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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 andhyperedges (i.e., subsets of nodes); and there have been a number of attemptsto extend the notion of $k$-cores, which proved useful with numerousapplications for pairwise graphs, to hypergraphs. However, the previousextensions are based on an unrealistic assumption that hyperedges are fragile,i.e., a high-order relation becomes obsolete as soon as a single member leavesit.  In this work, we propose a new substructure model, called ($k$,$t$)-hypercore, based on the assumption that high-order relations remain aslong as at least $t$ fraction of the members remain. Specifically, it isdefined as the maximal subhypergraph where (1) every node is contained in atleast $k$ hyperedges in it and (2) at least $t$ fraction of the nodes remain inevery 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 linearw.r.t the sum of the sizes of hyperedges. Then, we demonstrate that real-worldhypergraphs from the same domain share similar ($k$, $t$)-hypercore structures,which capture different perspectives depending on $t$. Lastly, we show thesuccessful applications of our model in identifying influential nodes, densesubstructures, and vulnerability in hypergraphs.

Hypercore Decomposition for Non-Fragile Hyperedges: Concepts, Algorithms, Observations, and Applications