Optimizing target node set for the control energy of directed complex networks

The energy needed in controlling a complex network is a problem of practical importance. Recent works have focused on the reduction of control energy via strategic placement of driver nodes, or by decreasing the cardinality of nodes to be controlled. However, how target nodes are arranged with respect to control energy has yet been explored. Here, we propose an iterative method based on stiefel manifold optimization of selectable target node matrix to reduce control energy. We derived the matrix derivative gradient needed for the search algorithm in a general way, and searched for target nodes which result in reduced control energy, given that driver nodes placement is fixed. Our findings reveal that when the path distances from driver nodes to target nodes are minimised, control energy is optimal. We also applied the algorithm to various model and real networks. The simulation results show that when compared to heuristic selection strategies of choosing target nodes, the control energy is reduced by a few orders of magnitude. Our work may be applicable to a social network voter model, where we are interested in optimizing the control effort needed to influence a fractional number of individuals' opinions.