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Abstract
Preferential attachment networks with power law exponent τ> 3 are known to exhibit a phase transition. There is a value ρ _{c}> 0 such that, for small edge densities ρ≤ ρ _{c} every component of the graph comprises an asymptotically vanishing proportion of vertices, while for large edge densities ρ> ρ _{c} there is a unique giant component comprising an asymptotically positive proportion of vertices. In this paper we study the decay in the size of the giant component as the critical edge density is approached from above. We show that the size decays very rapidly, like exp(c/ρρc) for an explicit constant c> 0 depending on the model implementation. This result is in contrast to the behaviour of the class of rankone models of scalefree networks, including the configuration model, where the decay is polynomial. Our proofs rely on the local neighbourhood approximations of Dereich and Mörters (Ann Probab 41(1):329–384, 2013) and recent progress in the theory of branching random walks (Gantert et al. in Ann Inst Henri Poincaré Probab Stat 47(1):111–129, 2011).
Original language  English 

Pages (fromto)  663703 
Number of pages  41 
Journal  Journal of Statistical Physics 
Volume  173 
Issue number  34 
Early online date  14 May 2018 
DOIs  
Publication status  Published  1 Nov 2018 
Keywords
 BarabásiAlbert model
 Killed branching random walk
 Percolation
 Preferential attachment
 Scalefree network
 Survival probability
ASJC Scopus subject areas
 Statistical and Nonlinear Physics
 Mathematical Physics
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 1 Finished

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