[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84163-en":3,"doc-seo-84163-105":28,"detail-sidebar-cat-0-en-105":90},{"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":11,"language":21,"language_code":22,"site_id":23,"html_lang":22,"table_of_contents":24,"faqs":25,"seo_title":13,"seo_description":14,"update_tm":26,"read_time":27},84163,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Subspace Consensus of Matrix Weighted Networks","This paper studies the subspace consensus problem in matrix-weighted multi-agent networks, where each agent holds a vector state in Rd and neighboring interactions are defined by matrix-valued edge weights. Instead of requiring agreement on all state components, the work targets agreement only on a selected subspace V, formalized by asymptotic convergence of projected state differences. Algebraic results give necessary and sufficient conditions via edge-weight null-space structure, while topological results use V-connectivity, V-spanning trees, and graph-cut conditions, including refined tree-network criteria.","Subspace Consensus of Matrix-Weighted Networks  \nYuhao Chen, Lulu Pan, Xiaohui Gong, Peng Wang, Haibin Shao, Member, IEEE  \narXiv :2607 .06970v1 [ ee ss . SY] 8 Jul 2026  \nAbstract—This paper investigates the subspace consensus problem of matrix-weighted multi-agent networks, where each agent possesses a vector-valued state in Rd and interactions between neighboring agents are characterized by matrix-valued edge weights. Besides all dimensions of the agent states achieve full-state consensus, many practical applications appeal that agents are required to agree only on certain dimensions while maintaining desired relative configurations in the remaining ones. To address this gap, we introduce the concept of subspace consensus. A matrix-weighted network is said to achieve subspace consensus on a subspace V ⊆ Rd if the projection of the agents’ state differences onto V asymptotically converges to zero. This definition renders the traditional consensus as a special case when V = Rd. From an algebraic perspective, we derive necessary and sufficient conditions for subspace consensus by analyzing the interplay between the null spaces of edge weights. From a topological perspective, we present sufficient conditions characterized by V-connectivity and the existence of a V-spanning tree, as well as necessary conditions based on graph cuts. Furthermore, we provide refined necessary and sufficient conditions specifically for tree networks. This work uncoversa fundamental capability inherent to matrix-weighted networksand establishes a systematic framework for analyzing agreement behaviors on prescribed subspaces.  \nIndex Terms—Subspace consensus, matrix-weighted networks, subspace connectivity, subspace spanning tree.  \nI. INTRODUCTION  \nOver the past two decades, the paradigm of distributed multi-agent coordination has emerged as a cornerstone in the study of networked systems, underpinning a wide range of applications including distributed estimation, control, optimization, and learning over networks [3], [4] . At the heart of this paradigm lies the consensus problem, wherein a group of agents, each equipped with a local state, interact through a communication network to asymptotically agree on a common value [6], [2] . The classical consensus framework, however, has long operated under the assumption that interagent interactions are captured by scalar-valued edge weights, thereby overlooking the potential complexity of interactions when agent states are vector-valued.  \nIn many applications, the state of each agent naturally resides in a higher-dimensional space. For instance, in formation control, an agent’s state may encode its position and orientation [16]; in opinion dynamics, it may represent stances on multiple topics [14]; in sensor networks, it may capture multidimensional measurements [1] . In such settings, interactions between agents are not merely scalar-weighted but can involve intricate couplings across different dimensions of the state vectors. This observation has motivated a growing body of research on matrix-weighted multi-agent networks, where each edge is endowed with a matrix-valued  \nThe authors are with the School of Automation and Intelligent Sensing, Shanghai Jiao Tong University, Shanghai 200240, China.  \nweight that modulates the influence between agents across the dimensions of their states [11], [7], [13], [5], [8] . Such matrix-weighted couplings naturally arise in scenarios such as generalized effective resistance in electrical networks [1], multi-topic opinion dynamics [14], bearing-based distributed formation control [16], and the dynamics of arrays of coupled oscillators [12] .  \nHowever, most existing results on matrix-weighted networks focus on a particular type of collective behavior, namely full-state consensus, in which all components of the agents’state vectors converge to a common value. These formulations, while important, do not fully capture the richness of behaviors that matrix-weighted inte","cbCaigv9Fm1cJbia","https://ap.wps.com/l/cbCaigv9Fm1cJbia","pdf",1141867,1,"English","en",105,"# Introduction\n## Background on consensus and matrix-weighted interactions\n## Motivation for subspace-specific agreement\n## Concept and problem statement of subspace consensus\n## Paper contributions and analysis perspectives","[{\"question\":\"What is subspace consensus in matrix-weighted networks?\",\"answer\":\"A matrix-weighted network achieves subspace consensus on a subspace V if the projection of agents’ state differences onto V asymptotically converges to zero. When V equals Rd, this reduces to traditional full-state consensus.\"},{\"question\":\"How does the paper characterize subspace consensus from an algebraic perspective?\",\"answer\":\"It derives necessary and sufficient conditions by analyzing how the null spaces of the matrix edge weights interact, linking consensus capability to the algebraic structure induced by these null spaces.\"},{\"question\":\"What topological conditions are used to guarantee or rule out subspace consensus?\",\"answer\":\"The paper provides sufficient conditions based on V-connectivity and the existence of a V-spanning tree, and it also presents necessary conditions using graph cuts. It further refines necessary and sufficient criteria for tree networks.\"}]",1784193574,20,{"code":4,"msg":29,"data":30},"ok",{"site_id":23,"language":22,"slug":31,"title":13,"keywords":32,"description":14,"schema_data":33,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":26},"subspace-consensus-of-matrix-weighted-networks","",{"@graph":34,"@context":84},[35,52,67],{"@type":36,"itemListElement":37},"BreadcrumbList",[38,42,46,49],{"item":39,"name":40,"@type":41,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":43,"name":44,"@type":41,"position":45},"https://docshare.wps.com/document/","Document",2,{"item":47,"name":12,"@type":41,"position":48},"https://docshare.wps.com/document/research-report/",3,{"item":50,"name":13,"@type":41,"position":51},"https://docshare.wps.com/document/subspace-consensus-of-matrix-weighted-networks/84163/",4,{"url":50,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":22,"description":14,"dateModified":60,"datePublished":61,"encodingFormat":59,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":39,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":20},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What is subspace consensus in matrix-weighted networks?","Question",{"text":74,"@type":75},"A matrix-weighted network achieves subspace consensus on a subspace V if the projection of agents’ state differences onto V asymptotically converges to zero. When V equals Rd, this reduces to traditional full-state consensus.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the paper characterize subspace consensus from an algebraic perspective?",{"text":79,"@type":75},"It derives necessary and sufficient conditions by analyzing how the null spaces of the matrix edge weights interact, linking consensus capability to the algebraic structure induced by these null spaces.",{"name":81,"@type":72,"acceptedAnswer":82},"What topological conditions are used to guarantee or rule out subspace consensus?",{"text":83,"@type":75},"The paper provides sufficient conditions based on V-connectivity and the existence of a V-spanning tree, and it also presents necessary conditions using graph cuts. It further refines necessary and sufficient criteria for tree networks.","https://schema.org",{"og:url":50,"og:type":86,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":88,"canonical":50},"index,follow",{"doc_id":7,"site_id":23},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,126,129,133],{"id":20,"doc_module":4,"doc_module_name":44,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":45,"doc_module":4,"doc_module_name":44,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":51,"doc_module":4,"doc_module_name":44,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":44,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":44,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":44,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":44,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":44,"category_name":124,"show_sort_weight":27,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":27,"doc_module":4,"doc_module_name":44,"category_name":127,"show_sort_weight":27,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":44,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":44,"category_name":135,"show_sort_weight":105,"slug":136},19,"General","general"]