The diameter of KPKVB random graphs

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