ScaleFree Networks Are Ultrasmall
Abstract
We study the diameter, or the mean distance between sites, in a scalefree 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 smallworld networks, which is known to be d∼ln(N, we show, using analytical arguments, that scalefree networks with 2<λ<3 have a much smaller diameter, behaving as d∼ln(ln(N. For λ=3, our analysis yields d∼ln(N/ln(ln(N, as obtained by Bollobas and Riordan, while for λ>3, d∼ln(N. We also show that, for any λ>2, one can construct a deterministic scalefree network with d∼ln(ln(N, which is the lowest possible diameter.
 Publication:

Physical Review Letters
 Pub Date:
 February 2003
 DOI:
 10.1103/PhysRevLett.90.058701
 arXiv:
 arXiv:condmat/0205476
 Bibcode:
 2003PhRvL..90e8701C
 Keywords:

 89.75.Hc;
 02.50.r;
 89.75.Da;
 Networks and genealogical trees;
 Probability theory stochastic processes and statistics;
 Systems obeying scaling laws;
 Condensed Matter  Disordered Systems and Neural Networks
 EPrint:
 Latex, 4 pages, 2 eps figures, small corrections, added explanations