The diameter of KPKVB random graphs

TL;DR

This work resolves the diameter question for the KPKVB hyperbolic random graphs in the regime $1/2<\alpha<1$ (any $\nu$) and for $\alpha=1$ with large $\nu$ by introducing a Poissonized, idealized Euclidean-embedded model $\Gamma$, discretizing it into boxes, and proving deterministic bounds via a box-walk framework. It couples the Poissonized KPKVB model to the Euclidean model and shows that short box-paths exist unless large inactive regions occur; probabilistic bounds then ensure such regions are unlikely, yielding an a.s. $O(\log N)$ diameter for all components. The results improve previous polylogarithmic bounds and establish a tight logarithmic scale for component diameters, with a method that translates to the original hyperbolic graph. The work thus reinforces the small-world nature of KPKVB graphs and provides a platform for further refinement of constants and extensions to other parameter regions.

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 $α, ν> 0$ 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.

Figures (9)

• Figure 1:
• Figure 2: An example of the Poissonized KPKVB random graph $G_{\mathop{\mathrm{Po}}\nolimits}$ (left) and the graph $\Gamma_{\alpha,\nu \alpha/\pi}$ (right), under the coupling of Lemma \ref{['lem:coupling']}. The graph $G_{\mathop{\mathrm{Po}}\nolimits}$ is drawn in the native model of the hyperbolic plane, where a point with hyperbolic polar coordinates $(r,\vartheta)$ is plotted with Euclidean polar coordinates $(r,\vartheta)$. Points are colored based on their angular coordinate. The edges for which the coupling fails are drawn in black in the picture of $G_{\mathop{\mathrm{Po}}\nolimits}$ and as dotted lines in the picture of $\Gamma_{\nu \alpha/\pi}$. The parameters used are $N=200$, $\alpha=0.8$ and $\nu = 1.3$.
• Figure 3:
• Figure 4:
• Figure 5:

Theorems & Definitions (18)

• Theorem 1
• Lemma 2: FM, Lemma 27
• Lemma 3: FM, Lemma 30
• Corollary 4
• Lemma 5
• Lemma 6
• Lemma 7: FM, Lemma 3
• Lemma 8
• Lemma 9
• Lemma 10