  We generate new mathematical tools with which to quantify the macroscopic
topological structure of large directed networks. This is achieved via a
statistical mechanical analysis of constrained maximum entropy ensembles of
directed random graphs with prescribed joint distributions for in- and
outdegrees and prescribed degree-degree correlation functions. We calculate
exact and explicit formulae for the leading orders in the system size of the
Shannon entropies and complexities of these ensembles, and for
information-theoretic distances. The results are applied to data on gene
regulation networks.
