Diameter in ultra-small scale-free random graphs

It is well known that many random graphs with infinite variance degrees are ultra-small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k −(τ − 1) with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to (Formula presented.) and (Formula presented.), respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order log log n precisely when the minimal forward degree d fwd of vertices is at least 2. We identify the exact constant, which equals that of the typical distances plus 2∕ logdfwd. Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.