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The diameter of KPKVB random graphs

We consider a model for complex networks that was recently proposed as a model for complex networks by Krioukov et al. 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 been previously 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 $α, ν> 0$ which we think of as constant. Roughly speaking $α$ controls the power law exponent of the degree sequence and $ν$ the average degree. Earlier work of Kiwi and Mitsche has shown that when $α< 1$ (which corresponds to the exponent of the power law degree sequence being $< 3$) then the diameter of the largest component is a.a.s.~polylogarithmic in $N$. Friedrich and Krohmer have shown it is a.a.s.~$Ω(\log N)$ and they 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.