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)proportional tok(-lambda), 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 dsimilar tolnN, we show, using analytical arguments, that scale-free networks with 2<lambda<3 have a much smaller diameter, behaving as dsimilar tolnlnN. For lambda=3, our analysis yields dsimilar tolnN/lnlnN, as obtained by Bollobas and Riordan, while for lambda>3, dsimilar tolnN. We also show that, for any lambda>2, one can construct a deterministic scale-free network with dsimilar tolnlnN, which is the lowest possible diameter.