Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Robustness of scale-free spatial networks   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.

13
The categories of this paper are: quant-ph
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
22
The categories of this paper are: chao-dyn
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
20
The categories of this paper are: acc-phys
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph