On the diameter of hyperbolic random graphs

TL;DR

This paper investigates the diameter of hyperbolic random graphs, providing simpler proofs for an improved upper bound and establishing a lower bound, enhancing understanding of their structural properties.

Contribution

The paper introduces two simpler proofs for the upper bound of hyperbolic random graph diameters and establishes a new lower bound, improving previous bounds significantly.

Findings

Upper bound of diameter is 60a;(7e2; \u221a; log n)^{2/(3-eta)}

Lower bound of diameter is 44d; log n

Simpler proofs for diameter bounds

Abstract

Large real-world networks are typically scale-free. Recent research has shown that such graphs are described best in a geometric space. More precisely, the internet can be mapped to a hyperbolic space such that geometric greedy routing performs close to optimal (Bogun\'a, Papadopoulos, and Krioukov. Nature Communications, 1:62, 2010). This observation pushed the interest in hyperbolic networks as a natural model for scale-free networks. Hyperbolic random graphs follow a power-law degree distribution with controllable exponent $\beta$ and show high clustering (Gugelmann, Panagiotou, and Peter. ICALP, pp. 573-585, 2012). For understanding the structure of the resulting graphs and for analyzing the behavior of network algorithms, the next question is bounding the size of the diameter. The only known explicit bound is $\mathcal O((\log n)^{32/((3-\beta)(5-\beta)) + 1})$ (Kiwi and Mitsche. ANALCO, pp. 26-39, 2015). We present two much simpler proofs for an improved upper bound of $\mathcal O((\log n)^{2/(3-\beta)})$ and a lower bound of $\Omega(\log n)$.