Sparsity-Inspired Optimal Topology Control of Complex Networks

So far there are still limited results on the role of network topology matrix in controlling complex networks with minimum control cost. In this article, we study this problem and propose an optimization model by considering the topology matrix as a variable to minimize the control cost. As sparsity nature commonly exists in many real-world networks, sparse, and non-negative constraints are included in the optimization model. Sparse projected gradient with momentum (SPGM) is proposed to solve the problem numerically. The convergence property of SPGM is theoretically established. As the optimization model is non-convex, a possible optimality condition is further derived to determine whether the converged solution is global optimal or not. Through numerous extensive simulations, we investigate the characteristics of the optimal topology with minimum control cost. It is found that several identical/similar stems are naturally evolved and the number of nodes in each stem is evenly distributed, which is also verified with examples from the cooperation problem in social networks. These findings provide a comprehensive understanding and explanation in controlling real-life networks from the control energy point of view, suggesting that some underlying universal mechanism in the formation and evolution of complex networks.