Network heterogeneity: a focus on links and types

Network heterogeneity: a focus on links and types | 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Research Article Network heterogeneity: a focus on links and types Li Li, Ronald Rousseau This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8231251/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 Network heterogeneity is a fundamental concept in defining the structural complexity of interacting systems, yet traditional definitions based on degree distribution often overlook the diversity of connections between distinct entity types. In this paper, we propose a novel theoretical framework centered on "link heterogeneity" within the context of bipartite and layered networks. By employing a matrix-based approach to transform bipartite structures into layered representations, we demonstrate that heterogeneity is driven exclusively by "external links"—connections between different node layers. We introduce the HE index to rigorously quantify this property, establishing that networks lacking cross-layer connections are mathematically equivalent to homogeneous systems. Our findings indicate that a single bipartite network generates two distinct weighted one-mode projections, each possessing a unique heterogeneity profile. However, this structural diversity presents a paradox: while high heterogeneity, characterized by pronounced modularity, facilitates information diffusion, it simultaneously introduces systemic risks by increasing vulnerability to the removal of key hubs. While currently limited to undirected projections and requiring further empirical validation on large-scale datasets, this study provides a mathematically grounded tool for analyzing bibliographic coupling, co-citation, and affiliation networks. The findings suggest that effective R&D policy management should balance the efficiency of connectivity with the protection of critical nodes to mitigate structural fragility. complex networks layered networks bipartite networks 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-8231251","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":562501276,"identity":"d782a68e-2293-42ac-b4ad-5d0ec1a0c048","order_by":0,"name":"Li 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types","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, layered networks, bipartite networks","lastPublishedDoi":"10.21203/rs.3.rs-8231251/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8231251/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eNetwork heterogeneity is a fundamental concept in defining the structural complexity of interacting systems, yet traditional definitions based on degree distribution often overlook the diversity of connections between distinct entity types. In this paper, we propose a novel theoretical framework centered on \"link heterogeneity\" within the context of bipartite and layered networks. By employing a matrix-based approach to transform bipartite structures into layered representations, we demonstrate that heterogeneity is driven exclusively by \"external links\"—connections between different node layers. We introduce the \u003cem\u003eHE\u003c/em\u003e index to rigorously quantify this property, establishing that networks lacking cross-layer connections are mathematically equivalent to homogeneous systems. Our findings indicate that a single bipartite network generates two distinct weighted one-mode projections, each possessing a unique heterogeneity profile. However, this structural diversity presents a paradox: while high heterogeneity, characterized by pronounced modularity, facilitates information diffusion, it simultaneously introduces systemic risks by increasing vulnerability to the removal of key hubs. While currently limited to undirected projections and requiring further empirical validation on large-scale datasets, this study provides a mathematically grounded tool for analyzing bibliographic coupling, co-citation, and affiliation networks. The findings suggest that effective R\u0026amp;D policy management should balance the efficiency of connectivity with the protection of critical nodes to mitigate structural fragility.\u003c/p\u003e","manuscriptTitle":"Network heterogeneity: a focus on links and types","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-12-22 07:50:51","doi":"10.21203/rs.3.rs-8231251/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":"ac0236c2-f1ce-4978-a58e-2d164247d44e","owner":[],"postedDate":"December 22nd, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-12-22T07:50:51+00:00","versionOfRecord":[],"versionCreatedAt":"2025-12-22 07:50:51","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8231251","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8231251","identity":"rs-8231251","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}