Extended Stochastic Block Models with Application to Criminal Networks

Reliably learning group structure among nodes in network data is challenging in modern applications. We are motivated by covert networks encoding relationships among criminals. These data are subject to measurement errors and exhibit a complex combination of an unknown number of core-periphery, assortative and disassortative structures that may unveil the internal architecture of the criminal organization. The coexistence of such noisy block structures limits the reliability of community detection algorithms routinely applied to criminal networks, and requires extensions of model-based solutions to realistically characterize the node partition process, incorporate information from node attributes, and provide improved strategies for estimation, uncertainty quantification, model selection and prediction. To address these goals, we develop a novel class of extended stochastic block models (ESBM) that infer groups of nodes having common connectivity patterns via Gibbs-type priors on the partition process. This choice encompasses several realistic priors for criminal networks, covering solutions with fixed, random and infinite number of possible groups, and facilitates inclusion of node attributes in a principled manner. Among the new alternatives in our class, we focus on the Gnedin process as a realistic prior that allows the number of groups to be finite, random and subject to a reinforcement process coherent with the modular structures in organized crime. A collapsed Gibbs sampler is proposed for the whole ESBM class, and refined strategies for estimation, prediction, uncertainty quantification and model selection are outlined. ESBM performance is illustrated in realistic simulations and in an application to an Italian Mafia network, where we learn key block patterns revealing a complex hierarchical structure of the organization, mostly hidden from state-of-the-art alternative solutions.