[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85561-en":3,"doc-seo-85561-105":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},85561,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Approximate Simulation-Based Verification of Compatibility of the Friedkin–Johnsen Model with Binary Observations","Verification of opinion dynamics from limited, discrete information focuses on binary observations rather than continuous agent opinions. The Friedkin–Johnsen (FJ) model generates a continuous opinion trajectory, while each agent’s observed output is binarized by thresholding. The work formulates the FJ model as a transition system and defines an approximate simulation relation using proximity of opinion trajectories and output sequences. Finite abstract models are built by simplifying influence parameters and discretizing stubbornness and initial opinions. Results guarantee approximation consistency checks over the finite abstraction, complemented by numerical experiments.","Approximate Simulation-Based Verification of Compatibility of the Friedkin–Johnsen Model with Binary Observations*  \nYu Xing 1 , Aneesh Raghavan2 , Michael T. Schaub 1 , and Karl H. Johansson2  \narXiv :2604 .05196v3 [ ee ss . SY] 11 Jul 2026  \nAbstract—We consider a verification problem for opinion dynamics based on binary observations. The opinion dynamics is governed by a Friedkin-Johnsen (FJ) model, where only a sequence of binary outputs is available instead of the agents’continuous opinions. At every time-step we observe a binarized output for each agent depending on whether the opinion exceedsa fixed threshold. The objective is to verify whether an FJ model with a given set of stubbornness parameters and initial opinions can generate the observed binary outputs up to a small error. The FJ model is formulated as a transition system, and an approximate simulation relation of two transition systems is defined in terms of the proximity of their opinion trajectories and output sequences. We then construct a finite set of abstract FJ models by simplifying the influence matrix and discretizing the stubbornness parameters and the initial opinions. It is shown that the abstraction approximately simulates any concrete FJ model with continuous parameters and initial opinions, and is itself approximately simulated by some concrete FJ model. These results ensure that consistency verification can be performed over the finite abstraction. Specifically, by checking whether an abstract model satisfies the observation constraints, we can conclude whether the corresponding family of concrete FJ models is consistent with the binary observations. Finally, numerical experiments are presented to illustrate the proposed verification framework.  \nI. INTRODUCTION  \nThe study of social opinion dynamics has attracted significant interest across disciplines [1], due to its wide range of applications, such as in marketing and recommender systems [2] . In recent years, an increasing number of studies have focused on learning such dynamics from data [3], [4],[5] . However, existing methods often require collecting along sequence of data from a single trajectory or snapshots from multiple trajectories. In practice, typically only a few opinion samples are available for each individual, and the data may be discrete-valued rather than continuous [6],[7], [8] . It is thus important to investigate how to infer information about social dynamics from such limited data.  \nA. Related Work  \nOpinion dynamics with continuous states include linear averaging models such as the DeGroot model and the  \n*This work was supported by the Alexander von Humboldt Foundation, Knut and Alice Wallenberg Foundation (Wallenberg Scholar grant), the Swedish Research Council (Distinguished Professor grant 2017-01078), and the Swedish Foundation for Strategic Research (SUCCESS FUS21-0026) .  \n1YX and MTS are with the Faculty of Computer Science, RWTH Aachen University, Germany. MTS is also with the Department of Biology, RWTH Aachen University. {yu .xing, [michael.schaub](michael.schaub}@rwth-aachen.de)[}](michael.schaub}@rwth-aachen.de)[@rwth-aachen.de](michael.schaub}@rwth-aachen.de)  \n2AR and KHJ are with Department of Decision and Control Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, and also with Digital Futures, Stockholm, Sweden.{aneesh,[kallej](kallej}@kth.se)[}](kallej}@kth.se)[@kth.se](kallej}@kth.se)  \nFriedkin–Johnsen (FJ) model [9] . With the rise of online social network platforms, researchers have increasingly investigated learning algorithms for opinion dynamics [3] . Learning sparse networks based on steady states has been studied in [10] for the DeGroot model, and in [11] for the FJ model. Learning network structure from discretevalued dynamical observations has been studied in e.g., [12],[13] . Inferring community structures is another approach to reducing sample complexity for large networked dynamics, such as ","cbCaiotj8WedOZI9","https://ap.wps.com/l/cbCaiotj8WedOZI9","pdf",492274,1,7,"English","en",105,"# Abstract\n# Introduction\n## Related Work\n## Contribution","[{\"question\":\"What problem does the paper address in opinion dynamics verification?\",\"answer\":\"The paper verifies whether an FJ model with specified stubbornness parameters and initial opinions can reproduce observed binary outputs within a small error.\"},{\"question\":\"How are binary observations generated from the Friedkin–Johnsen model?\",\"answer\":\"At each time step, each agent’s continuous opinion is compared to a fixed threshold, and the observed output is binarized accordingly.\"},{\"question\":\"How does the framework make the verification computationally feasible?\",\"answer\":\"It constructs finite abstract FJ models by simplifying the influence matrix and discretizing stubbornness parameters and initial opinions, then performs consistency checking on the finite abstraction.\"}]",1784204588,18,{"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},"approximate-simulation-based-verification-of-compatibility-of-the-friedkinjohnsen-model-with-binary-observations","",{"@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/approximate-simulation-based-verification-of-compatibility-of-the-friedkinjohnsen-model-with-binary-observations/85561/",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 problem does the paper address in opinion dynamics verification?","Question",{"text":75,"@type":76},"The paper verifies whether an FJ model with specified stubbornness parameters and initial opinions can reproduce observed binary outputs within a small error.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are binary observations generated from the Friedkin–Johnsen model?",{"text":80,"@type":76},"At each time step, each agent’s continuous opinion is compared to a fixed threshold, and the observed output is binarized accordingly.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the framework make the verification computationally feasible?",{"text":84,"@type":76},"It constructs finite abstract FJ models by simplifying the influence matrix and discretizing stubbornness parameters and initial opinions, then performs consistency checking on the finite abstraction.","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},[93,97,101,105,110,115,119,122,127,130,134],{"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":21,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]