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Abstract
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 ddimensional torus and are connected to existing vertices with a
probability favouring short spatial distances and high degrees. In this model
of a scalefree 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 onedimensional space we also show that the network is not robust if tau<2+1/(delta1). This implies that robustness of a
scalefree network depends not only on its powerlaw exponent but also on its
clustering features. Other than the classical models of scalefree networks our
model is not locally treelike, and hence we need to develop novel methods for
its study, including, for example, a surprising application of the BKinequality.
Original language  English 

Pages (fromto)  16801722 
Number of pages  35 
Journal  Annals of Probability 
Volume  45 
Issue number  3 
Early online date  15 May 2017 
DOIs  
Publication status  Published  31 May 2017 
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 1 Finished

Emergence of Condensation in Stochastic Systems
Morters, P.
Engineering and Physical Sciences Research Council
1/08/13 → 31/08/16
Project: Research council