A stochastic block model for hypergraphs

Over the past few decades a broad variety of models has been developed for graphs. However, modern applications in various fields highlighted the need to account for higher-order interactions, to include the information deriving from groups of three or more nodes. Simple examples include group interactions in social networks, scientific co-authorship, interactions between more than two species in ecological models or high-order correlations between neurons in brain networks. Hypergraphs provide the most general formalization of higher-order interactions: similarly to a graph, a hypergraph is defined as a set of nodes and a set of hyperedges, the latter specifying nodes taking part in each interaction. We propose a stochastic block model for hypergraphs to perform model-based clustering, capturing the information deriving from higher-order interactions. The formulation is sufficiently flexible to account for possible simplified latent structures. A variational expectation-maximization algorithm is developed to perform parameter estimation and model selection is explored using the ICL criterion. The model is applied to both simulated and real data, and the performance of the proposal is assessed in terms of parameter estimation and ability to recover the clusters. The estimation algorithm was implemented in C++ language and it was made available for the R software.