Hypercore Decomposition for Non-Fragile Hyperedges: Concepts, Algorithms, Observations, and Applications
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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 has degree at least k 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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