[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85805-en":3,"doc-seo-85805-105":29,"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":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},85805,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Beyond Worst-Case Distortions Distance-Preserving Embeddings in Inhomogeneous Random Graphs","Graph machine learning seeks node representations that preserve structural information across local and global scales, especially shortest-path distances. Existing theoretical guarantees for distance-preserving embeddings rely on worst-case analyses and yield overly pessimistic dimension–distortion trade-offs that do not reflect typical large heterogeneous networks. This work studies landmark-based shortest-path approximation on inhomogeneous random graphs with type-dependent edge probabilities, producing tighter, structure-aware bounds for both local and component-wide averages and extending them via a metric-sandwiching framework for L2 kernel models, including heavy-tailed networks.","arXiv :2607 . 10074v 1 [ cs .LG] 11 Jul 2026  \nBeyond Worst-Case Distortions: Distance-Preserving Embeddings in Inhomogeneous Random Graphs  \nMy Le [mle19@jh.edu](mle19@jh.edu)  \n[Department of Applied Mathematics & Statistics](Department of Applied Mathematics & Statistics)[ ](Department of Applied Mathematics & Statistics)Johns Hopkins University  \nBaltimore, MD 21218, USA  \nLuana Ruiz [lrubini1@jh.edu](lrubini1@jh.edu)  \n[Department of Applied Mathematics & Statistics](Department of Applied Mathematics & Statistics)[ ](Department of Applied Mathematics & Statistics)Johns Hopkins University  \nBaltimore, MD 21218, USA  \nSouvik Dhara [sdhara@gatech.edu](sdhara@gatech.edu)  \nSchool of Industrial and Systems Engineering Georgia Institute of Technology  \nAtlanta, GA 30332, USA  \nAbstract  \nGraph machine learning provides powerful tools for understanding complex networks and learning meaningful node representations. A central challenge, however, is designing embeddings with minimal distortion of both local and global functionals, such as shortest path lengths. Prior distortion guarantees for distance-preserving embeddings are worstcase in nature, producing overly pessimistic bounds that fail to capture the structure of typical large-scale networks. To address this, we analyze shortest-path approximation via landmark-based embeddings on inhomogeneous random graphs (IHGs), a general model with type-dependent edge probabilities. By retaining shortest paths to a small set of reference nodes called landmarks, landmark-based methods effectively function as virtual graph spanners, where structural heterogeneity and controlled neighborhood expansion modeled via multi-type branching processes enable significantly tighter dimension–distortion trade-offs, i.e. Ω 􀀀n1−ε log n 􀀁, than classical worst-case bounds, i.e. Ω 􀀐n 2(21εε) log n􀀑 for (1 − ε)-distortion and Ω 􀀐n 2ε log n􀀑 for (1 + ε)-distortion. We extend these guarantees to global, component-wide averages and unify the analysis across finite-type and continuous latent spaces through a novel metric sandwiching framework, establishing universal distortion bounds for general L2 kernel models, including heavy-tailed and power-law networks.  \nFinally, we introduce a GNN-augmented variant that replaces rigid, computationally expensive exact shortest-path queries with flexible, structure-aware neural surrogates. By leveraging the inherent alignment between graph neural message-passing and the dynamic programming principles of shortest-path algorithms, our approach demonstrates that models trained on small-scale random graphs learn to extract universal distance-preserving features, achieving robust generalization to large-scale, real-world networks that match or exceed the fidelity of classical, exact landmark-based embeddings.  \nKeywords: shortest path, distance-preserving embeddings, landmarks, graph spanners, inhomogeneous random graphs, heterogeneity, graph neural networks, transferability  \n©2026 My Le, Luana Ruiz, and Souvik Dhara.  \nLicense: CC-BY 4.0, see [https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/) .  \nLe, Ruiz, and Dhara  \n1 Introduction  \nA central challenge in graph learning is to represent networked data in a form that faithfully captures its essential structural characteristics, including local connectivity patterns, mesoscopic organization, and global topology. A common strategy is to map nodes into lowdimensional metric spaces so that distances between embedded points reflect the graph’s original structure. These embeddings provide a compact and mathematically tractable representation of the network, enabling both statistical analysis and efficient computation at scale in a wide range of inference tasks, including node classification, link prediction, clustering, and routing (Hamilton et al. 2017b, Grover and Leskovec 2016, Belkin and Niyogi 2003) .  \nDespite significant progress, node embedding techniques are primarily designed to","cbCaic4Za0qBelcK","https://ap.wps.com/l/cbCaic4Za0qBelcK","pdf",1965455,1,53,"English","en",105,"# Abstract\n# Keywords\n# Introduction\n## Background on graph embeddings\n## Limitations of global functional preservation\n## Landmark-based distance embeddings\n## Motivation for sharper guarantees","[{\"question\":\"What problem does the paper address in distance-preserving graph embeddings?\",\"answer\":\"It targets the gap between worst-case theoretical distortion guarantees and the typically better behavior of distance preservation in large, heterogeneous networks, where shortest-path distances are often poorly captured.\"},{\"question\":\"How does the paper improve analysis for shortest-path approximation?\",\"answer\":\"It analyzes landmark-based embeddings on inhomogeneous random graphs, leveraging type-dependent heterogeneity and neighborhood growth modeled by multi-type branching processes to obtain tighter dimension–distortion trade-offs.\"},{\"question\":\"What additional model extension is proposed beyond classical landmark methods?\",\"answer\":\"It introduces a GNN-augmented variant that replaces exact shortest-path queries with neural surrogates, using the alignment between message passing and shortest-path dynamic programming to improve scalability and generalization.\"}]",1784206367,134,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"beyond-worst-case-distortions-distance-preserving-embeddings-in-inhomogeneous-random-graphs","",{"@graph":35,"@context":85},[36,53,68],{"@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/beyond-worst-case-distortions-distance-preserving-embeddings-in-inhomogeneous-random-graphs/85805/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 in distance-preserving graph embeddings?","Question",{"text":75,"@type":76},"It targets the gap between worst-case theoretical distortion guarantees and the typically better behavior of distance preservation in large, heterogeneous networks, where shortest-path distances are often poorly captured.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper improve analysis for shortest-path approximation?",{"text":80,"@type":76},"It analyzes landmark-based embeddings on inhomogeneous random graphs, leveraging type-dependent heterogeneity and neighborhood growth modeled by multi-type branching processes to obtain tighter dimension–distortion trade-offs.",{"name":82,"@type":73,"acceptedAnswer":83},"What additional model extension is proposed beyond classical landmark methods?",{"text":84,"@type":76},"It introduces a GNN-augmented variant that replaces exact shortest-path queries with neural surrogates, using the alignment between message passing and shortest-path dynamic programming to improve scalability and generalization.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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