The diameter of KPKVB random graphs

TL;DR

This paper proves that the maximum diameter of all components in the KPKVB hyperbolic random graph model is almost surely bounded above by a constant times log N, confirming tight bounds for network diameter.

Contribution

It establishes an asymptotically tight O(log N) upper bound on the maximum component diameter in the hyperbolic random graph model, extending prior polylogarithmic bounds.

Findings

Maximum diameter over all components is O(log N) almost surely.

The bound is tight up to a constant factor.

Supports the model's relevance to complex network properties.

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.