  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.
