Core organization of directed complex networks

The recursive removal of leaves (dead end vertices) and their neighbors from an undirected network results, when this pruning algorithm stops, in a so-called core of the network. This specific subgraph should be distinguished from $k$-cores, which are principally different subgraphs in networks. If the vertex mean degree of a network is sufficiently large, the core is a giant cluster containing a finite fraction of vertices. We find that generalization of this pruning algorithm to directed networks provides a significantly more complex picture of cores. By implementing a rate equation approach to this pruning procedure for directed uncorrelated networks, we identify a set of cores progressively embedded into each other in a network and describe their birth points and structure.