Controlling Directed Networks With Evolving Topologies

Exploring how network topologies affect the cost of controlling the networks is an important issue in both theory and application. However, its solution still remains open due to the difficulty in analyzing the characteristics of networks. In this paper, a matrix function optimization model is proposed to study how the network topology evolves when the objective is to achieve optimal control of directed networks. By introducing an l-chain rule to obtain the direction of network topology evolution, a normalized and projected gradient-descent method (NPGM) is developed to solve the proposed optimization model. It is proven that the NPGM linearly converges to a local minimum point. We further derive an optimality condition to determine whether a converged solution is global minimum or not, and such a condition is also verified through numerous experimental tests on directed networks. We find that a network adaptively changes its topology in such a way that many subnetworks are gradually evolved toward a preestablished control target. Our finding enables us to model and explain how real-world complex networks adaptively self-organize themselves to many similar subnetworks during a relatively long evolution process.