Geometric Representations of Random Hypergraphs

TL;DR

This paper introduces a geometric parametrization of hypergraphs in Euclidean space, enabling flexible prior distributions and advanced inference methods for hypergraph structures, with applications demonstrated through simulations.

Contribution

It develops a novel geometric approach to hypergraph representation that allows for richer prior modeling and inference of complex multivariate dependencies.

Findings

Supports inference of hypergraph-based Markov structures

Enables control over graph feature distributions

Introduces new MCMC algorithms for hypergraph sampling

Abstract

A parametrization of hypergraphs based on the geometry of points in $\mathbf{R}^d$ is developed. Informative prior distributions on hypergraphs are induced through this parametrization by priors on point configurations via spatial processes. This prior specification is used to infer conditional independence models or Markov structure of multivariate distributions. Specifically, we can recover both the junction tree factorization as well as the hyper Markov law. This approach offers greater control on the distribution of graph features than Erd\"os-R\'enyi random graphs, supports inference of factorizations that cannot be retrieved by a graph alone, and leads to new Metropolis\slash Hastings Markov chain Monte Carlo algorithms with both local and global moves in graph space. We illustrate the utility of this parametrization and prior specification using simulations.