The diameter of KPKVB random graphs

We consider a random graph model that was recently proposed as a model for complex networks by Krioukov et al. (2010). In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has previously been shown that this model has various properties associated with complex networks, including a power-law degree distribution and a strictly positive clustering coefficient. The model is specified using three parameters: the number of nodes N, which we think of as going to infinity, and alpha, nu > 0, which we think of as constant. Roughly speaking, alpha controls the power-law exponent of the degree sequence and nu the average degree. Earlier work of Kiwi and Mitsche (2015) has shown that, when alpha < 1 (which corresponds to the exponent of the power law degree sequence being < 3), the diameter of the largest component is asymptotically almost surely (a.a.s.) at most polylogarithmic in N. Friedrich and Krohmer (2015) showed it was a.a.s. Omega(log N) and improved the exponent of the polynomial in log N in the upper bound. Here we show the maximum diameter over all components is a.a.s. O(log N), thus giving a bound that is tight up to a multiplicative constant.