面向非脆弱超边的超核分解：概念、算法、观察与应用
Hypercore Decomposition for Non-Fragile Hyperedges: Concepts, Algorithms, Observations, and Applications
AI总结：
针对现有超图$k$-核扩展假设超边脆弱的局限，提出基于成员保留比例的$(k,t)$-超核模型，证明其线性时间复杂度，验证真实超图的结构相似性，并展示其在多类超图分析任务中的应用。
摘要：
超图是建模高阶关系的强大抽象工具，这类关系在众多领域中普遍存在。超图由节点和超边（即节点的子集）组成；已有诸多研究尝试将在成对图中应用广泛且被证实有效的$k$-核概念扩展到超图中。然而，此前的扩展均基于一个不切实际的假设：超边是脆弱的，即只要有一个成员离开，高阶关系就会失效。\n在本研究中，我们基于“只要至少$t$比例的成员保留，高阶关系就仍然成立”的假设，提出了一种名为$(k, t)$-超核的新型子结构模型。具体而言，它被定义为满足以下两个条件的极大子超图：（1）每个节点至少包含在其中的$k$条超边中；（2）每条超边中至少保留$t$比例的节点。我们首先证明，在给定$t$（或$k$）的情况下，针对所有可能的$k$（或$t$）求解$(k, t)$-超核的时间复杂度与超边大小之和呈线性关系。随后，我们证实来自同一领域的真实超图具有相似的$(k, t)$-超核结构，这些结构会根据$t$的不同捕捉到不同视角的信息。最后，我们展示了该模型在识别超图中的影响力节点、稠密子结构以及脆弱性方面的成功应用。
英文摘要：
Hypergraphs are a powerful abstraction for modeling high-order relations, which are ubiquitous in many fields. A hypergraph consists of nodes and hyperedges (i.e., subsets of nodes); and there have been a number of attempts to extend the notion of $k$-cores, which proved useful with numerous applications for pairwise graphs, to hypergraphs. However, the previous extensions are based on an unrealistic assumption that hyperedges are fragile, i.e., a high-order relation becomes obsolete as soon as a single member leaves it. In this work, we propose a new substructure model, called ($k$, $t$)-hypercore, based on the assumption that high-order relations remain as long as at least $t$ fraction of the members remain. Specifically, it is defined as the maximal subhypergraph where (1) every node is contained in at least $k$ hyperedges in it and (2) at least $t$ fraction of the nodes remain in every hyperedge. We first prove that, given $t$ (or $k$), finding the ($k$, $t$)-hypercore for every possible $k$ (or $t$) can be computed in time linear w.r.t the sum of the sizes of hyperedges. Then, we demonstrate that real-world hypergraphs from the same domain share similar ($k$, $t$)-hypercore structures, which capture different perspectives depending on $t$. Lastly, we show the successful applications of our model in identifying influential nodes, dense substructures, and vulnerability in hypergraphs.
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