Core percolation on complex networks

TL;DR

This paper develops an analytical framework for studying core percolation in complex networks with arbitrary degree distributions, revealing differences between directed and undirected networks and applying findings to real-world data.

Contribution

It extends core percolation theory beyond Erdős-Rényi networks to arbitrary degree distributions, including scale-free networks, and analyzes the nature of the transition.

Findings

Purely scale-free networks lack a core for any degree exponent.

Core percolation is continuous in undirected networks and discontinuous in directed networks with different in- and out-degree distributions.

Real-world directed networks often have larger cores than random models.

Abstract

As a fundamental structural transition in complex networks, core percolation is related to a wide range of important problems. Yet, previous theoretical studies of core percolation have been focusing on the classical Erd\H{o}s-R\'enyi random networks with Poisson degree distribution, which are quite unlike many real-world networks with scale-free or fat-tailed degree distributions. Here we show that core percolation can be analytically studied for complex networks with arbitrary degree distributions. We derive the condition for core percolation and find that purely scale-free networks have no core for any degree exponents. We show that for undirected networks if core percolation occurs then it is always continuous while for directed networks it becomes discontinuous when the in- and out-degree distributions are different. We also apply our theory to real-world directed networks and find, surprisingly, that they often have much larger core sizes as compared to random models. These findings would help us better understand the interesting interplay between the structural and dynamical properties of complex networks.