A Bayesian Nonparametric Stochastic Block Model for Directed Acyclic Graphs

Random graphs have been widely used in statistics, for example in network analysis and graphical models. In some applications, the data may contain an inherent hierarchical ordering among its vertices, which prevents directed edges 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 extend the Stochastic Block Model (SBM) to account for the presence of such ordering in the data, ignoring which can lead to biased estimates of the number of blocks. The proposed approach includes in the model likelihood a topological ordering, which 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 real-world datasets from bibliometrics is presented. Supplementary materials for this article are available online.