Latent Space Representations of Hypergraphs

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 latent space distance model of Hoff et al. (2002) and, by drawing a connection to constructs from computational topology, we develop a model whose likelihood is inexpensive to compute. We obtain posterior samples via an MCMC scheme and we rely on Bookstein coordinates to remove the identifiability issues associated with the latent representation. We demonstrate that the latent space construction imposes desirable properties on the hypergraphs generated in our framework and provides a convenient visualisation of the data. Furthermore, through simulation, we investigate the flexibility of our model and consider estimating predictive distributions. Finally, we explore the application of our model to a real world co-occurrence dataset.