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.
Highlights
Background and MotivationScale-free networks are characterised by the fact that, as the network size goes to infinity, the asymptotic proportion of nodes with degree at least k behaves like k−τ+o(1) for some power law exponent τ
We look at models from both classes when the asymptotic proportion of vertices with degree at least k scales like k−2(log k)2α+o(1) and show that for preferential attachment networks the typical distance is in probability as the number N of vertices goes to infinity
There are a number of mathematical models for scale-free networks, in the class of rank-one models the probability that two vertices are directly connected is asymptotically equivalent to the product of suitably defined weights wv associated to the vertices v in a network GN with vertex set [N ] := {1, . . . , N }
Summary
Scale-free networks are characterised by the fact that, as the network size goes to infinity, the asymptotic proportion of nodes with degree at least k behaves like k−τ+o(1) for some power law exponent τ. It came as a surprise when a finer analysis in [DMM12] showed that in the ultrasmall regime, i.e. when the power law exponent is in the range 2 < τ < 3, distances in preferential attachment models are twice as long as in the rank one models above when they have the same tail of the degree distribution This is due to the fact that two vertices of high degree in the preferential attachment model are much more likely to be connected by a path of length two, rather than a single edge as in the rank one models. Our main result shows that typical distances in the preferential attachment networks are bigger by an asymptotic factor of (1 + 2α)/(1 + α), which converges to two as α ↑ ∞
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