Scale-Free Networks Are Ultrasmall

We study the diameter, or the mean distance between sites, in a scale-free network, having [Formula presented] sites and degree distribution [Formula presented], i.e., the probability of having [Formula presented] links outgoing from a site. In contrast to the diameter of regular random networks or small-world networks, which is known to be [Formula presented], we show, using analytical arguments, that scale-free networks with [Formula presented] have a much smaller diameter, behaving as [Formula presented]. For [Formula presented], our analysis yields [Formula presented], as obtained by Bollobas and Riordan, while for [Formula presented], [Formula presented]. We also show that, for any [Formula presented], one can construct a deterministic scale-free network with [Formula presented], which is the lowest possible diameter.