[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117547-en":3,"doc-seo-117547-105":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":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":27,"seo_description":14,"update_tm":28,"read_time":29},117547,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Weisfeiler and Leman go Machine Learning - The Story so far","Algorithms and neural architectures inspired by the Weisfeiler–Leman graph isomorphism heuristic have become central tools for machine learning on graphs and relational data. The work provides a comprehensive, supervised-focused overview of the algorithm’s role, including the theoretical foundations and practical use for supervised graph and node representation learning. It reviews recent extensions, clarifies connections to (permutation-)equivariant neural architectures, and summarizes current applications and future research directions.","Weisfeiler and Leman go Machine Learning: The Story so far  \nChristopher Morris [morris@cs.rwth-aachen.de](morris@cs.rwth-aachen.de)  \nDepartment of Computer Science RWTH Aachen University Aachen, Germany  \nYaron Lipman [yaron.lipman@weizmann.ac.il](yaron.lipman@weizmann.ac.il)  \nMeta AI Research  \nDepartment of Computer Science and Applied Mathematics Weizmann Institute of Science  \nRehovot, Israel  \nHaggai Maron [hmaron@nvidia.com](hmaron@nvidia.com)  \nNVIDIA Research Tel Aviv, Israel  \nBastian Rieck [bastian.rieck@helmholtz-muenchen.de](bastian.rieck@helmholtz-muenchen.de)  \nAIDOS Lab, Institute of AI for Health  \nHelmholtz Zentrum M􀁿unchen and Technical University of Munich Munich, Germany  \nNils M. Kriege [nils.kriege@univie.ac.at](nils.kriege@univie.ac.at)  \nFaculty of Computer Science, University of Vienna, Vienna, Austria Research Network Data Science, University of Vienna, Vienna, Austria  \nMartin Grohe [grohe@informatik.rwth-aachen.de](grohe@informatik.rwth-aachen.de)  \nDepartment of Computer Science RWTH Aachen University Aachen, Germany  \nMatthias Fey [matthias@kumo.ai](matthias@kumo.ai)  \nKumo.AI  \nMountain View, CA  \nKarsten Borgwardt􀀃 [karsten.borgwardt@bsse.ethz.ch](karsten.borgwardt@bsse.ethz.ch)  \nMachine Learning & Computational Biology Lab Department of Biosystems Science and Engineering ETH Z􀁿urich, Basel, Switzerland and  \nSwiss Institute of Bioinformatics, Lausanne, Switzerland  \nEditor: David Wipf  \n􀀃 . Karsten Borgwardt is now at the Max Planck Institute of Biochemistry in Martinsried, Germany.  \n􀀍c2023 Christopher Morris, Yaron Lipman, Haggai Maron, Bastian Rieck, Nils M. Kriege, Martin Grohe, Matthias Fey, Karsten Borgwardt.  \nLicense: CC-BY 4.0, see [https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/. Attribution)[. Attribution](https://creativecommons.org/licenses/by/4.0/. Attribution) requirements are provided  \nat [http://jmlr.org/papers/v24/22-0240.html](http://jmlr.org/papers/v24/22-0240.html).  \nMorris, Lipman, Maron, Rieck, Kriege, Grohe, Fey, Borgwardt  \nAbstract  \nIn recent years, algorithms and neural architectures based on the Weisfeiler{Leman algorithm, a well-known heuristic for the graph isomorphism problem, have emerged as a powerful tool for machine learning with graphs and relational data. Here, we give a comprehensive overview of the algorithm's use in a machine-learning setting, focusing on the supervised regime. We discuss the theoretical background, show how to use it for supervised graph and node representation learning, discuss recent extensions, and outline the algorithm's connection to (permutation-)equivariant neural architectures. Moreover, we give an overview of current applications and future directions to stimulate further research.  \nKeywords: Machine learning for graphs, Graph neural networks, Weisfeiler{Leman algorithm, expressivity, equivariance  \n1. Introduction  \nGraph-structured data is ubiquitous across application domains, ranging from chemo-and bioinformatics (Barabasi and Oltvai, 2004; Jumper et al., 2021; Stokes et al., 2020) to computer vision (Simonovsky and Komodakis, 2017), and social network analysis (Easley and Kleinberg, 2010); see Figure 1 for an overview of application areas. We need techniques exploiting the rich graph structure and feature information within nodes and edges to develop successful machine-learning models in these domains. Due to the highly non-regular structure of real-world graphs, most approaches 􀀌rst generate a vectorial representation of each graph or node, so-called node or graph embeddings, respectively, to apply standard machine learning tools such as linear regression, random forests, or neural networks.  \nFor successful (supervised) machine learning with graphs, node and graph embeddings need to address the following key challenges:  \n1. The graph embedding needs to be invariant to any permutation of the graph's nodes, i.e., the output of the graph embedding must not change for di􀀋ere","cbCainHT9mTy0fCw","https://ap.wps.com/l/cbCainHT9mTy0fCw","pdf",3631579,1,59,"English","en",105,"# Introduction\n## Graph embeddings and key challenges\n## Weisfeiler–Leman algorithm and graph neural networks\n## 1-WL color refinement basics\n## Supervised representation learning workflow","[{\"question\":\"What is the main purpose of the document’s overview of Weisfeiler–Leman in machine learning?\",\"answer\":\"It explains how Weisfeiler–Leman-inspired algorithms and neural architectures are used for supervised machine learning on graphs, covering theory, representation learning, extensions, and research directions.\"},{\"question\":\"Which invariance and equivariance requirements do graph embeddings need to satisfy?\",\"answer\":\"Graph embeddings must be permutation-invariant at the graph level, while node embeddings must be permutation-(equivariant) to node orderings so that reordering inputs yields corresponding reordering of outputs.\"},{\"question\":\"How does the (1-dimensional) Weisfeiler–Leman (1-WL) color refinement algorithm work at a high level?\",\"answer\":\"Starting from an initial node labeling, each iteration updates node labels based on the multiset counts of neighbor labels; nodes with differing neighbor-label distributions receive different labels.\"}]","Weisfeiler and Leman go Machine Learning - The Story so far | PDF",1785676909,149,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"weisfeiler-and-leman-go-machine-learning-the-story-so-far","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/weisfeiler-and-leman-go-machine-learning-the-story-so-far/117547/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main purpose of the document’s overview of Weisfeiler–Leman in machine learning?","Question",{"text":75,"@type":76},"It explains how Weisfeiler–Leman-inspired algorithms and neural architectures are used for supervised machine learning on graphs, covering theory, representation learning, extensions, and research directions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which invariance and equivariance requirements do graph embeddings need to satisfy?",{"text":80,"@type":76},"Graph embeddings must be permutation-invariant at the graph level, while node embeddings must be permutation-(equivariant) to node orderings so that reordering inputs yields corresponding reordering of outputs.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the (1-dimensional) Weisfeiler–Leman (1-WL) color refinement algorithm work at a high level?",{"text":84,"@type":76},"Starting from an initial node labeling, each iteration updates node labels based on the multiset counts of neighbor labels; 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