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 $k$k -core, the maximal subgraph in which every vertex has at least $k$k neighbors. To identify critical users, we propose the collapsed $k$k -core problem: given a graph $G$G , a positive integer $k$k and a budget $b$b , we aim to find $b$b vertices in $G$G such that the deletion of the $b$b vertices leads to the smallest $k$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 $k$k -truss which further considers tie strength by conducting additional computation w.r.t. $k$k -core. We prove the corresponding collapsed $k$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 nine real-life social networks demonstrate the effectiveness and efficiency of our proposed methods.