Tailored graph ensembles as proxies or null models for real networks II:   results on directed graphs
  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. 
E.S. Roberts, A.C.C. Coolen, and T. Schlitt
q-bio.QM cond-mat.dis-nn cs.SI physics.soc-ph