Scale-Free Networks Are Ultrasmall

We study the diameter, or the mean distance between sites, in a scale-free network, having N sites and degree distribution p(k)∝k^-λ, 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<λ<3 have a much smaller diameter, behaving as d~lnlnN. For λ=3, our analysis yields d~lnN/lnlnN, as obtained by Bollobas and Riordan, while for λ>3, d~lnN. We also show that, for any λ>2, one can construct a deterministic scale-free network with d~lnlnN, which is the lowest possible diameter.