[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82679-en":3,"doc-seo-82679-105":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},82679,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Scaling Weisfeiler-Leman Expressiveness Analysis to Massive Graphs with GPUs","Stable coloring of the Weisfeiler–Leman (1-WL) test is a key theoretical tool for Graph Neural Networks, giving an upper bound on the expressive power of message-passing models. Computing it faces two bottlenecks: classic refinement methods are sequential and cannot leverage massively parallel hardware, and they are global, requiring the full graph in memory. A randomized linear-algebraic refinement with probabilistic guarantees and a correctness-preserving batching scheme enable GPU-efficient computation, achieving up to two orders of magnitude speedups and scaling to web-scale graphs with 30+ billion edges.","arXiv :2607 .02603v 1 [ cs .DC] 1 Jul 2026  \nScaling Weisfeiler–Leman Expressiveness Analysis to Massive Graphs with GPUs  \nFilippo Biondi 1 (􀀀), Mirco Tribastone1 , and Max Tschaikowski2  \n1 IMT Lucca, Italy  \n{filippo.biondi,[mirco.tribastone}@imtlucca.it](mirco.tribastone}@imtlucca.it)  \n2 Sapienza University Rome, Italy  \n[max.tschaikowski@uniroma1.it](max.tschaikowski@uniroma1.it)  \nAbstract The stable coloring of the Weisfeiler-Leman (1-WL) test is a cornerstone of Graph Neural Networks because it provides an upper bound to the expressive power of message-passing architectures. Unfortunately, computing it presents two fundamental bottlenecks. First, classic algorithms are inherently sequential and cannot exploit modern massively parallel hardware. Second, these are global algorithms, i.e., they require availability in memory of the full graph, severely limiting applicability to real-world instances. We leverage a linear-algebraic interpretation of 1-WL stable coloring and introduce two key contributions:  \n(i) a randomized refinement algorithm with tight probabilistic guarantees and (ii) a correctness-preserving batching scheme that decomposes the graph into independently processable subgraphs while provably returning a stable coloring of the original graph. This approach maps directly to GPU-efficient primitives. In numerical experiments, our CUDA implementation delivers speedups up to two orders of magnitude over classical CPU-based partition refinement and, for the first time, successfully computes stable colorings on web-scale graphs with over 30 billion edges, where CPU baselines time out or fail.  \nKeywords: Weisfeiler-Leman Test (1-WL) Computation · Randomized Parallel Algorithm · Linear Algebra Characterization  \n1 Introduction  \nGraph Neural Networks (GNNs) have emerged as powerful tools for learning on graph-structured data, achieving state-of-the-art performance in domains such as chemistry, biology, social networks, and recommender systems [27,25,19,24] . A central question in their theoretical analysis is expressiveness: which structural distinctions can a GNN architecture capture? The Weisfeiler-Leman (WL) graph isomorphism test—particularly its first-order variant (1-WL)—has become the standard lens for characterizing GNN expressiveness [23,29] . The 1-WL stable coloring is relevant because it provides an upper bound to the expressive power of message-passing GNNs—two nodes assigned the same color will always receive identical embeddings, meaning the network cannot distinguish them.  \n2 Filippo Biondi (􀀀), Mirco Tribastone, and Max Tschaikowski  \nComputing a stable coloring amounts to determining a partition of a graph such that nodes in the same block share identical neighborhood counts [22,23] . Classical algorithms based on partition refinement compute the coarsest stable coloring, i.e., the partition with the fewest blocks, in optimal O (mlog n) complexity [32], a bound that is asymptotically tight [3] . Yet they suffer from two fundamental limitations: they are sequential and global, requiring the entire input graph to reside in memory. These characteristics make them ill-suited for modern architectures such as GPUs, which excel at massively parallel workloads.  \nAs a consequence, despite its centrality, computing stable colorings at web scale remains practically infeasible, limiting empirical WL-based expressiveness analysis to small or medium graphs. In this paper we address both limitations of the state of the art with a parallel algorithm that can execute locally on independent subgraphs.  \nRandomized linear-algebraic refinement. Our approach leverages a key invariance property: if a vector is symmetric on an equitable partition (that is, it has equal values on its blocks of nodes), then so is its image under the adjacency matrix [26,38,16,34] . As a result, stable coloring can be expressed as a sequence of matrix–vector multiplications (computing images until a fixed point) that naturally maps to","cbCaivhdIPFTMbpN","https://ap.wps.com/l/cbCaivhdIPFTMbpN","pdf",580771,1,22,"English","en",105,"# Introduction\n# Randomized linear-algebraic refinement\n# Correctness-preserving batching for large-scale graphs\n# Experimental results","[{\"question\":\"Why is 1-WL stable coloring important for Graph Neural Networks?\",\"answer\":\"It provides an upper bound on the expressive power of message-passing GNNs: nodes with the same 1-WL color must receive identical embeddings, limiting what the network can distinguish.\"},{\"question\":\"What prevents current stable-coloring algorithms from scaling to modern hardware and web-scale graphs?\",\"answer\":\"They are inherently sequential and global, so they cannot exploit massive parallelism and they require the full input graph to be available in memory.\"},{\"question\":\"How do the proposed methods address compute and memory bottlenecks while preserving correctness?\",\"answer\":\"A randomized linear-algebraic refinement converts stable coloring into GPU-friendly matrix–vector operations, while a batching scheme processes disjoint edge batches and uses a quotient-based merge that provably preserves the original graph’s stable coloring.\"}]",1784182242,55,{"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},"scaling-weisfeiler-leman-expressiveness-analysis-to-massive-graphs-with-gpus","",{"@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/scaling-weisfeiler-leman-expressiveness-analysis-to-massive-graphs-with-gpus/82679/",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},"Why is 1-WL stable coloring important for Graph Neural Networks?","Question",{"text":75,"@type":76},"It provides an upper bound on the expressive power of message-passing GNNs: nodes with the same 1-WL color must receive identical embeddings, limiting what the network can distinguish.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What prevents current stable-coloring algorithms from scaling to modern hardware and web-scale graphs?",{"text":80,"@type":76},"They are inherently sequential and global, so they cannot exploit massive parallelism and they require the full input graph to be available in memory.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the proposed methods address compute and memory bottlenecks while preserving correctness?",{"text":84,"@type":76},"A randomized linear-algebraic refinement converts stable coloring into GPU-friendly matrix–vector operations, while a batching scheme processes disjoint edge batches and uses a quotient-based merge that provably 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