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.

翻译：超图是对高阶关系进行建模的强大抽象，此类关系在诸多领域中普遍存在。超图由节点和超边（即节点的子集）构成；已有大量研究尝试将$k$-核的概念（该概念在成对图中因众多应用而被证明非常有用）扩展到超图中。然而，先前的扩展基于一个不现实的假设，即超边是脆弱的——一旦单个成员离开，高阶关系即失效。在本工作中，我们提出一种新的子结构模型，称为$(k, t)$-超核，其基于如下假设：只要至少$t$比例的成员保留，高阶关系就得以维持。具体而言，该模型定义为满足以下条件的最大子超图：(1) 其中每个节点至少包含在$k$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，给定$t$（或$k$），对于所有可能的$k$（或$t$），求解$(k, t)$-超核可在与超边大小之和呈线性关系的时间复杂度内完成。随后，我们展示来自同一领域的真实超图共享相似的$(k, t)$-超核结构，这些结构根据$t$的不同捕捉不同的视角。最后，我们展示该模型在识别超图中的影响力节点、稠密子结构及脆弱性方面的成功应用。