Perfect Clustering in Nonuniform Hypergraphs

TL;DR

The paper addresses the lack of tractable statistical models for hypergraphs by proposing an interaction hypergraph, an interaction-centric framework that permits multiplicities and varying hyperedge sizes. It develops latent embeddings for interactions and a spectral clustering-based estimator that can achieve perfect clustering when enough interactions are observed. This approach captures nonuniform hypergraphs and intra-interaction dependence, providing a principled alternative to traditional pairwise methods. Theoretical results (proved for a restricted class) and empirical validation via simulations and real data illustrate the method's potential for high-order network inference in domains like neuroscience and communication networks. Overall, the work delivers a new exchangeable, hyperedge-centric model and a scalable spectral technique for exact clustering in complex hypergraphs.

Abstract

While there has been tremendous activity in the area of statistical network inference on graphs, hypergraphs have not enjoyed the same attention, on account of their relative complexity and the lack of tractable statistical models. We introduce a hyper-edge-centric model for analyzing hypergraphs, called the interaction hypergraph, which models natural sampling methods for hypergraphs in neuroscience and communication networks, and accommodates interactions involving different numbers of entities. We define latent embeddings for the interactions in such a network, and analyze their estimators. In particular, we show that a spectral estimate of the interaction latent positions can achieve perfect clustering once enough interactions are observed.

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