Diameters in preferential attachment models

In this paper, we investigate the diameter in preferential attachment (PA-) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PA-models. There is a substantial amount of literature proving that, quite generally, PA-graphs possess power-law degree sequences with a power-law exponent t>2. We prove that the diameter of the PA-model is bounded above by a constant times log¿t, where t is the size of the graph. When the power-law exponent t exceeds 3, then we prove that log¿t is the right order, by proving a lower bound of this order, both for the diameter as well as for the typical distance. This shows that, for t>3, distances are of the order log¿t. For t¿(2,3), we improve the upper bound to a constant times log¿log¿t, and prove a lower bound of the same order for the diameter. Unfortunately, this proof does not extend to typical distances. These results do show that the diameter is of order log¿log¿t. These bounds partially prove predictions by physicists that the typical distance in PA-graphs are similar to the ones in other scale-free random graphs, such as the configuration model and various inhomogeneous random graph models, where typical distances have been shown to be of order log¿log¿t when t¿(2,3), and of order log¿t when t>3. Keywords Small-world networks - Preferential attachment models - Distances in random graphs - Universality