k-core organization of complex networks

TL;DR

This paper analytically investigates the structure and robustness of k-cores in complex networks, revealing phase transition behaviors and conditions for ultra-robustness against random damage.

Contribution

It provides a detailed analytical description of k-core structures, their emergence thresholds, and robustness properties in uncorrelated networks.

Findings

K-cores are successively enclosed substructures with specific thresholds.

Emergence of k-cores can be a hybrid phase transition.

Networks with diverging second-nearest neighbors contain infinitely many ultra-robust k-cores.

Abstract

We analytically describe the architecture of randomly damaged uncorrelated networks as a set of successively enclosed substructures -- k-cores. The k-core is the largest subgraph where vertices have at least k interconnections. We find the structure of k-cores, their sizes, and their birth points -- the bootstrap percolation thresholds. We show that in networks with a finite mean number z_2 of the second-nearest neighbors, the emergence of a k-core is a hybrid phase transition. In contrast, if z_2 diverges, the networks contain an infinite sequence of k-cores which are ultra-robust against random damage.