J. Ding, C. Wen, G. Li and Z. Chen, "Key Nodes Selection in Controlling Complex Networks via Convex Optimization," in IEEE Transactions on Cybernetics. doi: 10.1109/TCYB.2018.2888953
Key nodes are the nodes connected with a given number of external source controllers that result in minimal control cost. Finding such a subset of nodes is a challenging task since it impossible to list and evaluate all possible solutions unless the network is small. In this paper, we approximately solve this problem by proposing three algorithms step by step. By relaxing the Boolean constraints in the original optimization model, a convex problem is obtained. Then inexact alternating direction method of multipliers (IADMMs) is proposed and convergence property is theoretically established. Based on the degree distribution, an extension method named degree-based IADMM (D-IADMM) is proposed such that key nodes are pinpointed. In addition, with the technique of local optimization employed on the results of D-IADMM, we also develop LD-IADMM and the performance is greatly improved. The effectiveness of the proposed algorithms is validated on different networks ranging from Erdős-Rényi networks and scale-free networks to some real-life networks.