2002/05/31 by Reuven Cohen, Shlomo Havlin · 2 citations
Physics and Astronomy · #cond-mat.dis-nn
paper · pdf · doi:10.1103/physrevlett.90.058701
published as Phys. Rev. Lett 90, 058701 (2003) · Latex, 4 pages, 2 eps figures, small corrections, added explanations
arxiv created 2003/02/05 · arxiv updated 2009/11/30
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.