[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86219-en":3,"doc-seo-86219-105":30,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86219,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Hyperbolic embeddings for graph compression","Network theorists argue that real-world networks follow an underlying geometric structure, and hyperbolic geometry has proven effective for modeling scale-free graphs. This study presents a fast lossless graph compression algorithm built on modern hyperbolic embedders. Experiments on both real and synthetic networks show up to 42% improvement over state-of-the-art methods. The approach also leverages embedding-based vertex ordering to strengthen WebGraph performance.","arXiv :2607 . 1 1379v 1 [ cs . SI] 13 Jul 2026  \nHyperbolic embeddings for graph compression  \nDorota Celi´nska-Kopczy´nska, Eryk Kopczy´nski Institute of Informatics, University of Warsaw, Warsaw, Poland  \nJuly 14, 2026  \nAbstract  \nNetwork theoreticians hypothesize that the structure of real-world networks has a geometric origin. Especially, hyperbolic geometry was proven insightful in representing and modeling of scale-free networks. Embedders are algorithms used to find a geometric representation of a network. In this study, we introduce a fast lossless graph compression algorithm based on modern hyperbolic embedders. Experimental validation on real-world and generated networks shows that our algorithm beats state-of-the-art by up to 42% on real-world graphs.  \n1 Introduction  \nUbiquity of large graphs in today’s world prompts a quest to compress such data; that is, for a given graph G = (V, E) where E ⊆ V 2 , construct a minimal bit sequence s that encodes E losslessly. Beyond minimizing the lenght of sequence s, practical requirements include fast compression and decompression and efficient query support (e.g. , what are all the successors of v ∈ V ) with a low working memory footprint [29] . Designing such representations is central to graph algorithms and succinct data structures.  \nThe dominant practical approach to lossless graph compression is WebGraph [7, 19] that relies on the observation that successors of v are often similar to eachother, so they are likely to have similar indices (in some natural enumeration of V ) . Instead of writing every successor of v as a full index, we only encode gaps between them using a variable-length encoding. However, WebGraph’s effectiveness depends on the ordering of V. Consequently, it is typically preceded by computationally expensive reordering algorithms, such as Layered Label Propagation (LLP) [6] or recursive graph bisection (BP) [15] .  \nWebGraph exploits locality from vertex ordering, but does not take into account the latent geometric structure that has proved useful for describing many real-world networks. According to geometric network-generation models, two nodes v and w are likely to be connected if they are in a close neighborhood in a given metric space. However, the knowledge of the locations of the nodes in the given (often latent) metric space is needed. A geometric embedding of a graph G to a metric space X is a mapping m : V → X. This way we can  \noperationalize the probability of the connection between two nodes v and w asa function of the distance between m (v) and m (w) . The embedding needs tobe good, typically, we are interested in embeddings maximizing likelihood, which measures how good the embedding is at predicting edges between nodes. An embedder is an algorithm which finds a good embedding m : V → X.  \nHyperbolic geometry was found to be a promising choice in visualization and modeling of real-world graphs characterized by similar properties, such as powerlaw scaling behavior [33] . In Random Hyperbolic Graph model (RHG) [27], the metric space X is a disk of radius R in the hyperbolic plane H2 . Every node gets two polar coordinates, r (radial) and ϕ (angular) . The angular coordinate ϕ corresponds to similarity (nodes with close ϕ are considered similar) and the radial coordinate r proxies popularity (nodes with small r are considered popular and thus connect to more other nodes, even if they are less similar) . For more than a decade, hyperbolic embedders (finding good embeddings m : V → H2 ) have been a vivid research area not only in network theory but algorithmic and machine learning communities as well [13] .  \nIn this paper, we propose a new approach to lossless compression of graphs, based on hyperbolic embeddings. Our contributions are as follows:  \n• First practical lossless graph compression algorithm based on hyperbolic embeddings, running in time O (nBb Bd + mlog n), where Bb and Bd are compression parameters. We use entropy coding to e","cbCaihhmaX4KXVOX","https://ap.wps.com/l/cbCaihhmaX4KXVOX","pdf",2626441,3,1,28,"English","en",105,"# Introduction\n# Prerequisites","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper targets lossless compression of large graphs, aiming to encode edges with a minimal bit sequence while keeping compression/decompression fast and enabling efficient queries with low memory usage.\"},{\"question\":\"How does hyperbolic geometry relate to the proposed compression method?\",\"answer\":\"It uses hyperbolic embeddings to represent vertices in the hyperbolic plane, turning edge existence into a distance-based connection likelihood that can be entropy-coded for compression.\"},{\"question\":\"What performance improvement is reported compared with existing methods?\",\"answer\":\"Experimental validation on real-world and generated networks shows the algorithm outperforms state-of-the-art by up to 42% on real-world graphs, and embedding-based ordering can further improve WebGraph.\"}]",1784209569,71,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"hyperbolic-embeddings-for-graph-compression","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/hyperbolic-embeddings-for-graph-compression/86219/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address?","Question",{"text":75,"@type":76},"The paper targets lossless compression of large graphs, aiming to encode edges with a minimal bit sequence while keeping compression/decompression fast and enabling efficient queries with low memory usage.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does hyperbolic geometry relate to the proposed compression method?",{"text":80,"@type":76},"It uses hyperbolic embeddings to represent vertices in the hyperbolic plane, turning edge existence into a distance-based connection likelihood that can be entropy-coded for compression.",{"name":82,"@type":73,"acceptedAnswer":83},"What performance improvement is reported compared with existing methods?",{"text":84,"@type":76},"Experimental validation on real-world and generated networks shows the algorithm outperforms state-of-the-art by up to 42% on real-world graphs, and embedding-based ordering can further improve WebGraph.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]