The Hierarchical Stochastic Block Model for Multiple Networks

Abstract In many research fields, there is an increased availability of network data arising as multiple networks. However, most statistical models for network data in the literature are designed for a single network. Among these, the Stochastic Block Model is arguably the most popular model to perform vertex clustering and community detection. We propose the Hierarchical Stochastic Block Model, a generalization of the SBM to the setting of multiple networks. This model uses a Hierarchical Pitman-Yor prior for the block allocation vector of each graph. The proposed model has two main advantages: 1) it allows different networks to share the same latent blocks and the level of sharing is learnt from the data; 2) the number of blocks in each graph and the overall number of blocks are learnt from the data too, hence avoiding complicated model selection procedures. We derive both MCMC and Variational Inference algorithms. The former targets the correct posterior and is tuning-free, while the latter relies on an approximation of the posterior distribution, but is potentially more scalable than MCMC. We apply the HSBM to a co-authorship network and a brain connectomic network, to illustrate how the model is able to capture different levels of block sharing.