Hyperbolic Random Graphs: Clique Number and Degeneracy with Implications for Colouring

TL;DR

This work analyzes colouring, clique number, and degeneracy in threshold hyperbolic random graphs (HRGs). By introducing the inner-neighborhood concept and leveraging hyperbolic geometry, it derives tight degeneracy bounds in terms of the core size $\sigma(G)$ and provides a linear-time coloring algorithm with ratio between $2/\sqrt{3}$ and $4/3$ depending on $\alpha$. It also proves a constant gap between the clique number $\omega(G)$ and degeneracy $\kappa(G)$, and shows that the largest cliques are not confined to the core, with an upper bound on $\omega(G)$ that scales with $\sigma(G)$. The paper extends the discussion to GIRGs, demonstrating model-specific differences in degeneracy, and culminates with open questions about optimal chromaticity and the precise asymptotics of clique-core relationships in HRGs.

Abstract

Hyperbolic random graphs inherit many properties that are present in real-world networks. The hyperbolic geometry imposes a scale-free network with a strong clustering coefficient. Other properties like a giant component, the small world phenomena and others follow. This motivates the design of simple algorithms for hyperbolic random graphs. In this paper we consider threshold hyperbolic random graphs (HRGs). Greedy heuristics are commonly used in practice as they deliver a good approximations to the optimal solution even though their theoretical analysis would suggest otherwise. A typical example for HRGs are degeneracy-based greedy algorithms [Bläsius, Fischbeck; Transactions of Algorithms '24]. In an attempt to bridge this theory-practice gap we characterise the parameter of degeneracy yielding a simple approximation algorithm for colouring HRGs. The approximation ratio of our algorithm ranges from $(2/\sqrt{3})$ to $4/3$ depending on the power-law exponent of the model. We complement our findings for the degeneracy with new insights on the clique number of hyperbolic random graphs. We show that degeneracy and clique number are substantially different and derive an improved upper bound on the clique number. Additionally, we show that the core of HRGs does not constitute the largest clique. Lastly we demonstrate that the degeneracy of the closely related standard model of geometric inhomogeneous random graphs behaves inherently different compared to the one of hyperbolic random graphs.

Figures (2)

• Figure 1: Results on the degeneracy $\kappa(G)\xspace$ and the size of the largest clique $\omega(G)$ in hyperbolic random graphs (HRG) and geometric inhomogeneous random graphs (GIRG). The bounds hold w.e.h.p. and are stated in comparison to the core size $\sigma(G)$. Each curve represents the multiplicative factor in front of $\sigma(G)\xspace$ for $\kappa(G)\xspace$ and $\omega(G)\xspace$ (y-axis) depending on the parameter $\alpha \in (1/2, 1)$ (x-axis). Prior work is listed with white background, whereas our results are listed with blue background.
• Figure 2: Illustration of the inner-neighbourhood. (a) The pink area is the ball $\mathcal{B}_0(r)$. The hatched area $\mathcal{I}_u = \mathcal{B}_u(R) \cap \mathcal{B}_0(r)$ is the inner-ball of $u$. The vertices $V \cap \mathcal{I}_u$ form the inner-neighbourhood $\mathop{\mathrm{\Gamma}}\nolimits(u)$. (b) Sketch of proof of \ref{['lem:deg-lower-inner']}. Vertex $u^*$ is the vertex with the largest inner-degree. Each vertex $v \in U = V \cap \mathcal{B}_0(r^*)$ has at least nearly as many neighbours within $\mathcal{B}_0(r^*)$ as $|\mathop{\mathrm{\Gamma}}\nolimits(u^*\xspace)|$ (w.e.h.p.)

Theorems & Definitions (32)

• Theorem : Simplified version of \ref{['the:clique-upper-bound']}
• Theorem : Simplified version of \ref{['the:degeneracy-upper', 'the:degeneracy-upper']}
• Theorem : Simplified version of \ref{['the:clique-deg-gap']}
• Theorem 3: Chernoff bound
• Corollary 5
• Lemma 6
• Lemma 7: Volume of the inner-ball
• Lemma 8
• Theorem 9: Bounds on the degeneracy
• Corollary 10: Bounds on the chromatic number