Perfect Clustering in Nonuniform Hypergraphs

2025-04-11

AI Summary

To address the lack of theoretical foundations and tractable models for perfect clustering in non-uniform hypergraphs, this paper proposes an edge-centric interaction hypergraph model that supports variable-order interaction modeling and latent variable embedding estimation. Methodologically, it integrates spectral analysis, latent embedding, and non-uniform statistical modeling to capture heterogeneous higher-order interactions arising naturally in neuroscience and communication networks. Its key contribution is the first formal definition of “interaction latent positions” and a rigorous proof that their spectral estimation achieves exact node clustering recovery under mild observability conditions. Theoretical analysis establishes precise sufficient conditions for exact clustering, overcoming the reliance of existing hypergraph clustering methods on uniformity assumptions or low-order structures. This work introduces a new paradigm for modeling complex higher-order relational data.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Modeling hypergraphs in neuroscience and communication networks
• Estimating latent embeddings for hypergraph interactions
• Achieving perfect clustering with spectral estimation

Innovation

• Hyper-edge-centric model for hypergraphs
• Latent embeddings for interaction analysis
• Spectral estimate achieves perfect clustering