[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83424-en":3,"doc-seo-83424-105":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},83424,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Dimensionality Reduction Meets Network Science: Sensemaking on UMAP’s kNN Graph","UMAP is widely used for exploring high-dimensional data, yet most workflows focus on its 2D embedding and ignore the internal k-nearest-neighbor (kNN) graph. This graph captures manifold connectivity in the original high-dimensional space before projection distortion. The work demonstrates graph-based sensemaking using PageRank to find representative points, k-core decomposition to separate dense cores from sparse periphery, and clustering coefficient to detect tightly connected neighborhoods. Results on MNIST and Fashion MNIST show practical, competitive value versus methods like k-medoids and HDBSCAN.","Dimensionality Reduction Meets Network Science: Sensemaking on UMAP’s kNN Graph  \nDuen Horng (Polo) Chau*  \nApple  \nDonghao Ren†  \nApple  \nFred Hohman‡  \nApple  \nDominik Moritz§  \nApple  \narXiv :2607 .08746v 1 [ cs .LG] 9 Jul 2026  \nFigure 1: Standard graph algorithms applied to UMAP’s internal kNN graph reveal structure lost in 2D scatter plot layouts. A. PageRank on the Fashion MNIST kNN graph identifies representative data points. The highest-scoring points exhibit prototypical appearances, while the lowest-scoring points display atypical variations. Top 500 points shown in high saturation. B. k-core decomposition reveals distinct sub-categories in the “bag” class (e.g., messenger bags, waist packs, heavy textures) through filtering for a coreness of 6 from the dense mass of points in the 2D scatter plot layout. Grayscale images inverted for clarity.  \nABSTRACT  \nWhile UMAP is widely used for exploring high-dimensional data, typical workflows focus on its lower-dimensional embedding, largely overlooking the rich k-nearest-neighbor (kNN) graph that UMAP constructs internally. This graph encodes the data manifold in its original high-dimensional space, before the distortion that UMAP’s 2D projection introduces. We demonstrate the untapped potential of this internal representation, showing how standard graph algorithms applied to this graph enhance data sensemaking: (1) PageRank identifies representative data points, (2) kcore decomposition reveals dense core regions versus sparse periphery, and (3) clustering coefficient detects tight-knit neighborhoods with highly-similar data points. Through quantitative and qualitative evaluation on MNIST and Fashion MNIST, we show that these graph-based analyses are not only practical but also competitive with or complementary to purpose-built methods (e.g., k-medoids for exemplar selection, HDBSCAN for density-based clustering) .  \nIndex Terms: Dimensionality reduction, graph algorithms, UMAP, kNN graph, sensemaking.  \n* e-mail: [polochau@apple.com](polochau@apple.com)  \n†e-mail: [donghao@apple.com](donghao@apple.com)  \n‡e-mail: [fredhohman@apple.com](fredhohman@apple.com)  \n§ e-mail: [domoritz@apple.com](domoritz@apple.com)  \n1 INTRODUCTION  \nUMAP [15] is among the most widely used tools for visually exploring high-dimensional data. Yet the 2D scatter plot output by UMAP is typically treated as the sole analytical artifact [10, 11] .  \nUMAP’s discarded kNN graph. Before producing a 2D layout, UMAP builds a weighted directed graph that models the data manifolds’ local geometry (Fig. 2) . For each point, it finds k nearest neighbors in high-dimensional space, then applies a densityadaptive normalization: each point’s bandwidth σi is calibrated to its local density, transforming raw distances into membership strengths in [0, 1] that are comparable across sparse and dense regions [15] . Every point has exactly k outgoing edges, but in-degree varies: points deemed similar by many others receive more incoming edges, while dissimilar ones are nominated by few. We call this the kNN graph for short. This kNN graph encodes the manifold’s connectivity far more faithfully than the 2D scatter plot layout that UMAP subsequently optimizes from it [9, 11]—yet after layout optimization, the graph is typically set aside. We argue it should be retained as a first-class analytical resource.  \nSensemaking from kNN graph. Because the kNN graph faithfully reflects the high-dimensional manifold, standard graph algorithms can be applied to answer sensemaking questions—about representativeness, density structure, and local cohesion—that the 2D scatter plot alone cannot.  \nOur work makes two major contributions:  \n1. Elevating UMAP’s kNN graph to a first-class analytical resource, bridging dimensionality reduction and network science. Instead of treating this graph as a disposable intermediate, this perspective enables direct application of standard graph algo-  \nFigure 2: UMAP’s 2D layout discards structural inf","cbCaimO4OYBLapHM","https://ap.wps.com/l/cbCaimO4OYBLapHM","pdf",2532854,3,1,5,"English","en",105,"# Abstract\n# Introduction\n## UMAP’s discarded kNN graph\n## Sensemaking from kNN graph\n# Related Work","[{\"question\":\"What key resource does the document propose keeping from UMAP?\",\"answer\":\"The internal weighted directed k-nearest-neighbor (kNN) graph. It encodes manifold connectivity in the original high-dimensional space, which is otherwise discarded after producing the 2D layout.\"},{\"question\":\"How does PageRank help with sensemaking in the kNN graph?\",\"answer\":\"PageRank identifies globally representative data points through transitive centrality, ranking nodes highly when many high-ranking nodes nominate them as neighbors.\"},{\"question\":\"What do k-core decomposition and clustering coefficient reveal?\",\"answer\":\"k-core decomposition distinguishes dense core regions from sparse periphery, while clustering coefficient detects tight-knit micro-neighborhoods where points are mutually nearest neighbors and highly similar.\"}]",1784187515,13,{"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},"dimensionality-reduction-meets-network-science-sensemaking-on-umaps-knn-graph","",{"@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/dimensionality-reduction-meets-network-science-sensemaking-on-umaps-knn-graph/83424/",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","2026-07-26","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 key resource does the document propose keeping from UMAP?","Question",{"text":75,"@type":76},"The internal weighted directed k-nearest-neighbor (kNN) graph. It encodes manifold connectivity in the original high-dimensional space, which is otherwise discarded after producing the 2D layout.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does PageRank help with sensemaking in the kNN graph?",{"text":80,"@type":76},"PageRank identifies globally representative data points through transitive centrality, ranking nodes highly when many high-ranking nodes nominate them as neighbors.",{"name":82,"@type":73,"acceptedAnswer":83},"What do k-core decomposition and clustering coefficient reveal?",{"text":84,"@type":76},"k-core decomposition distinguishes dense core regions from sparse periphery, while clustering coefficient detects tight-knit micro-neighborhoods where points are mutually nearest neighbors and highly similar.","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},[93,97,101,105,109,114,119,122,127,130,134],{"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":22,"doc_module":4,"doc_module_name":46,"category_name":106,"show_sort_weight":107,"slug":108},"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":22,"slug":137},19,"General","general"]