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 . of the degree distribution and the rate ..d at which the connection probability decreases with the distance of two vertices. We show that the network is robust if .<2+1., but fails to be robust if .>3. In the case of one-dimensional space, we also show that the network is not robust if .>2+1..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.