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Abstract
Scalefree networks with moderate edge dependence experience a phase transition between ultrasmall and small world behaviour when the power law exponent passes the critical value of three. Moreover, there are laws of large numbers for the graph distance of two randomly chosen vertices in the giant component. When the degree distribution follows a pure power law these show the same asymptotic distances of logN/log logN at the critical value three, but in the ultrasmall regime reveal a difference of a factor two between the moststudied rankone and preferential attachment model classes. In this paper we identify the critical window where this factor emerges. We look at models from both classes when the asymptotic proportion of vertices with degree at least k scales like k2(log k)2α+o(1) and show that for preferential attachment networks the typical distance is (1/1+α + o(1)) logN/log logN in probability as the number N of vertices goes to infinity. By contrast the typical distance in a rank one model with the same asymptotic degree sequence is (1/1+2α + o(1)) logN/log logN. As α → ∞ we see the emergence of a factor two between the length of shortest paths as we approach the ultrasmall regime
Original language  English 

Article number  77 
Number of pages  38 
Journal  Electronic Journal of Probability 
Volume  22 
DOIs  
Publication status  Published  31 Dec 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