Robustness of scale-free spatial networks

Emmanuel Jacob, Peter Mörters
2017 Annals of Probability  
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 τ of the degree distribution
more » ... degree distribution and the rate −δd at which the connection probability decreases with the distance of two vertices. We show that the network is robust if τ < 2 + 1 δ , but fails to be robust if τ > 3. In the case of one-dimensional space we also show that the network is not robust if τ > 2 + 1 δ−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. MSc Classification: Primary 05C80 Secondary 60C05, 90B15.
doi:10.1214/16-aop1098 fatcat:qe5n5xikgnc5fnujfbzcenpluu