Key-nodes selection problem for minimum cost control of directed networks

The key-nodes selection problem is to determine key nodes from all those connected to external control sources so that the minimum cost is achieved when they are used to control a directed network. Clearly, it is an important issue in both theory and application. But its solution still remains open because of the difficulty in analyzing the graphical properties of key nodes. We present a method, called normalized projected gradient method (NPGM), to address this critical issue. An index notation arrangement-based chain rule is proposed to obtain the gradient of a defined cost function where matrix-by-matrix derivatives are involved. In addition, projection and normalization operators are used to establish the convergence property of NPGM. Simulation results also demonstrate its satisfactory performance. We believe that the presented results of NPGM not only provide a technical breakthrough, but also our discussions on them reveal certain important physical insights regarding key nodes selection in controlling complex directed networks.