Testing for Repeated Motifs and Hierarchical Structure in Stochastic Blockmodels

The rise in complexity of network data in neuroscience, social networks, and protein-protein interaction networks has been accompanied by several efforts to model and understand these data at different scales. A key multiscale network modeling technique posits hierarchical structure in the network, and by treating networks as multiple levels of subdivisions with shared statistical properties we can efficiently discover smaller subgraph primitives with manageable complexity. One such example of hierarchical modeling is the Hierarchical Stochastic Block Model, which seeks to model complex networks as being composed of community structures repeated across the network. Incorporating repeated structure allows for parameter tying across communities in the SBM, reducing the model complexity compared to the traditional blockmodel. In this work, we formulate a framework for testing for repeated motif hierarchical structure in the stochastic blockmodel framework. We describe a model which naturally expresses networks as a hierarchy of sub-networks with a set of motifs repeating across it, and we demonstrate the practical utility of the test through theoretical analysis and extensive simulation and real data.