Interfaces and the edge percolation map of random directed networks

The traditional node percolation map of directed networks is reanalyzed in terms of edges. In the percolated phase, edges can mainly organize into five distinct giant connected components, interfaces bridging the communication of nodes in the strongly connected component and those in the in and out components. Formal equations for the relative sizes in the number of edges of these giant structures are derived for arbitrary joint degree distributions in the presence of local and two-point correlations. The uncorrelated null model is fully solved analytically and compared against simulations, finding an excellent agreement. Interfaces, and their particular conformations giving place from "hairy ball" percolation landscapes to bottleneck straits, could bring new light to the discussion of how a structure is interwoven with functionality in flow networks.