Hypercore decomposition for non-fragile hyperedges: concepts, algorithms, observations, and applications

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 \({\varvec{k}}\)-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 \({\varvec{(k,t)}}\)-hypercore, based on the assumption that high-order relations remain as long as at least t fraction of the members remains. Specifically, it is defined as the maximal subhypergraph where (1) every node is contained in at least \({\varvec{k}}\) hyperedges in it and (2) at least \({\varvec{t}}\) fraction of the nodes remain in every hyperedge. We first prove that, given \({\varvec{t}}\) (or \({\varvec{k}}\)), finding the \({\varvec{(k,t)}}\)-hypercore for every possible \({\varvec{k}}\) (or \({\varvec{t}}\)) 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 \({\varvec{(k,t)}}\)-hypercore structures, which capture different perspectives depending on \({\varvec{t}}\). Lastly, we show the successful applications of our model in identifying influential nodes, dense substructures, and vulnerability in hypergraphs.