Energy–Bridge Transfer Law: A Quantitative Bridge Between Topology and Dynamics in Graph Systems

Energy–Bridge Transfer Law: A Quantitative Bridge Between Topology and Dynamics in Graph Systems | 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 Energy–Bridge Transfer Law: A Quantitative Bridge Between Topology and Dynamics in Graph Systems Atulya Thakur This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8010502/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 The study introduces the Energy–Bridge Transfer Law (EBTL), a finite-time diffusion framework that models energy exchange between connected clusters in weighted graphs through a quantifiable spectral bridge. Departing from classical Laplacian diffusion \(\:\varvec{E}(\varvec{t}+1)=\varvec{P}\text{\hspace{0.17em}}\varvec{E}\left(\varvec{t}\right)\) , the formulation incorporates reactivity \(\:\varvec{\beta\:}\) and self-retention \(\:\varvec{s}\) , producing the generalized operator \(\:{\varvec{P}}_{\varvec{s}}=(1-\varvec{s})\varvec{I}+\varvec{s}{\varvec{D}}^{-1}\varvec{A}\) and the dynamic update \(\:{\varvec{E}}^{(\varvec{t}+1)}=(1-\varvec{\beta\:}){\varvec{E}}^{\left(\varvec{t}\right)}+\varvec{\beta\:}{\varvec{P}}_{\varvec{s}}{\varvec{E}}^{\left(\varvec{t}\right)}\) . Analytical derivation yields the stability constraint \(\:\varvec{\rho\:}\left[\right(1-\varvec{\beta\:})\varvec{I}+\varvec{\beta\:}{\varvec{P}}_{\varvec{s}}]\le\:1\) and a closed-form finite-time law \(\:{\varvec{f}}_{\varvec{t}}\left({\varvec{w}}_{\varvec{b}}\right)={\varvec{f}}_{\varvec{\infty\:}}\left({\varvec{w}}_{\varvec{b}}\right)(1-{\varvec{e}}^{-\varvec{t}/\varvec{\tau\:}\left({\varvec{w}}_{\varvec{b}}\right)})\) , unifying diffusion, spectral gap, and topological coupling in one relation. Simulations across four canonical topologies confirm near-perfect agreement with theory ( \(\:\varvec{r}=0.9945\) , RMSE = 0.0139), validating both the steady-state and transient predictions. The Energy–Bridge framework thus extends graph diffusion theory to the finite-time domain while demonstrating a transparent, reproducible model of human–AI scientific collaboration. Finite-time diffusion graph neural dynamics spectral stability energy transfer law bridge coupling reactivity coefficient human–AI research collaboration 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. 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Systems\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","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":"Finite-time diffusion, graph neural dynamics, spectral stability, energy transfer law, bridge coupling, reactivity coefficient, human–AI research collaboration","lastPublishedDoi":"10.21203/rs.3.rs-8010502/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8010502/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe study introduces the Energy\u0026ndash;Bridge Transfer Law (EBTL), a finite-time diffusion framework that models energy exchange between connected clusters in weighted graphs through a quantifiable spectral bridge. Departing from classical Laplacian diffusion \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{E}(\\varvec{t}+1)=\\varvec{P}\\text{\\hspace{0.17em}}\\varvec{E}\\left(\\varvec{t}\\right)\\)\u003c/span\u003e\u003c/span\u003e, the formulation incorporates reactivity \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{\\beta\\:}\\)\u003c/span\u003e\u003c/span\u003eand self-retention \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{s}\\)\u003c/span\u003e\u003c/span\u003e, producing the generalized operator \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{P}}_{\\varvec{s}}=(1-\\varvec{s})\\varvec{I}+\\varvec{s}{\\varvec{D}}^{-1}\\varvec{A}\\)\u003c/span\u003e\u003c/span\u003eand the dynamic update \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{E}}^{(\\varvec{t}+1)}=(1-\\varvec{\\beta\\:}){\\varvec{E}}^{\\left(\\varvec{t}\\right)}+\\varvec{\\beta\\:}{\\varvec{P}}_{\\varvec{s}}{\\varvec{E}}^{\\left(\\varvec{t}\\right)}\\)\u003c/span\u003e\u003c/span\u003e. Analytical derivation yields the stability constraint \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{\\rho\\:}\\left[\\right(1-\\varvec{\\beta\\:})\\varvec{I}+\\varvec{\\beta\\:}{\\varvec{P}}_{\\varvec{s}}]\\le\\:1\\)\u003c/span\u003e\u003c/span\u003eand a closed-form finite-time law \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{f}}_{\\varvec{t}}\\left({\\varvec{w}}_{\\varvec{b}}\\right)={\\varvec{f}}_{\\varvec{\\infty\\:}}\\left({\\varvec{w}}_{\\varvec{b}}\\right)(1-{\\varvec{e}}^{-\\varvec{t}/\\varvec{\\tau\\:}\\left({\\varvec{w}}_{\\varvec{b}}\\right)})\\)\u003c/span\u003e\u003c/span\u003e, unifying diffusion, spectral gap, and topological coupling in one relation. Simulations across four canonical topologies confirm near-perfect agreement with theory (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{r}=0.9945\\)\u003c/span\u003e\u003c/span\u003e, RMSE\u0026thinsp;=\u0026thinsp;0.0139), validating both the steady-state and transient predictions. The Energy\u0026ndash;Bridge framework thus extends graph diffusion theory to the finite-time domain while demonstrating a transparent, reproducible model of human\u0026ndash;AI scientific collaboration.\u003c/p\u003e","manuscriptTitle":"Energy–Bridge Transfer Law: A Quantitative Bridge Between Topology and Dynamics in Graph Systems","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-11-04 05:45:48","doi":"10.21203/rs.3.rs-8010502/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":"ff61bc3f-a450-44f0-8646-b97d70125480","owner":[],"postedDate":"November 4th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-11-04T05:45:48+00:00","versionOfRecord":[],"versionCreatedAt":"2025-11-04 05:45:48","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8010502","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8010502","identity":"rs-8010502","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}