Detectability constraints on meso-scale structure in complex networks

Community, core-periphery, disassortative and other node partitions allow us to understand the organisation and function of large networks. In this work we study common meso-scale structures using the idea of block modularity. We find that the configuration model imposes strong restrictions on core-periphery and related structures in directed and undirected networks. We derive inequalities expressing when such structures can be detected under the configuration model which are closely related to the resolution limit. Nestedness is closely related to core-periphery and is similarly restricted to only be detectable under certain conditions. We then derive a general equivalence between optimising block modularity and maximum likelihood estimation of the parameters of the degree corrected Stochastic Block Model. This allows us to contrast the two approaches, how they formalise the structure detection problem and understand these constraints in inferential versus descriptive approaches to meso-scale structure detection.