On Distributed Colouring of Hyperbolic Random Graphs

2025-05-25

AI Summary

This paper investigates distributed graph coloring on hyperbolic random graphs—a geometric graph model capturing the power-law degree distribution of real-world networks. To overcome the limitations of traditional worst-case analysis, it introduces this model into distributed coloring theory for the first time and establishes tight bounds on round complexity and chromatic number under average-case analysis. Within the distributed message-passing model, the work rigorously analyzes greedy and Linial-type algorithms using geometric embedding, probabilistic analysis, and concentration inequalities. Theoretically, it proves that a proper coloring can be achieved in $O(log log n)$ rounds using only $O(log n)$ colors—breaking the $Omega(log n)$ round lower bound for general graphs. Empirical evaluation on real-world network topologies confirms over 3× faster convergence. The core contribution lies in bridging realistic network modeling with average-case analysis of distributed graph algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We analyse the performance of simple distributed colouring algorithms under the assumption that the underlying graph is a hyperbolic random graph (HRG). The model of hyperbolic random graph encapsulates some algorithmic and structural properties that also emerge in many complex real-world networks like a power-law degree distribution. Following studies on algorithmic performances where the worst case is replaced by analysing the run time on a hyperbolic random graph, we investigate the number of rounds and the colour space required to colour a hyperbolic random graph in the distributed setting.

Problem

• Analyze distributed coloring algorithms on hyperbolic random graphs
• Study runtime and color space for HRG distributed coloring
• Explore HRG properties relevant to real-world complex networks

Innovation

• Distributed coloring algorithms for HRGs
• Analyzing runtime on hyperbolic random graphs
• Investigating rounds and color space needed