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

2024-10-15

arXiv.org

AI Summary

This work addresses the theoretical gap between clique number and degeneracy in hyperbolic random graphs (HRGs) and its implications for graph coloring. Using hyperbolic geometric modeling, stochastic analysis of random graphs, and constructive degeneracy ordering, we establish the first rigorous proof that HRGs exhibit asymptotically strictly greater degeneracy than clique number—demonstrating that their core subgraph structure is not governed by maximum cliques. We derive a precise asymptotic characterization of degeneracy and tighten the upper bound on clique number to an optimal form. Leveraging these results, we design a greedy coloring algorithm achieving an approximation ratio of $(2/sqrt{3}) - 4/3 approx 1.154$, the best known for HRGs. Furthermore, we uncover a fundamental structural distinction between HRGs and geometric inhomogeneous random graphs in terms of degeneracy, revealing the unique role of hyperbolic geometry in shaping degeneracy in sparse networks.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 [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.

Problem

• Characterize degeneracy for coloring hyperbolic random graphs.
• Provide new insights on the clique number of HRGs.
• Compare degeneracy between HRGs and geometric inhomogeneous random graphs.

Innovation

• Hyperbolic random graphs analysis
• Degeneracy-based greedy algorithms
• Improved upper bound clique number