Core organization of directed complex networks

TL;DR

This paper investigates the core structure of directed complex networks using a pruning algorithm, revealing a complex hierarchy of cores and their formation points through a rate equation approach.

Contribution

It generalizes the leaf-removal core concept to directed networks and characterizes the emergence and structure of nested cores using a novel analytical method.

Findings

Directed networks exhibit a hierarchy of embedded cores.

The core formation depends on the mean degree of the network.

Analytical description of core emergence in directed networks.

Abstract

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.