[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84410-en":3,"doc-seo-84410-105":29,"detail-sidebar-cat-0-en-105":83},{"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},84410,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","A Cheeger Inequality for Size-Specific Conductance","µ-conductance by Lovász and Simonovits measures size-specific conductance by seeking the smallest-conductance set subject to a volume constraint: sets with volume below a µ fraction of the whole graph are ignored. This manuscript analyzes a modified spectral cut for µ-conductance, formulating it as a natural relaxation of the associated integer program. The study establishes a two-sided Cheeger inequality that relates the spectral optimum and µ-conductance, enabling spectral characterization of size-restricted clustering structure.","A Cheeger Inequality for Size-Specific Conductance  \nYufan Huang Purdue University [2019hyf@gmail. com](2019hyf@gmail. com)  \nDavid F. Gleich Purdue University [dgleich@purdue. edu](dgleich@purdue. edu)  \narXiv :2303 . 11452v3 [ cs .DM] 13 Jul 2026  \nJuly 14, 2026  \nAbstract  \nThe µ-conductance measure proposed by Lov´asz and Simonovits is a size-specific conductance score that identifies the set with smallest conductance while disregarding those sets with volume smaller than a µ fraction of the whole graph. Using µ-conductance enables us to study the network structures in new ways. In this manuscript we study a modified spectral cut for µ-conductance that is a natural relaxation of the integer program of µ-conductance and show that the optimum of this program has a two-sided Cheeger inequality with µ-conductance.  \n1 Introduction  \nGraph clustering is one of the most fundamental problems in graph analysis and has many practical applications (Schaeffer, 2007) . A major class of clustering methods is based on finding small cuts, motivated by the observation that meaningful clusters tend to be internally wellconnected while being sparsely connected to the rest of the graph. The minimum cut problem is perhaps the most classical example, but it often produces highly unbalanced solutions, such as isolating a single low-degree vertex (Ford and Fulkerson, 1963) . To address this issue, the sparsest cut (or low-conductance cut) objective instead minimizes the ratio between the size of the cut, |∂S| , and the size of the smaller side of the partition, |S| (or, in the conductance case, the smaller volume side), and has achieved notable success (Leighton and Rao, 1999; Chung, 1996) .  \nHowever, a limitation of the sparsest cut and low-conductance cut objectives is that they capture only a single structure in a graph—the set with the worst bottleneck. In reality, the relationship between cut size |∂S| and set size |S| can be much more complex. Leskovec et al.(2009) conducted a systematic study of the so-called network community profile, which measures the minimum conductance over a range of different set sizes, across a large collection of realworld graphs. They observed that, in many graphs, the minimum-conductance sets at different size scales can have dramatically different conductance values. For example, the minimum conductance set of size roughly half of the graph can have conductance several orders of magnitude larger than that of the unconstrained minimum-conductance set.  \nThe work was funded in part by NSF CCF-1909528, IIS-2007481, DOE DE-SC0023162, and the IARPA Agile program. We thank C. Seshadhri for discussions on this topic.  \nA related size-specific cut measure is µ-conductance, originally introduced by Lov´asz and Simonovits in the study of Markov chains for sampling convex bodies (Lov´asz and Simonovits, 1990) . µ-conductance seeks the minimum-conductance set subject to a size constraint: the volume of the set S, denoted Vol(S), must be at least a µ fraction of the total graph volume. By varying µ, one can reveal richer information about the cut structure of a graph while ignoring sets of extremely small volume.  \nDespite their descriptive power, such size-specific cut profiles cannot be computed exactly, as computing minimum-conductance sets is NP-hard ( ˇS´ıma and Schaeffer, 2006) . A common heuristic approach is to generate many graph cuts using different low-conductance algorithmsand aggregate the results (Leskovec et al. , 2009) . This approach faces two main challenges. First, existing approximation algorithms for low-conductance sets often provide guarantees only in asymptotic Big-O form, for example O(log n) or O ( √log n), which obscures the quality of the resulting cuts in practice (Leighton and Rao, 1999; Khandekar, Rao, and Vazirani, 2009; Arora, Rao, and Vazirani, 2009) . Second, algorithms for finding low-conductance sets under size or volume constraints are typically bicriteria, meaning they do not enforce","cbCaijxm6plHUkT5","https://ap.wps.com/l/cbCaijxm6plHUkT5","pdf",402219,1,16,"English","en",105,"# Abstract\n# Introduction\n## Background: cuts and conductance objectives\n## Limitations of single bottleneck structures\n## µ-conductance and size-specific cut profiles\n## Computational challenges and prior bounds\n## Cheeger-type inequalities and spectral programs","[{\"question\":\"How does this work connect to classical Cheeger’s inequality?\",\"answer\":\"Cheeger’s inequality bounds conductance using eigenvalues of the normalized Laplacian. The paper presents a size-specific version that replaces global conductance with µ-conductance and shows an analogous spectral characterization through a tailored spectral relaxation.\"}]",1784195456,40,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":27},"a-cheeger-inequality-for-size-specific-conductance","",{"@graph":35,"@context":77},[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/a-cheeger-inequality-for-size-specific-conductance/84410/",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],{"name":72,"@type":73,"acceptedAnswer":74},"How does this work connect to classical Cheeger’s inequality?","Question",{"text":75,"@type":76},"Cheeger’s inequality bounds conductance using eigenvalues of the normalized Laplacian. The paper presents a size-specific version that replaces global conductance with µ-conductance and shows an analogous spectral characterization through a tailored spectral relaxation.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,111,114,119,122,126],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":45,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":45,"category_name":109,"show_sort_weight":28,"slug":110},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":112,"slug":113},30,"research-report",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":45,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":98,"slug":129},19,"General","general"]