Localized Flow-Based Clustering in Hypergraphs

Local graph clustering algorithms are designed to efficiently detect small clusters of nodes that are biased to a localized region of a large graph. Although many techniques have been developed for local clustering in graphs, very few algorithms have been designed to detect local clusters in hypergraphs, which better model complex systems involving multiway relationships between data objects. In this paper we present a framework for local clustering in hypergraphs based on minimum cuts and maximum flows. Our approach extends previous research on flow-based local graph clustering, but has been generalized in a number of key ways. First of all, we demonstrate how to incorporate recent results on generalized hypergraph $s$-$t$ cut problems. This allows us to accommodate a wide range of different hypergraph cut functions, which can assign different penalties based on how each hyperedge is split across different clusters. Furthermore, our algorithm comes with a number of attractive theoretical properties in terms of recovering nodes sets with low hypergraph conductance and hypergraph normalized cut scores. Finally, and most importantly, our method is strongly-local, meaning that its runtime depends only on the size of an input set. In practice this allows our method to quickly find localized clusters without exploring an entire input hypergraph. We demonstrate the power of our method in local cluster detection experiments on an Amazon product hypergraph and a Stack Overflow question hypergraph. Although both datasets involve millions of nodes, millions of edges, and a large average hyperedge size, we are able to detect local clusters in a matter of a few seconds or a few minutes, depending on the size of the cluster.