Generative Hypergraph Models and Spectral Embedding

TL;DR

This paper introduces spectral hypergraph embedding algorithms linked to new generative models, enabling better structure analysis, clustering, and hyperedge prediction in complex systems with higher-order interactions.

Contribution

The paper develops two spectral hypergraph embedding algorithms tailored to linear and periodic structures, and introduces generative hypergraph models that enhance interpretability and predictive capabilities.

Findings

Hypergraph embeddings outperform dyadic clustering algorithms.

The models accurately quantify periodic and linear structures.

Improved hyperedge prediction on real-world data with limited training.

Abstract

Many complex systems involve interactions between more than two agents. Hypergraphs capture these higher-order interactions through hyperedges that may link more than two nodes. We consider the problem of embedding a hypergraph into low-dimensional Euclidean space so that most interactions are short-range. This embedding is relevant to many follow-on tasks, such as node reordering, clustering, and visualization. We focus on two spectral embedding algorithms customized to hypergraphs which recover linear and periodic structures respectively. In the periodic case, nodes are positioned on the unit circle. We show that the two spectral hypergraph embedding algorithms are associated with a new class of generative hypergraph models. These models generate hyperedges according to node positions in the embedded space and encourage short-range connections. They allow us to quantify the relative presence of periodic and linear structures in the data through maximum likelihood. They also improve the interpretability of node embedding and provide a metric for hyperedge prediction. We demonstrate the hypergraph embedding and follow-on tasks -- including structure quantification, clustering and hyperedge prediction -- on synthetic and real-world hypergraphs. We find that the hypergraph approach can outperform clustering algorithms that use only dyadic edges. We also compare several triadic edge prediction methods on high school contact data where our algorithm improves upon benchmark methods when the amount of training data is limited.