Scalable Sample-to-Population Estimation of Hyperbolic Space Models for Hypergraphs

Hypergraphs are useful mathematical representations of overlapping and nested subsets of interacting units, including groups of genes or brain regions, economic cartels, political or military coalitions, and groups of products that are purchased together. Despite the vast range of applications, the statistical analysis of hypergraphs is challenging: There are many hyperedges of small and large sizes, and hyperedges can overlap or be nested. We develop a novel statistical approach to hypergraphs with overlapping and nested hyperedges of varying sizes and levels of sparsity, which is amenable to scalable sample-to-population estimation with non-asymptotic theoretical guarantees. First, we introduce a probabilistic framework that embeds the units of a hypergraph in an unobserved hyperbolic space capturing core-periphery structure along with local structure in hypergraphs. Second, we develop scalable manifold optimization algorithms for learning hyperbolic space models based on samples from a hypergraph. Third, we show that the positions of units are identifiable (up to rotations) and provide non-asymptotic theoretical guarantees based on samples from a hypergraph. We use the framework to detect core-periphery structure along with proximity among U.S. politicians based on historical media reports.