  A key topic in network science is the detection of intermediate or meso-scale
structures. Community, core-periphery, disassortative and other partitions
allow us to understand the organisation and function of large networks. In this
work we study under what conditions certain common meso-scale structures are
detectable using the idea of block modularity. We find that the configuration
model imposes strong restrictions on core-periphery and related structures in
directed networks. We derive inequalities expressing when such structures can
be detected under the configuration model. Nestedness is closely related to
core-periphery and is similarly restricted to only be detectable under certain
conditions. We show that these conditions are a generalisation of the
resolution limit to structures other than assortative communities. We show how
block modularity is related to the degree corrected Stochastic Block Model and
that optimisation of one can be made equivalent to the other in general.
Finally, we discuss these issues in inferential versus descriptive approaches
to meso-scale structure detection.
