## The Annals of Probability

### Robustness of scale-free spatial networks

#### 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.

#### Article information

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

Dates
Revised: January 2016
First available in Project Euclid: 15 May 2017

https://projecteuclid.org/euclid.aop/1494835228

Digital Object Identifier
doi:10.1214/16-AOP1098

Mathematical Reviews number (MathSciNet)
MR3650412

Zentralblatt MATH identifier
1367.05194

Subjects
Secondary: 60C05: Combinatorial probability 90B15: Network models, stochastic

#### Citation

Jacob, Emmanuel; Mörters, Peter. Robustness of scale-free spatial networks. Ann. Probab. 45 (2017), no. 3, 1680--1722. doi:10.1214/16-AOP1098. https://projecteuclid.org/euclid.aop/1494835228

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