Network and interaction models for data with hierarchical granularity via fragmentation and coagulation

We introduce a nested family of Bayesian nonparametric models for network and interaction data with a hierarchical granularity structure that naturally arises through finer and coarser population labelings. In the case of network data, the structure is easily visualized by merging and shattering vertices, while respecting the edge structure. We further develop Bayesian inference procedures for the model family, and apply them to synthetic and real data. The family provides a connection of practical and theoretical interest between the Hollywood model of Crane and Dempsey, and the generalized-gamma graphex model of Caron and Fox. A key ingredient for the construction of the family is fragmentation and coagulation duality for integer partitions, and for this we develop novel duality relations that generalize those of Pitman and Dong, Goldschmidt and Martin. The duality is also crucially used in our inferential procedures.