[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82234-en":3,"doc-seo-82234-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82234,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Distributed Symmetry Breaking on Hyperbolic Random Graphs","Real-world networks often exhibit power-law degree distributions and strong clustering, which are captured by hyperbolic random graphs (HRGs) as a generative model. Motivated by algorithms performing better on real networks than worst-case guarantees, the work develops and analyzes distributed algorithms assuming the input graph is an HRG. It proves MIS and maximal matching are far harder than Δ+1 coloring, giving Ω(log log n) lower bounds, and designs HRG-tailored algorithms with matching polynomial tightness, plus an embedding-aware speedup for maximal matching.","arXiv :2607 .09170v1 [ cs .DC] 10 Jul 2026  \nDistributed Symmetry Breaking on Hyperbolic Random Graphs  \nYannic Maus∗\\#1, Janosch Ruff†\\#2, Sonia Simons\\#2, and George Skretas\\#2  \n1 TU Graz, Austria  \n2Hasso Plattner Institute, University of Potsdam, Germany  \nAbstract  \nReal-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] .  \nIn stark contrast to this 2-round algorithm for Δ + 1 colouring, we prove that the related symmetrybreaking problems of maximal independent set (MIS) and maximal matching (MM) are substantially harder: we establish a lower bound of Ω  log log 􀁮  for MIS and MM on HRGs. Our lower bound  \n(log log log 􀁮 )  \ntechniques rely on new structural insights that may be of independent interest: we show that HRGs contain 􀁤-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.  \nWe also show that these lower bounds are polynomial tight: we design algorithms tailored to HRGs that solve MIS and MM in ̃􀁏(log5/3 log 􀁮) rounds with high probability in the LOCAL model, improving over the general worst-case lower bound of Ω (min {log Δ, √log 􀁮 }) rounds [Khoury and Schild; FOCS’25] .  \nFinally, we show that access to geometric information can significantly reduce the complexity of maximal matching: if vertices know their respective geometric embeddings, then MM can be solved in 􀁏(log log log 􀁮) rounds on HRGs. This reveals a separation between the standard LOCAL model and an embedding-aware variant on hyperbolic random graphs.  \n∗This research was funded in whole or in part by the Austrian Science Fund (FWF) [https://doi. org/10.55776/P36280](https://doi. org/10.55776/P36280), [https://doi. org/10.55776/I6915](https://doi. org/10.55776/I6915. For open access purposes)[. For open access purposes](https://doi. org/10.55776/I6915. For open access purposes), the author has applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission.  \n†This research was partially funded by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG)– project number 390859508 .  \nContents  \n1 Introduction 1  \n1. 1 Our Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2  \n1. 2 Further Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4  \n2 Technical Overview 5  \n2.1 Efficient Algorithms for MIS and MM .............................. 6  \n2. 2 Lower Bounds for MIS and MM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8  \n2.3 Embedding-Aware Algorithms .................................. 10  \n3 Conclusion 11  \n4 Hyperbolic Random Graphs 12  \n5 Efficient Algorithms for MIS/MM on HRGs (Theorem 1) 14  \n5.1 Efficient Maximal Independent Set Algorithm (MIS part of Theorem 1) ........... 14  \n5.2 Efficient Maximal Matching Algorithm (MM part of Theorem 1) ............... 20  \n6 Lower Bounds for MIS/MM & Substructures in HRGs (Theorem 2) 23  \n6.1 On 􀁤-ary Trees in Hyperbolic Random Graphs (Proof of Theorem 18) ............ 23  \n6.2 Lower Bounds for MM and MIS (Proof of Theorem 2) ..................... 32  \n7 Embedding-Aware Symmetry Breaking (Theorem 3) 35  \n7.1 Embedding-Aware Maximal Independent Set (MIS part of Theorem 3) ............ 37  \n7","cbCaidcofJnrHDdn","https://ap.wps.com/l/cbCaidcofJnrHDdn","pdf",1616833,1,58,"English","en",105,"# Introduction\n## Our Contributions\n## Further Related Work\n# Technical Overview\n## Efficient Algorithms for MIS and MM\n## Lower Bounds for MIS and MM\n## Embedding-Aware Algorithms\n# Conclusion\n# Hyperbolic Random Graphs\n# Efficient Algorithms for MIS/MM on HRGs (Theorem 1)\n## Efficient Maximal Independent Set Algorithm\n## Efficient Maximal Matching Algorithm\n# Lower Bounds for MIS/MM & Substructures in HRGs (Theorem 2)\n## On K-ary Trees in Hyperbolic Random Graphs\n## Lower Bounds for MM and MIS\n# Embedding-Aware Symmetry Breaking (Theorem 3)","[{\"question\":\"What graph model underlies the distributed algorithms in this paper?\",\"answer\":\"The analysis assumes the input graph is a hyperbolic random graph (HRG), using its structural properties to study distributed symmetry breaking tasks.\"},{\"question\":\"How do the hardness results for MIS and maximal matching compare to Δ+1 coloring?\",\"answer\":\"While prior work shows Δ+1 coloring can be solved in 2 rounds on HRGs, this paper proves MIS and maximal matching are substantially harder, establishing Ω(log log n) lower bounds on HRGs.\"},{\"question\":\"What impact does geometric embedding information have on maximal matching?\",\"answer\":\"If vertices know their geometric embeddings, maximal matching can be solved in ~O(log log log n) rounds on HRGs, showing a complexity separation between the standard LOCAL model and an embedding-aware variant.\"}]",1784179018,146,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"distributed-symmetry-breaking-on-hyperbolic-random-graphs","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/distributed-symmetry-breaking-on-hyperbolic-random-graphs/82234/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What graph model underlies the distributed algorithms in this paper?","Question",{"text":74,"@type":75},"The analysis assumes the input graph is a hyperbolic random graph (HRG), using its structural properties to study distributed symmetry breaking tasks.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How do the hardness results for MIS and maximal matching compare to Δ+1 coloring?",{"text":79,"@type":75},"While prior work shows Δ+1 coloring can be solved in 2 rounds on HRGs, this paper proves MIS and maximal matching are substantially harder, establishing Ω(log log n) lower bounds on HRGs.",{"name":81,"@type":72,"acceptedAnswer":82},"What impact does geometric embedding information have on maximal matching?",{"text":83,"@type":75},"If vertices know their geometric embeddings, maximal matching can be solved in ~O(log log log n) rounds on HRGs, showing a complexity separation between the standard LOCAL model and an embedding-aware 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