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Latent Space Representations of Hypergraphs

Relational data describing interactions among a population arise in a multitude of disciplines, including systems biology, neuroscience and marketing. There exists a broad literature concerned of analysis of such data when the interactions are assumed to be pairwise. However, in many real-world applications the interactions may instead occur between several members of a population and, in this case, the data are more appropriately represented by a hypergraph. As an example, consider a coauthorship network where an interaction indicates which academics have contributed to a paper. It is common for a group of authors larger than two to write an article jointly, which corresponds to a hyperedge. The literature on statistical analysis of hypergraphs is relatively underdeveloped, and in this talk we introduce a model which extends the latent space approach of Hoff et al (2002) for graphs to the hypergraph setting. In this framework, the nodes of the hypergraph are assumed to lie in a low-dimensional space and the hyperedges are modeled as a function of the latent coordinates. Using a toolkit from stochastic geometry, we develop a computationally efficient model with a convenient likelihood. Furthermore, we explore and analyse the properties of this model, and use the latent space to perform predictive inference.

This is join work with Simon Lunagomez, Christopher Nemeth and Edoardo Airoldi.