Distributed Symmetry Breaking on Hyperbolic Random Graphs

2026-07-10

AI Summary

This work investigates the complexity of distributed algorithms for computing maximal independent sets (MIS) and maximal matchings (MM) on hyperbolic random graphs (HRGs). Leveraging the distinctive power-law degree distribution and high clustering inherent to HRGs, the authors uncover a novel geometric property: the presence of tall d-ary trees. They establish the first Ω(log log n / log log log n) round lower bound for both MIS and MM in the LOCAL model. Building on this insight, they design tailored symmetry-breaking algorithms that solve MIS and MM with high probability in Õ(log^{5/3} log n) rounds—significantly outperforming the Ω(min{log Δ, √log n}) lower bounds known for general graphs. This result offers a new paradigm for efficient distributed computation on real-world networks exhibiting hyperbolic geometry.

Abstract

Real-world networks like the internet share patterns like a power law degree distribution and a high clustering coefficient. Many of these properties are captured by the generative model of hyperbolic random graphs (HRGs), which provides a theoretical framework for studying such networks. Motivated by the observation that several algorithms perform better on real-world networks than their worst-case guarantees suggest, we design and analyse distributed algorithms under the assumption that the input graph is an HRG. Indeed, prior work has shown that the classical symmetry-breaking problem of $Δ+1$ colouring, where $Δ$ is the maximum degree of the graph, can be solved in 2 rounds on HRGs [Maus and Ruff; SODA'26]. In stark contrast to this 2-round algorithm for $Δ+1$ colouring, we prove that the related symmetry-breaking problems of maximal independent set (MIS) and maximal matching (MM) are substantially harder: we establish a lower bound of $Ω\left(\frac{\log\log n}{\log\log\log n}\right)$ for MIS and MM on HRGs. Our lower bound techniques rely on new structural insights that may be of independent interest: we show that HRGs contain $d$-ary trees with large height and degree which enables us to adapt and lift prior impossibility results for distributed algorithms to the setting of HRGs. We also show that these lower bounds are polynomial tight: we design algorithms tailored to HRGs that solve MIS and MM in $\tilde{\mathcal{O}}(\log^{5/3}\log n)$ rounds with high probability in the LOCAL model, improving over the general worst-case lower bound of $Ω\left(\min\left\{\log Δ, \sqrt{\log n}\right\}\right)$ rounds [Khoury and Schild; FOCS'25].

Problem

• Distributed Symmetry Breaking
• Hyperbolic Random Graphs
• Maximal Independent Set
• Maximal Matching

Innovation

• hyperbolic random graphs
• distributed algorithms
• symmetry breaking
• maximal independent set
• lower bounds