Robustness of scale-free spatial networks

TL;DR

This paper investigates the robustness of spatially embedded scale-free networks with clustering, showing how robustness depends on the degree distribution exponent and spatial decay rate, and introduces new analytical methods for such non-tree-like networks.

Contribution

It introduces a spatial scale-free network model with tunable parameters and analyzes its robustness, revealing the interplay between degree distribution, spatial decay, and clustering.

Findings

Network is robust if τ<2+1/δ

Network fails to be robust if τ>3

Robustness depends on both degree exponent and clustering features

Abstract

A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the $d$-dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent $\tau$ of the degree distribution and the rate $\delta d$ at which the connection probability decreases with the distance of two vertices. We show that the network is robust if $\tau<2+1/\delta$, but fails to be robust if $\tau>3$. In the case of one-dimensional space we also show that the network is not robust if $\tau<2+1/(\delta-1)$. This implies that robustness of a scale-free network depends not only on its power-law exponent but also on its clustering features. Other than the classical models of scale-free networks our model is not locally tree-like, and hence we need to develop novel methods for its study, including, for example, a surprising application of the BK-inequality.