A bound for the diameter of random hyperbolic graphs

TL;DR

This paper establishes an upper bound on the diameter of random hyperbolic graphs for certain parameters, providing insights into their structure and answering open questions about component sizes.

Contribution

It proves a logarithmic bound on the diameter of components in random hyperbolic graphs for 1/2 < α < 1, and analyzes the size of the second largest component.

Findings

Graph diameter is O(log^{C_0+1+o(1)} n) for vertices in the same component.

Second largest component size is O(log^{2C_0+1+o(1)} n).

Existence of isolated components forming long paths of length Ω(log n).

Abstract

Random hyperbolic graphs were recently introduced by Krioukov et. al. [KPKVB10] as a model for large networks. Gugelmann, Panagiotou, and Peter [GPP12] then initiated the rigorous study of random hyperbolic graphs using the following model: for $\alpha> \tfrac{1}{2}$, $C\in\mathbb{R}$, $n\in\mathbb{N}$, set $R=2\ln n+C$ and build the graph $G=(V,E)$ with $|V|=n$ as follows: For each $v\in V$, generate i.i.d. polar coordinates $(r_{v},\theta_{v})$ using the joint density function $f(r,\theta)$, with $\theta_{v}$ chosen uniformly from $[0,2\pi)$ and $r_{v}$ with density $f(r)=\frac{\alpha\sinh(\alpha r)}{\cosh(\alpha R)-1}$ for $0\leq r< R$. Then, join two vertices by an edge, if their hyperbolic distance is at most $R$. We prove that in the range $\tfrac{1}{2} < \alpha < 1$ a.a.s. for any two vertices of the same component, their graph distance is $O(\log^{C_0+1+o(1)}n)$, where $C_0=2/(\tfrac{1}{2}-\frac{3}{4}\alpha+\tfrac{\alpha^2}{4})$, thus answering a question raised in [GPP12] concerning the diameter of such random graphs. As a corollary from our proof we obtain that the second largest component has size $O(\log^{2C_0+1+o(1)}n)$, thus answering a question of Bode, Fountoulakis and M\"{u}ller [BFM13]. We also show that a.a.s. there exist isolated components forming a path of length $\Omega(\log n)$, thus yielding a lower bound on the size of the second largest component.