Optimizing the controllability index of directed networks with the fixed number of control nodes

Abstract

The studies on the controllability of complex networks, arising from natural, social, and man-made-engineered systems, have attracted great attention from both network community and control community. With the fixed number of control nodes, it is of great significance in both academic research and industrial applications to design the optimal control configurations of a directed network to make its controllability index (i.e., the maximum dimension of the controllable subnetwork) maximum. In this paper, a design strategy for the optimal control configurations is proposed, and the results of the experiments conducted on multiple real and model networks show the effectiveness of this design strategy compared to other commonly-used design strategies. Moreover, we have two interesting findings in the macroscopic level and the microscopic level, respectively: (1) the dense and homogeneous networks have larger controllability indexes than the sparse and heterogeneous ones; (2) the average in-degree of the controlled state nodes in the optimal control configurations is far less than that of the network, which provides us a heuristic way to design a sub-optimal control configuration. These findings are helpful to further our understanding on the interplay between the network structure and its control.