Learning from a Teacher: Andrew the First-Called and Thomas the Doubter

Learning from a Teacher: Andrew the First-Called and Thomas the Doubter | 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 Learning from a Teacher: Andrew the First-Called and Thomas the Doubter Dimitri Volchenkov This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6710850/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 20 Jan, 2026 Read the published version in Nonlinear Dynamics → Version 1 posted 14 You are reading this latest preprint version Abstract We present a nonlinear reaction–diffusion model of belief propagation in hierarchical networks, incorporating logistic self-amplification and degenerating diffusion. Inspired by contrasting apostolic archetypes, Andrew the First-Called and Thomas the Doubter, the model captures both immediate social adoption and delayed, self-sustained belief. We show that when influence diffusivity is low, FKPP-type pulled waves fail to reach the network periphery, whereas degenerating diffusion (DD), despite being slower, enables spatially correlated and temporally synchronized activation in distal layers. Analytical traveling-wave solutions reveal that belief fronts in the DD regime retain memory of initial conditions and saturate without direct influence from the source. Numerical simulations confirm that belief saturation propagates through indirect coordination rather than sequential diffusion, illuminating the non-intuitive dynamics of large-scale influence. Applications to digital misinformation, political persuasion, and collective decision-making are discussed. Mathematics Subject Classification (2020) 35K57 · 91D30 · 60J28 Belief dynamics in networks Degenerating diffusion Nonlinear reaction–diffusion systems Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 20 Jan, 2026 Read the published version in Nonlinear Dynamics → Version 1 posted Editorial decision: Revision requested 15 Jul, 2025 Reviews received at journal 15 Jul, 2025 Reviewers agreed at journal 08 Jul, 2025 Reviewers agreed at journal 08 Jul, 2025 Reviews received at journal 05 Jul, 2025 Reviewers agreed at journal 05 Jul, 2025 Reviews received at journal 24 Jun, 2025 Reviews received at journal 03 Jun, 2025 Reviewers agreed at journal 02 Jun, 2025 Reviewers agreed at journal 30 May, 2025 Reviewers invited by journal 30 May, 2025 Editor assigned by journal 29 May, 2025 Submission checks completed at journal 21 May, 2025 First submitted to journal 20 May, 2025 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-6710850","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":463803838,"identity":"d23a2b46-d26a-4466-a10b-bb2095ae683d","order_by":0,"name":"Dimitri Volchenkov","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABAElEQVRIiWNgGAWjYDCCwwwMBxgYJIAs5gZmIGkAZByASB3Ao+UAWAsjTAtbYsMBfFoOwOXgWngM8WrhO86dePjjHgsGg+ONjZ8LKu4Y80vkfH/8sY1Bju9GAlYtkod5Nxw48EyCweDMwWbpGWeemUnOyN3YcLCNwVgShxYDsJYDEgySMxLbmHnbDtsYnDkL1pK4gTgt/0BazjwEaaknqIVfAqSl4bCZwfEeRpCWBAN8fjlzQIKHnwfoF55jh40l29sMZ5w5J2E488wD7CF2/uzmDxUH6uTY2JsPfuapOWzYz8z84ENFmY0833HstsAAD7qABF7lo2AUjIJRMArwAwAyfW0butjXfgAAAABJRU5ErkJggg==","orcid":"","institution":"Texas Tech University","correspondingAuthor":true,"prefix":"","firstName":"Dimitri","middleName":"","lastName":"Volchenkov","suffix":""}],"badges":[],"createdAt":"2025-05-20 21:08:09","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6710850/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6710850/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11071-025-11995-3","type":"published","date":"2026-01-20T15:57:59+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":101152847,"identity":"a3cabcb2-27da-4dab-9c8c-d4e6adf05005","added_by":"auto","created_at":"2026-01-26 16:13:21","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":910198,"visible":true,"origin":"","legend":"","description":"","filename":"NDSUBMITTEDVOLCHENKOVTENREIROMemorial.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6710850/v1_covered_a793174e-9a37-4fc0-a924-f4f69a79d868.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Learning from a Teacher: Andrew the First-Called and Thomas the Doubter","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nonlinear-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nody","sideBox":"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)","snPcode":"11071","submissionUrl":"https://submission.nature.com/new-submission/11071/3","title":"Nonlinear Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Belief dynamics in networks, Degenerating diffusion, Nonlinear reaction–diffusion systems","lastPublishedDoi":"10.21203/rs.3.rs-6710850/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6710850/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe present a nonlinear reaction–diffusion model of belief propagation in hierarchical networks, incorporating logistic self-amplification and degenerating diffusion. Inspired by contrasting apostolic archetypes, Andrew the First-Called and Thomas the Doubter, the model captures both immediate social adoption and delayed, self-sustained belief. We show that when influence diffusivity is low, FKPP-type pulled waves fail to reach the network periphery, whereas degenerating diffusion (DD), despite being slower, enables spatially correlated and temporally synchronized activation in distal layers. Analytical traveling-wave solutions reveal that belief fronts in the DD regime retain memory of initial conditions and saturate without direct influence from the source. Numerical simulations confirm that belief saturation propagates through indirect coordination rather than sequential diffusion, illuminating the non-intuitive dynamics of large-scale influence. Applications to digital misinformation, political persuasion, and collective decision-making are discussed.\u003c/p\u003e\n\u003cp\u003eMathematics Subject Classification (2020) 35K57 · 91D30 · 60J28\u003c/p\u003e","manuscriptTitle":"Learning from a Teacher: Andrew the First-Called and Thomas the Doubter","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-05-30 05:09:17","doi":"10.21203/rs.3.rs-6710850/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-07-15T21:56:41+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-07-15T17:49:05+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"225119258799852484835891924461299660407","date":"2025-07-08T13:47:19+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"284146301697360679137379590537580575922","date":"2025-07-08T09:36:15+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-07-05T21:24:17+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"72915282741320881875362787582818854093","date":"2025-07-05T21:08:05+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-06-24T19:46:30+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-06-03T19:54:05+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"308574589692462837992262058053737957969","date":"2025-06-02T08:42:30+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"94063531668989579082972655207100068967","date":"2025-05-30T16:53:36+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-05-30T10:55:58+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-05-29T18:43:26+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-05-21T14:56:42+00:00","index":"","fulltext":""},{"type":"submitted","content":"Nonlinear Dynamics","date":"2025-05-20T20:57:45+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nonlinear-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nody","sideBox":"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)","snPcode":"11071","submissionUrl":"https://submission.nature.com/new-submission/11071/3","title":"Nonlinear Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"7801d9db-3aec-4bf0-a9da-df07ccc43603","owner":[],"postedDate":"May 30th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2026-01-26T16:10:20+00:00","versionOfRecord":{"articleIdentity":"rs-6710850","link":"https://doi.org/10.1007/s11071-025-11995-3","journal":{"identity":"nonlinear-dynamics","isVorOnly":false,"title":"Nonlinear Dynamics"},"publishedOn":"2026-01-20 15:57:59","publishedOnDateReadable":"January 20th, 2026"},"versionCreatedAt":"2025-05-30 05:09:17","video":"","vorDoi":"10.1007/s11071-025-11995-3","vorDoiUrl":"https://doi.org/10.1007/s11071-025-11995-3","workflowStages":[]},"version":"v1","identity":"rs-6710850","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6710850","identity":"rs-6710850","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}