The diameter of KPKVB random graphs

Abstract

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 $\alpha ,\nu >0$ 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 has shown that when $\alpha <1$ (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.~$\Omega (\log N)$ 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.

very minor corrections since the last version

References in corpus (2)

• Hyperbolic Geometry of Complex Networks
• Sustaining the Internet with Hyperbolic Mapping

Cited by in corpus (9)

• Network Geometry
• Explosion in weighted Hyperbolic Random Graphs and Geometric Inhomogeneous Random Graphs
• Emergence of geometric Turing patterns in complex networks
• A geometry-induced topological phase transition in random graphs
• Limit theory of isolated and extreme points in hyperbolic random geometric graphs
• Network Renormalization
• Random graphs and real networks with weak geometric coupling
• Efficiently Approximating Vertex Cover on Scale-Free Networks with Underlying Hyperbolic Geometry
• Cover and Hitting Times of Hyperbolic Random Graphs