Optimal control of complex networks based on matrix differentiation

Finding the key node set to be connected to external control sources so as to minimize the energy for controlling a complex network, known as the minimum-energy control problem, is of critical importance but remains open. We address this critical problem where matrix differentiation is involved. To this end, the differentiation of energy/cost function with respect to the input matrix is obtained based on tensor analysis, and the Hessian matrix is compressed from a fourth-order tensor. Normalized projected gradient method (NPGM) normalized projected trust-region method (NPTM) are proposed with established convergence property. We show that NPGM is more computationally efficient than NPTM. Simulation results demonstrate satisfactory performance of the algorithms, and reveal important insights as well. Two interesting phenomena are observed. One is that the key node set tends to divide elementary paths equally. The other is that the low-degree nodes may be more important than hubs from a control point of view, indicating that controlling hub nodes does not help to lower the control energy. These results suggest a way of achieving optimal control of complex networks, and provide meaningful insights for future researches.