[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84783-en":3,"doc-seo-84783-105":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},84783,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Pathological Regimes of Closed-Loop Recommendation Systems over Social Networks","This paper studies the design of recommendation systems for social networks and e-commerce through a control-theoretic lens. Recommendation design is modeled as an infinite-horizon state-feedback optimal control problem with a cost that rewards alignment/engagement while penalizing polarization, deviation from an uncontrolled baseline, recommendation mismatch, control effort, and exposure disagreement across neighboring users. Explicit spectral conditions are derived to guarantee strict positive-definiteness; violating them yields pathological behaviors such as unstable free modes, failure to attain the infimum, or breakdown of stationary affine synthesis.","arXiv :2607 .05010v1 [ ee ss . SY] 6 Jul 2026  \nPathological Regimes of Closed-Loop Recommendation Systems over Social Networks ⋆  \nSimone Mariano ∗ ,∗∗ Paolo Frasca ∗  \n∗ Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-lab, 38000 Grenoble, France (e-mail: [simone.mariano@grenoble-inp.fr](simone.mariano@grenoble-inp.fr), paolo.frasca@gipsa-lab.fr) .∗∗ Univ. Grenoble Alpes, CNRS, Sciences Po Grenoble-UGA, Pacte, 38000 Grenoble, France.  \nAbstract: This paper addresses the problem of designing recommendation systems for social networks and e-commerce platforms from a control-theoretic perspective. We formulate recommendation design as an infinite-horizon state-feedback optimal control problem whose performance index rewards alignment/engagement while penalizing polarization, large deviations from an uncontrolled baseline, recommendation mismatch, control effort, and exposure disagreement across neighboring users. We derive explicit spectral conditions under which the reduced quadratic stage cost is strictly positive-definite, and we show that the failure of these conditions makes the resulting recommendation design exhibit pathological behaviors, such as unstable free modes, non-attainment of the infimum, or failure of the stationary affine synthesis.  \nKeywords: Social networks and opinion dynamics; Control of networks; Applications of optimal control; Recommendation systems  \n1. INTRODUCTION  \nRecommendation systems increasingly mediate how users encounter information, products, and social content in online spaces, such as social media and e-commerce platforms. While recommendations are often optimized to improve short-term engagement, the resulting feedback loop between recommendations and user behavior may amplify confirmation bias and contribute to undesirable collective phenomena, including polarization, echo chambers, and extreme opinions (Gausen et al., 2022; Husz´ar et al., 2022; Bail et al., 2018) . Moreover, combined with insufficient exploration, maximizing short-term engagement may lead to preferences to drift toward degenerate or extreme contents (Jiang et al., 2019), and repeatedly training on selfinfluenced logs may reduce diversity and long-run utility (Chaney et al., 2018) . A basic mechanism behind these effects is that engagement-oriented recommendations can push users toward more extreme or polarized positions, while extreme positions may themselves generate higher engagement and monopolize users’ attention. This mechanism is captured in closed-loop form in the single-user model by Rossi et al. (2021), which provides a useful baseline for studying recommendation-induced opinion dynamics.  \nThe perspective developed by Dean et al. (2025) highlights the limitations discussed above and argues that recommendation systems should be designed with explicit models of how users and algorithms shape one another. This point  \n⋆ This work has been supported in part by ANR projects Feeding Bias (ANR-22-CE380017-01) and MIAI Cluster BEAR (ANR-23-IACL-0006) .  \nis particularly relevant because several common design choices, including memoryless architectures (Covington et al., 2016), simplified models of user dynamics, omitted creator adaptation, and aggressive engagement-oriented optimization (Chen et al., 2019; Immorlica et al. , 2024), can obscure long-term causal effects. Usually ad hoc, aposteriori mitigation strategies for ill-behaved systems are often introduced only after deployment, using logged data generated by the same platform dynamics they are meant to correct, and may fail to prevent exposure shifts, reduced content diversity, feedback-loop amplification, and polarization (Sinha et al., 2016; Chaney et al., 2018; Mansouryet al., 2020; Nguyen et al., 2014) .  \nIn contrast, the recent survey paper De Pasquale et al.(2026) presents a control-theoretic perspective, in which recommender systems are interpreted as feedback systems and fairness-related objectives are analyzed as long-term dynamical propertie","cbCaisFO3SlHZHmX","https://ap.wps.com/l/cbCaisFO3SlHZHmX","pdf",448680,1,14,"English","en",105,"# Introduction\n## Control-theoretic view of recommendation design\n## Problem setup and objectives\n## Main results: spectral conditions and well-posedness","[{\"question\":\"How does the paper model recommendation design for social networks?\",\"answer\":\"It formulates recommendation design as an infinite-horizon state-feedback optimal control problem where recommendations act as control inputs on a continuous-time opinion dynamics model over a network of users.\"},{\"question\":\"What does the performance index in the control formulation include?\",\"answer\":\"It rewards alignment/engagement while penalizing polarization, deviations from an uncontrolled baseline, recommendation mismatch, control effort, and disagreement of exposure across neighboring users.\"},{\"question\":\"What are the key consequences when the derived spectral conditions fail?\",\"answer\":\"Failure of the conditions leads to pathological behaviors such as unstable free modes, non-attainment of the infimum, or failure of the stationary affine synthesis.\"}]",1784198201,35,{"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},"pathological-regimes-of-closed-loop-recommendation-systems-over-social-networks","",{"@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/pathological-regimes-of-closed-loop-recommendation-systems-over-social-networks/84783/",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},"How does the paper model recommendation design for social networks?","Question",{"text":75,"@type":76},"It formulates recommendation design as an infinite-horizon state-feedback optimal control problem where recommendations act as control inputs on a continuous-time opinion dynamics model over a network of users.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does the performance index in the control formulation include?",{"text":80,"@type":76},"It rewards alignment/engagement while penalizing polarization, deviations from an uncontrolled baseline, recommendation mismatch, control effort, and disagreement of exposure across neighboring users.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the key consequences when the derived spectral conditions fail?",{"text":84,"@type":76},"Failure of the conditions leads to pathological behaviors such as unstable free modes, non-attainment of the infimum, or failure of the stationary affine synthesis.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]