Finding Critical Users in Social Communities: The Collapsed Core and Truss Problems

In social networks, the leave of critical users may significantly break network engagement, i.e., lead a large number of other users to drop out. A popular model to measure social network engagement is k-core, the maximal subgraph in which every vertex has at least k neighbors. To identify critical users, we propose the collapsed k-core problem: given a graph G, a positive integer k and a budget b, we aim to find b vertices in G such that the deletion of the b vertices leads to the smallest k-core. We prove the problem is NP-hard and inapproximate. An efficient algorithm is proposed, which significantly reduces the number of candidate vertices. We also study the user leave towards the model of k-truss which further considers tie strength by conducting additional computation w.r.t. k-core. We prove the corresponding collapsed k-truss problem is also NP-hard and inapproximate. An efficient algorithm is proposed to solve the problem. The advantages and disadvantages of the two proposed models are experimentally compared. Comprehensive experiments on 9 real-life social networks demonstrate the effectiveness and efficiency of our proposed methods.