  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.
