双曲随机图上的分布式对称性破缺

中文摘要 AI 辅助

像互联网这样的现实世界网络具有幂律度分布和高聚类系数等模式，双曲随机图（HRGs）生成模型可捕捉这些特性。鉴于一些算法在现实网络上表现优于最坏情况保证，本文在输入图为HRG的假设下设计并分析分布式算法。此前工作表明经典的\(\Delta + 1\)着色对称性破缺问题在HRGs上可两轮解决，而本文证明最大独立集（MIS）和最大匹配（MM）相关对称性破缺问题更难，建立了下界。下界技术依赖新结构见解，还表明这些下界是多项式紧的，设计了在LOCAL模型中以高概率在\(\tilde{\mathcal{O}}(\log^{5/3}\log n)\)轮解决MIS和MM的算法。

英文摘要

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].

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58 pages, 9 figures