The Annals of Probability

Robustness of scale-free spatial networks

Emmanuel Jacob and Peter Mörters

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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.

Article information

Ann. Probab., Volume 45, Number 3 (2017), 1680-1722.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60C05: Combinatorial probability 90B15: Network models, stochastic

Spatial network scale-free network clustering Barabási–Albert model preferential attachment geometric random graph power-law giant component robustness phase transition continuum percolation disjoint occurrence BK-inequality


Jacob, Emmanuel; Mörters, Peter. Robustness of scale-free spatial networks. Ann. Probab. 45 (2017), no. 3, 1680--1722. doi:10.1214/16-AOP1098.

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