  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.
