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.

中文翻译：

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在社交社区中寻找关键用户：崩溃的核心和桁架问题

在社交网络中，关键用户的离开可能会严重破坏网络参与度，即导致大量其他用户退出。衡量社交网络参与度的流行模型是$千$克-core，每个顶点至少有的最大子图 $千$克邻居。为了识别关键用户，我们建议折叠$千$克-核心问题：给定一个图 $G$G, 一个正整数 $千$克 和预算 $b$乙，我们的目标是找到 $b$乙 顶点在 $G$G 从而删除 $b$乙 顶点导致最小 $千$克-核。我们证明这个问题是 NP-hard 和不近似的。提出了一种有效的算法，显着减少了候选顶点的数量。我们还研究了用户离开的模型$千$克-truss 通过进行额外的计算 wrt 进一步考虑了连接强度 $千$克-核。我们证明相应的折叠$千$克-truss 问题也是 NP-hard 和不近似的。提出了一种有效的算法来解决这个问题。通过实验比较了两种提出的模型的优缺点。对九个现实生活社交网络的综合实验证明了我们提出的方法的有效性和效率。