Generative hypergraph models and spectral embedding.

Area of Science:

• Graph theory
• Network science
• Data analysis

Background:

• Complex systems often involve interactions among more than two agents, a phenomenon not fully captured by traditional graph theory.
• Hypergraphs provide a framework to model these higher-order interactions using hyperedges connecting multiple nodes.
• Embedding hypergraphs into low-dimensional space is crucial for tasks like node reordering, clustering, and visualization.

Purpose of the Study:

• To develop and analyze spectral embedding algorithms tailored for hypergraphs.
• To associate these algorithms with generative hypergraph models that promote short-range interactions.
• To enable quantification of linear and periodic structures and improve downstream tasks like hyperedge prediction.

Main Methods:

• Customized two spectral embedding algorithms for hypergraphs, one for linear and one for periodic structures.
• Development of generative hypergraph models linked to spectral embedding algorithms.
• Application of maximum likelihood estimation to quantify structural components.

Main Results:

• Demonstrated effectiveness of hypergraph embedding on synthetic and real-world data.
• Showcased improved performance in clustering and hyperedge prediction compared to dyadic (pairwise) methods.
• Achieved superior results in triadic edge prediction on contact hypergraphs, especially with limited training data.

Conclusions:

• Spectral hypergraph embedding offers a powerful approach for analyzing complex systems with higher-order interactions.
• The associated generative models enhance interpretability and provide metrics for structural analysis and prediction.
• Hypergraph-based methods outperform traditional dyadic approaches in specific network analysis and prediction tasks.