Scale-Free Networks are Ultrasmall

TL;DR

This paper analyzes the diameter of scale-free networks, revealing they are ultrashort with diameters scaling as lnlnN for 2<a<3, which is significantly smaller than traditional networks, and provides deterministic constructions achieving minimal diameter.

Contribution

It analytically characterizes the diameter scaling in scale-free networks for different degree exponents and introduces a deterministic construction with minimal diameter.

Findings

Scale-free networks with 2<a<3 have diameter ~ lnlnN.

For a=3, diameter scales as lnN/lnlnN.

Deterministic networks can achieve diameter ~ lnlnN for a>2.

Abstract

We study the diameter, or the mean distance between sites, in a scale-free network, having N sites and degree distribution p(k) ~ k^-a, i.e. the probability of having k links outgoing from a site. In contrast to the diameter of regular random networks or small world networks which is known to be d ~ lnN, we show, using analytical arguments, that scale free networks with 2<a<3 have a much smaller diameter, behaving as d ~ lnlnN. For a=3, our analysis yields d ~ lnN/lnlnN, as obtained by Bollobas and Riordan, while for a>3, d ~ lnN. We also show that, for any a>2, one can construct a deterministic scale free network with d ~ lnlnN, and this construction yields the lowest possible diameter.