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 [Bl\"asius, 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.

翻译：双曲随机图继承了现实世界网络中的诸多特性。双曲几何结构使其具有无标度网络特性及强聚类系数，同时衍生出巨连通分支、小世界现象等性质。这为设计针对双曲随机图的简洁算法提供了动机。本文研究阈值型双曲随机图（HRG）。贪婪启发式算法在实践中被广泛采用，因其能在提供良好近似解的同时，其理论分析却往往得出相反的结论。HRG的典型案例如基于退化性的贪婪算法[Bläsius, Fischbeck; Transactions of Algorithms '24]。为弥合理论与实践的鸿沟，我们刻画了退化性参数，并提出一种针对HRG染色问题的简洁近似算法。该算法的近似比在$(2/\sqrt{3})$至$4/3$之间浮动，具体取决于模型的幂律指数。我们进一步补充了关于双曲随机图团数的新发现：退化性与团数存在显著差异，并推导出改进的团数上界。此外，我们证明HRG的核心并不构成最大团。最后通过对比分析表明，与双曲随机图密切相关的几何非齐次随机图标准模型，其退化性行为与双曲随机图存在本质差异。