  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.
