Distances in scale free networks at criticality

Abstract: Scale-free 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 $\frac{\log N}{\log\log N}$ at the critical value three, but in the ultrasmall regime reveal a difference of a factor two between the most-studied rank-one 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 $k^{-2} (\log k)^{2\alpha + o(1)}$ and show that for preferential attachment networks the typical distance is $\big(\frac{1}{1+\alpha}+o(1)\big)\frac{\log N}{\log\log N}$ 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 $\big(\frac{1}{1+2\alpha}+o(1)\big)\frac{\log N}{\log\log N}.$ As $\alpha\to\infty$ we see the emergence of a factor two between the length of shortest paths as we approach the ultrasmall regime.