Latent Space Modelling of Hypergraph Data

TL;DR

This paper introduces a novel latent space model for hypergraph data, extending traditional graph models to better capture complex multi-member interactions, with efficient likelihood computation and practical application to real datasets.

Contribution

It develops a new hypergraph latent space model connecting to computational topology, with an efficient likelihood, MCMC inference, and solutions to identifiability issues.

Findings

Model accurately captures hypergraph degree distributions

Simulation shows high flexibility of the proposed model

Application to real data demonstrates practical utility

Abstract

The increasing prevalence of relational data describing interactions among a target population has motivated a wide literature on statistical network analysis. In many applications, interactions may involve more than two members of the population and this data is more appropriately represented by a hypergraph. In this paper, we present a model for hypergraph data which extends the well established latent space approach for graphs and, by drawing a connection to constructs from computational topology, we develop a model whose likelihood is inexpensive to compute. A delayed-acceptance MCMC scheme is proposed to obtain posterior samples and we rely on Bookstein coordinates to remove the identifiability issues associated with the latent representation. We theoretically examine the degree distribution of hypergraphs generated under our framework and, through simulation, we investigate the flexibility of our model and consider estimation of predictive distributions. Finally, we explore the application of our model to two real-world datasets.