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May 2017 Robustness of scale-free spatial networks
Emmanuel Jacob, Peter Mörters
Ann. Probab. 45(3): 1680-1722 (May 2017). DOI: 10.1214/16-AOP1098

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 $\tau<2+\frac{1}{\delta}$, 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 $\tau>2+\frac{1}{\delta-1}$. 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.

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Emmanuel Jacob. Peter Mörters. "Robustness of scale-free spatial networks." Ann. Probab. 45 (3) 1680 - 1722, May 2017. https://doi.org/10.1214/16-AOP1098

Information

Received: 1 April 2015; Revised: 1 January 2016; Published: May 2017
First available in Project Euclid: 15 May 2017

zbMATH: 1367.05194
MathSciNet: MR3650412
Digital Object Identifier: 10.1214/16-AOP1098

Subjects:
Primary: 05C80
Secondary: 60C05, 90B15

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 3 • May 2017
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