  Hypergraphs are a useful abstraction for modeling multiway relationships in
data, and hypergraph clustering is the task of detecting groups of closely
related nodes in such data. Graph clustering has been studied extensively, and
there are numerous methods for detecting small, localized clusters without
having to explore an entire input graph. However, there are only a few
specialized approaches for localized clustering in hypergraphs. Here we present
a framework for local hypergraph clustering based on minimizing localized ratio
cut objectives. Our framework takes an input set of reference nodes in a
hypergraph and solves a sequence of hypergraph minimum $s$-$t$ cut problems in
order to identify a nearby well-connected cluster of nodes that overlaps
substantially with the input set.
  Our methods extend graph-based techniques but are significantly more general
and have new output quality guarantees. First, our methods can minimize new
generalized notions of hypergraph cuts, which depend on specific configurations
of nodes within each hyperedge, rather than just on the number of cut
hyperedges. Second, our framework has several attractive theoretical properties
in terms of output cluster quality. Most importantly, our algorithm is
strongly-local, meaning that its runtime depends only on the size of the input
set, and does not need to explore the entire hypergraph to find good local
clusters. We use our methodology to effectively identify clusters in
hypergraphs of real-world data with millions of nodes, millions of hyperedges,
and large average hyperedge size with runtimes ranging between a few seconds
and a few minutes.
