[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82870-en":3,"doc-seo-82870-105":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},82870,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Active Learning on Adversarially Corrupted Graphs","Motivated by real-world attacks that tamper with networks, the paper models an adversary that embeds a set of corrupted vertices inside a graph G*. The adversary can add edges among corrupted vertices and between them and G*, with its power measured by how large the corrupted vertices’ neighborhood becomes in G*. The goal is an active learning algorithm that identifies the corrupted subset using only a small number of label queries, with query complexity depending polynomially on the adversary power and on the vertex expansion of G*, a key connectivity measure.","arXiv :2607 .04869v 1 [ cs .LG] 6 Jul 2026  \nActive Learning on Adversarially Corrupted Graphs  \nMarco Bressan  \nUniversità degli Studi di Milano, Italy  \nNicolò Cesa-Bianchi  \nUniversità degli Studi di Milano, Italy  \nTommaso d’Orsi  \nBocconi University, Italy  \nEmmanuel Esposito  \nUniversità degli Studi di Milano, Italy  \nSilvio Lattanzi  \nGoogle Research  \nMARCO . BRESSAN @UNIMI. IT NICOLO . CESA-BIANCHI@UNIMI. IT TOMMASO . DORSI @UNIBOCCONI. IT EMMANUEL @EMMANUELESPOSITO . IT  \nSILVIOL @ GOOGLE . COM  \nAbstract  \nMotivated by real-world scenarios where malicious entities tamper with existing networks, we define a model where an adversary seeks to hide a set of corrupted vertices inside a graph G∗ . To this end, the adversary can add edges between the corrupted vertices, as well as edges between the corrupted vertices and G∗ , and its power is then measured by the size of the neighborhood of the corrupted vertices in G∗ . Our goal is to design an active learning algorithm that efficiently finds the subset of corrupted vertices using a small number of label queries. We devise an efficient algorithm that approximately recovers the corrupted vertices with a query complexity that depends polynomiallyon both the power of the adversary and the vertex expansion of G∗ , a fundamental measure of graph connectivity. At the heart of this result is a polynomial-time algorithm, obtained by carefully adapting sum-of-squares algorithms for approximating minimum expansion, that finds a set with small vertex expansion subject to cardinality constraints. To the best of our knowledge, this is the first time that the vertex expansion is shown to play a key role in determining the query complexity of active learning algorithms robust to structural adversarial attacks.  \nKeywords: active learning, weak recovery, adversarial robustness  \n1. Introduction  \nGraph-based machine learning is a powerful paradigm for analyzing relational data across diverse domains such as social networks, bioinformatics, web analysis, and recommendation systems. A key application is node classification, aiming to infer node labels or attributes from the graph structure and any available node or edge features. While standard approaches typically depend on the integrity of the observed graph structure, this presumption is often undermined by adversarial interventions. Malicious actors can, in fact, manipulate the graph by creating deceptive nodes or engineering  \nAuthors are listed in alphabetical order.  \n© M. Bressan, N. Cesa-Bianchi, T. d’Orsi, E. Esposito & S. Lattanzi.  \nBRESSAN CESA-BIANCHI D’ORSI ESPOSITO LATTANZI  \nspurious connections. This threat is especially pronounced in real-world applications like anti-abuse systems, where adversaries strategically deploy fake entities and artificial links to propagate spam, misinformation, or engage in fraudulent activities (Yu et al., 2008a,b ; Danezis and Mittal, 2009 ; Tranet al., 2011 ; Alvisi et al., 2013) .  \nThis paper addresses the problem of active learning on graphs in the presence of such structural adversarial manipulations. Active learning in graphs aims to minimize the cost of data labeling by exploiting the graph structure to select which nodes to query for their true labels (Guillory and Bilmes, 2009) . Most previous work focuses on adversaries who choose the node labeling, rather than adversaries who modify the graph structure itself. In this paper, we focus instead on adversarial structural changes and on algorithms with formal guarantees for general graphs.  \nTowards this end, we introduce an adversary that captures many practical safety scenarios. Given an(2arbdis)niartrintibargalitiluigrayshraprilymaVhn(Gcoy eG)n,dvet mGV(∗astTrybhenrgto(irofisceanithgibrnaphmo thawveshrtltiipegtsg| (,GaGn|nvd(er3h)cas vertTices ntundonnnetihdmt,hibt)a.dhavsearpyut thatsigniaficalnticieffoourstianctor cordera easily crecorrupt anate neexistiwngcovrrupttexedof Gsotoandal nsmneorkr ntaer.rsasurbitars rarymeo-rrldureelye","cbCaie8F0c7azels","https://ap.wps.com/l/cbCaie8F0c7azels","pdf",555284,1,37,"English","en",105,"# Introduction\n## Problem setting and motivation\n## Adversary model and goal\n## Main contribution and theoretical link","[{\"question\":\"What does the adversary do in the proposed model of corrupted graphs?\",\"answer\":\"The adversary hides a set of corrupted vertices inside the graph by adding edges among corrupted vertices and also between corrupted vertices and the original graph G*. Its strength is captured by the size of the corrupted vertices’ neighborhood in G*.\"},{\"question\":\"How is success of the active learning algorithm defined?\",\"answer\":\"The algorithm aims to recover a high fraction of the corrupted nodes by querying the true labels of only a small number of vertices.\"},{\"question\":\"Why does vertex expansion matter for query complexity?\",\"answer\":\"The paper proves that the number of label queries needed to reach high accuracy is fundamentally linked to the graph’s vertex expansion and the adversary budget. Low vertex expansion lets corrupted vertices be hidden more effectively within sparsely connected regions.\"}]",1784183571,93,{"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},"active-learning-on-adversarially-corrupted-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/active-learning-on-adversarially-corrupted-graphs/82870/",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 does the adversary do in the proposed model of corrupted graphs?","Question",{"text":75,"@type":76},"The adversary hides a set of corrupted vertices inside the graph by adding edges among corrupted vertices and also between corrupted vertices and the original graph G*. Its strength is captured by the size of the corrupted vertices’ neighborhood in G*.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is success of the active learning algorithm defined?",{"text":80,"@type":76},"The algorithm aims to recover a high fraction of the corrupted nodes by querying the true labels of only a small number of vertices.",{"name":82,"@type":73,"acceptedAnswer":83},"Why does vertex expansion matter for query complexity?",{"text":84,"@type":76},"The paper proves that the number of label queries needed to reach high accuracy is fundamentally linked to the graph’s vertex expansion and the adversary budget. 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