{"paper_id":"382234f0-10a1-445a-bbce-2a9a8e512a53","body_text":"Networks Multiscale Entropy Analysis | 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 Article Networks Multiscale Entropy Analysis Andres Abeliuk, Sebastián Brzovic, Cristóbal Rojas This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7623618/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted You are reading this latest preprint version Abstract Understanding the structural complexity and predictability of complex networks is a central challenge in network science. Although recent studies have revealed a relationship between compression-based entropy and link prediction performance, existing methods focus on single-scale representations. This approach often overlooks the rich hierarchical patterns that can exist in real-world networks. In this study, we introduce a multiscale entropy framework that extends previous entropy-based approaches by applying spectral graph reduction. This allows us to quantify how structural entropy evolves as the network is gradually coarsened, capturing complexity across multiple scales. We apply our framework to real-world networks across biological, economic, social, technological, and transportation domains. The results uncover consistent entropy profiles across network families, revealing three structural regimes---stable, increasing, and hybrid---that align with domain-specific behaviors. Compared to single-scale models, multiscale entropy significantly improves our ability to determine network predictability. This shows that considering structural information across scales provides a more complete characterization of network complexity. Together, these results position multiscale entropy as a powerful and scalable tool for characterizing, classifying, and assessing the structure of complex networks. Physical sciences/Mathematics and computing/Applied mathematics Physical sciences/Mathematics and computing/Computational science Graph reduction Graph entropy Structural complexity Link prediction Full Text Additional Declarations There is NO Competing Interest. Supplementary Files MultiscaleEntropy5appendix.pdf Networks Multiscale Entropy Analysis Cite Share Download PDF Status: Under Review 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. 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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-7623618\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":false,\"archivedVersions\":[],\"articleType\":\"Article\",\"associatedPublications\":[],\"authors\":[{\"id\":522412687,\"identity\":\"3202fc88-01b9-4760-baf4-fcd1df69dc27\",\"order_by\":0,\"name\":\"Andres Abeliuk\",\"email\":\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAxUlEQVRIiWNgGAWjYHACA2YGhgQ5EOvAA1K0GIO1JJCiJbEBxCRKC//s5o2fCyrS0ueHHX4ItMVOTreBgBaJO8eKpWecycndeDvNAKgl2djsACFrbuQYSPO2VeRunJ0A0nIgcRshLfI3cox/8/6rSDecnf6BOC0GN3LMpHkbchLkpXOItMXwRlqZ9YxjaYYbpHMKDiQYEOEXuRvJm28X1CTLy89O3/zhQ4WdHGHvw10IVmlArHIQkG8gRfUoGAWjYBSMKAAArpFHa3sJ8L8AAAAASUVORK5CYII=\",\"orcid\":\"https://orcid.org/0000-0002-8158-4647\",\"institution\":\"University of Chile\",\"correspondingAuthor\":true,\"prefix\":\"\",\"firstName\":\"Andres\",\"middleName\":\"\",\"lastName\":\"Abeliuk\",\"suffix\":\"\"},{\"id\":522412688,\"identity\":\"a03a37c4-206b-45da-87e6-a438dc17b2d8\",\"order_by\":1,\"name\":\"Sebastián Brzovic\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"University of Chile\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Sebastián\",\"middleName\":\"\",\"lastName\":\"Brzovic\",\"suffix\":\"\"},{\"id\":522412689,\"identity\":\"ae88f0d9-2efd-4aa5-8b46-7d18f858845c\",\"order_by\":2,\"name\":\"Cristóbal Rojas\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"Pontifical Catholic University of Chile\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Cristóbal\",\"middleName\":\"\",\"lastName\":\"Rojas\",\"suffix\":\"\"}],\"badges\":[],\"createdAt\":\"2025-09-15 19:10:12\",\"currentVersionCode\":1,\"declarations\":\"\",\"doi\":\"10.21203/rs.3.rs-7623618/v1\",\"doiUrl\":\"https://doi.org/10.21203/rs.3.rs-7623618/v1\",\"draftVersion\":[],\"editorialEvents\":[],\"editorialNote\":\"\",\"failedWorkflow\":false,\"files\":[{\"id\":93370673,\"identity\":\"8d634ad9-c043-4489-9d20-ea0d7a9643c7\",\"added_by\":\"auto\",\"created_at\":\"2025-10-13 06:32:15\",\"extension\":\"pdf\",\"order_by\":1,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"manuscript-pdf\",\"size\":2295357,\"visible\":true,\"origin\":\"\",\"legend\":\"Article File\",\"description\":\"\",\"filename\":\"MultiscaleEntropy5mainpdf.pdf\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-7623618/v1_covered_7f5dc986-da20-488e-a891-32b3851308f4.pdf\"},{\"id\":93370392,\"identity\":\"0ee35cf8-05a0-4406-bfcf-4015ef5b4563\",\"added_by\":\"auto\",\"created_at\":\"2025-10-13 06:24:22\",\"extension\":\"pdf\",\"order_by\":1,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"supplement\",\"size\":5501923,\"visible\":true,\"origin\":\"\",\"legend\":\"Networks Multiscale Entropy Analysis\",\"description\":\"\",\"filename\":\"MultiscaleEntropy5appendix.pdf\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-7623618/v1/a3b1e28cf199b30431768feb.pdf\"}],\"financialInterests\":\"There is \\u003cb\\u003eNO\\u003c/b\\u003e Competing Interest.\",\"formattedTitle\":\"Networks Multiscale Entropy Analysis\",\"fulltext\":[],\"fulltextSource\":\"\",\"fullText\":\"\",\"funders\":[],\"hasAdminPriorityOnWorkflow\":false,\"hasManuscriptDocX\":false,\"hasOptedInToPreprint\":true,\"hasPassedJournalQc\":\"\",\"hasAnyPriority\":true,\"hideJournal\":false,\"highlight\":\"\",\"institution\":\"\",\"isAcceptedByJournal\":false,\"isAuthorSuppliedPdf\":true,\"isDeskRejected\":\"\",\"isHiddenFromSearch\":false,\"isInQc\":false,\"isInWorkflow\":false,\"isPdf\":true,\"isPdfUpToDate\":true,\"isWithdrawnOrRetracted\":false,\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"nature-portfolio\",\"isNatureJournal\":true,\"hasQc\":false,\"allowDirectSubmit\":false,\"externalIdentity\":\"\",\"sideBox\":\"\",\"snPcode\":\"\",\"submissionUrl\":\"\",\"title\":\"Nature Portfolio\",\"twitterHandle\":\"\",\"acdcEnabled\":false,\"dfaEnabled\":false,\"editorialSystem\":\"ejp\",\"reportingPortfolio\":\"\",\"inReviewEnabled\":true,\"inReviewRevisionsEnabled\":false},\"keywords\":\"Graph reduction, Graph entropy, Structural complexity, Link prediction\",\"lastPublishedDoi\":\"10.21203/rs.3.rs-7623618/v1\",\"lastPublishedDoiUrl\":\"https://doi.org/10.21203/rs.3.rs-7623618/v1\",\"license\":{\"name\":\"CC BY 4.0\",\"url\":\"https://creativecommons.org/licenses/by/4.0/\"},\"manuscriptAbstract\":\"Understanding the structural complexity and predictability of complex networks is a central challenge in network science. 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