Robustness of Spatial Preferential Attachment Networks

We study robustness under random attack for a class of networks, in which new nodes are given a spatial position and connect to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering one can independently tune the power law exponent tau > 2 of the degree distribution and a parameter delta > 1 determining the decay rate of the probability of long edges. We argue that the network is robust if omega < 2 + 1/delta, but fails to be robust if tau > 2 + 1/delta-1. Hence robustness depends not only on the power-law exponent but also on the clustering features of the network.