A Bayesian Nonparametric Stochastic Block Model for Directed Acyclic Graphs

Random graphs have been widely used in statistics, for example in network and social interaction analysis. In some applications, data may contain an inherent hierarchical ordering among its vertices, which prevents any directed edge between pairs of vertices that do not respect this order. For example, in bibliometrics, older papers cannot cite newer ones. In such situations, the resulting graph forms a Directed Acyclic Graph. In this article, we propose an extension of the popular Stochastic Block Model (SBM) to account for the presence of a latent hierarchical ordering in the data. The proposed approach includes a topological ordering in the likelihood of the model, which allows a directed edge to have positive probability only if the corresponding pair of vertices respect the order. This latent ordering is treated as an unknown parameter and endowed with a prior distribution. We describe how to formalize the model and perform posterior inference for a Bayesian nonparametric version of the SBM in which both the hierarchical ordering and the number of latent blocks are learnt from the data. Finally, an illustration with a real-world dataset from bibliometrics is presented. Additional supplementary materials are available online.