Phase transitions for scaling of structural correlations in directed networks

Analysis of degree-degree dependencies in complex networks, and their impact on processes on networks requires null models, i.e. models that generate uncorrelated scale-free networks. Most models to date however show structural negative dependencies, caused by finite size effects. We analyze the behavior of these structural negative degree-degree dependencies, using rank based correlation measures, in the directed Erased Configuration Model. We obtain expressions for the scaling as a function of the exponents of the distributions. Moreover, we show that this scaling undergoes a phase transition, where one region exhibits scaling related to the natural cut-off of the network while another region has scaling similar to the structural cut-off for uncorrelated networks. By establishing the speed of convergence of these structural dependencies we are able to asses statistical significance of degree-degree dependencies on finite complex networks when compared to networks generated by the directed Erased Configuration Model.