Modeling Real-World Networks Using Intrinsic Vertex Fitness ERGMs

Modeling Real-World Networks Using Intrinsic Vertex Fitness ERGMs | 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 Modeling Real-World Networks Using Intrinsic Vertex Fitness ERGMs Alfonso Zack Robles Saldaña, Alexander Ilyich Nesterov, Claudia Moreno Gonzalez This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8823459/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 In this work, we present a Fitness Exponential Random Graph Model (FERGMs) that brings together concepts from statistical mechanics and network science to describe complex networks beyond the traditional scale-free framework. By assigning each node an intrinsic ''fitness'' and viewing network formation as a thermodynamic process, we develop two complementary variants: a Configuration Model (CM) with bounded fitness and a Generalized Configuration Model (GCM) with unbounded fitness. These model are regulated by a temperature-like parameter $\tau$ and a chemical potential $\mu$, respectively. Together, these parameters control the balance between randomness and structural organization, as well as the overall link density in the network. We obtain closed-form expressions for the degree distributions across distinct thermodynamic regimes, showing that they become Poisson-like at high temperatures and transition to structured, weakly scale-free forms at low temperatures. We then examine 12 heterogeneous real-world networks—including social (YouTube, Orkut), technological (Skitter, web-Google), collaborative (DBLP), and infrastructural (road networks) systems—and find that FERGM surpasses conventional power-law models, lowering the relative error in average-degree prediction from as high as 768% to nearly zero, while also capturing clustering and higher-order statistics. This framework offers a principled, interpretable, and empirically supported alternative to scale-free models, opening up new opportunities for network design, inference, and analysis in domains ranging from epidemiology to computational social science. Complex networks Statistical mechanics Real-world networks Graph ensembles Vertex fitness models Hidden variables Network temperature ERGMs Full Text Additional Declarations No competing interests reported. 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-8823459","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":604678347,"identity":"2c9cce4a-536f-48e7-a5c4-c40cf5d55cdf","order_by":0,"name":"Alfonso Zack Robles Saldaña","email":"data:image/png;base64,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","orcid":"","institution":"University of Guadalajara","correspondingAuthor":true,"prefix":"","firstName":"Alfonso","middleName":"Zack Robles","lastName":"Saldaña","suffix":""},{"id":604678351,"identity":"9dbe18f2-77aa-4dad-a4f0-ac5f7824ec2c","order_by":1,"name":"Alexander Ilyich Nesterov","email":"","orcid":"","institution":"University of Guadalajara","correspondingAuthor":false,"prefix":"","firstName":"Alexander","middleName":"Ilyich","lastName":"Nesterov","suffix":""},{"id":604678354,"identity":"18d1598f-4d29-461e-882c-5ab3f072b2d8","order_by":2,"name":"Claudia Moreno Gonzalez","email":"","orcid":"","institution":"University of Guadalajara","correspondingAuthor":false,"prefix":"","firstName":"Claudia","middleName":"Moreno","lastName":"Gonzalez","suffix":""}],"badges":[],"createdAt":"2026-02-08 17:53:33","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8823459/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8823459/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109476472,"identity":"cb0a39d4-41d6-439f-9e9f-3d50a1ba95dc","added_by":"auto","created_at":"2026-05-18 14:11:16","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1890611,"visible":true,"origin":"","legend":"","description":"","filename":"SNtemplateSubmit.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8823459/v1_covered_cb570013-c90e-4d80-bdfa-2f1b9834b024.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Modeling Real-World Networks Using Intrinsic Vertex Fitness ERGMs","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"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":"Complex networks, Statistical mechanics, Real-world networks, Graph ensembles, Vertex fitness models, Hidden variables, Network temperature, ERGMs","lastPublishedDoi":"10.21203/rs.3.rs-8823459/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8823459/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In this work, we present a Fitness Exponential Random Graph Model (FERGMs) that brings together concepts from statistical mechanics and network science to describe complex networks beyond the traditional scale-free framework. By assigning each node an intrinsic ''fitness'' and viewing network formation as a thermodynamic process, we develop two complementary variants: a Configuration Model (CM) with bounded fitness and a Generalized Configuration Model (GCM) with unbounded fitness. These model are regulated by a temperature-like parameter $\\tau$ and a chemical potential $\\mu$, respectively. Together, these parameters control the balance between randomness and structural organization, as well as the overall link density in the network. We obtain closed-form expressions for the degree distributions across distinct thermodynamic regimes, showing that they become Poisson-like at high temperatures and transition to structured, weakly scale-free forms at low temperatures. We then examine 12 heterogeneous real-world networks—including social (YouTube, Orkut), technological (Skitter, web-Google), collaborative (DBLP), and infrastructural (road networks) systems—and find that FERGM surpasses conventional power-law models, lowering the relative error in average-degree prediction from as high as 768\\% to nearly zero, while also capturing clustering and higher-order statistics. 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