates energy-related controllability of complex networks. Specifically, our objective is to establish controllability characteristics on signed complex networks, where the network units interact via neighbor-based Laplacian feedback and the network admits positive and negative edges to capture cooperative and competitive interactions among these units. The network units can be classified into leaders and followers. This article focuses on characterizing the energy-related controllability in signed networks (i.e., the energy incurred by the leaders in the control of a network). To this end, controllability Gramian-based measures are exploited to quantify the difficulty of the control problem on signed networks in terms of the required control energy. Fundamental relationships between these measures and network topology are developed via graph Laplacian to characterize energy-related controllability. It is revealed that, for structurally unbalanced signed graphs, the energy-related controllability is closely related to the diagonal entries of the inverse of the graph Laplacian. It is also discovered that structurally balanced signed graphs and their corresponding unsigned graphs have the same energy-related controllability.