Consensus Measurement in Complex Space: Statistical Modeling and Empirical Validation on Social Network Data

Consensus Measurement in Complex Space: Statistical Modeling and Empirical Validation on Social Network Data | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Consensus Measurement in Complex Space: Statistical Modeling and Empirical Validation on Social Network Data Mushtaq K. Abdalrahem This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7802848/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Measurement of public opinion and consensus faces unprecedented challenges in the digital age, where attitudes are increasingly expressed as complex, multidimensional constructs rather than simple binary positions. Traditional analytical approaches based on sentiment analysis or unidimensional variance measures prove inadequate for capturing the nuanced relationships between different dimensions of opinion in these complex spaces. This study proposes a novel statistical framework for measuring consensus based on principles from multivariate analysis of variance (MANOVA) and distance metrics in multidimensional Euclidean space. The methodology applies this model to a comprehensive dataset of Twitter discourse concerning COVID-19 vaccination, utilizing both topic modeling and transformer-based embeddings to represent opinions in high-dimensional spaces. Our results demonstrate the model's effectiveness in quantifying consensus on a continuous scale from 0 to 1, revealing that online discussions frequently maintain multipolar opinion structures rather than trending toward either complete consensus or polarization. Furthermore, we identify strong correlations between network properties—particularly community structure and density—and internal consensus levels within subcommunities. The proposed Consensus Index provides a powerful quantitative tool for analyzing opinion dynamics in complex spaces, with significant applications in computational social science, social media marketing, and polarization research. Applied Mathematics Applied Statistics Consensus Measurement Multivariate Analysis Social Network Analysis Opinion Dynamics Computational Social Science Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7802848","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":526196926,"identity":"232f82c7-bb19-4d43-a146-c1e056812dfd","order_by":0,"name":"Mushtaq K. Abdalrahem","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABA0lEQVRIiWNgGAWjYHACNjApwd7AeADMApISYIRPC0ipBM8BBlK1SCSgaMENdGekP3v8ocaaQXLm4wcHfubYyPEdYD54m4fBQrYBhxazGznmBgeOpTNIS6cZHOzdlmYseYAt2ZqHQcIYjxY2iQNshxnkpBMMDvBuO5y44QCPmTRQSyJuLenPJA78A2qRPP7h4N9t/+s3HOD/RkBLgpnEwbbDDNISPAaHebcdANrFw4Zfy5k3ZhJn+9J5JHtyCg7Lbks2nHmYzdhyjgEevxwHOqzim7WcxPHjGx++3WYnz3e8+eGNNxV1OEMMCph5kNggwoCBkZAWTCFCWkbBKBgFo2DkAABP81sjGBdAbgAAAABJRU5ErkJggg==","orcid":"","institution":"University of Al-Ameed","correspondingAuthor":true,"prefix":"","firstName":"Mushtaq","middleName":"K.","lastName":"Abdalrahem","suffix":""}],"badges":[],"createdAt":"2025-10-07 22:52:17","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7802848/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7802848/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":93107571,"identity":"fc6ba652-35ec-4a84-8d90-86ffc8ef0478","added_by":"auto","created_at":"2025-10-09 06:56:19","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":795933,"visible":true,"origin":"","legend":"","description":"","filename":"ConsensusMeasurementinComplexSpace.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7802848/v1_covered_68a817d3-bb81-4bb4-8fc0-38be3e18635b.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eConsensus Measurement in Complex Space: Statistical Modeling and Empirical Validation on Social Network Data\u003c/strong\u003e\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"University of Al-Ameed","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Consensus Measurement, Multivariate Analysis, Social Network Analysis, Opinion Dynamics, Computational Social Science","lastPublishedDoi":"10.21203/rs.3.rs-7802848/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7802848/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eMeasurement of public opinion and consensus faces unprecedented challenges in the digital age, where attitudes are increasingly expressed as complex, multidimensional constructs rather than simple binary positions. Traditional analytical approaches based on sentiment analysis or unidimensional variance measures prove inadequate for capturing the nuanced relationships between different dimensions of opinion in these complex spaces. This study proposes a novel statistical framework for measuring consensus based on principles from multivariate analysis of variance (MANOVA) and distance metrics in multidimensional Euclidean space. The methodology applies this model to a comprehensive dataset of Twitter discourse concerning COVID-19 vaccination, utilizing both topic modeling and transformer-based embeddings to represent opinions in high-dimensional spaces. Our results demonstrate the model's effectiveness in quantifying consensus on a continuous scale from 0 to 1, revealing that online discussions frequently maintain multipolar opinion structures rather than trending toward either complete consensus or polarization. Furthermore, we identify strong correlations between network properties—particularly community structure and density—and internal consensus levels within subcommunities. The proposed Consensus Index provides a powerful quantitative tool for analyzing opinion dynamics in complex spaces, with significant applications in computational social science, social media marketing, and polarization research.\u003c/p\u003e","manuscriptTitle":"Consensus Measurement in Complex Space: Statistical Modeling and Empirical Validation on Social Network Data","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-09 06:48:11","doi":"10.21203/rs.3.rs-7802848/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"4710bfbe-6508-4076-896e-22cac2ec47ae","owner":[],"postedDate":"October 9th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":55927661,"name":"Applied Mathematics"},{"id":55927662,"name":"Applied Statistics"}],"tags":[],"updatedAt":"2025-10-09T06:48:11+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-09 06:48:11","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7802848","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7802848","identity":"rs-7802848","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}