Entropy-Based Indicators of Critical Transitions in Power-Law Networks Under Progressive Node Removal

Entropy-Based Indicators of Critical Transitions in Power-Law Networks Under Progressive Node Removal | 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 Entropy-Based Indicators of Critical Transitions in Power-Law Networks Under Progressive Node Removal Zachary Alexander Kraehling This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9204974/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 Power-law networks tolerate extensive random failures yet can collapse abruptly under targeted attacks, often with little warning from connectivity measures alone. We ask whether an information-theoretic rate of structural change can provide early warning of impending connectivity collapse in such networks, and under what conditions any such signal transfers across generative ensembles. We track EWMA-smoothed successive Kullback–Leibler divergence between adjacent empirical degree distributions in Chung–Lu scale-free graphs with 2 < γ < 3 under both random and targeted node removal. Using fixed, pre-specified detection rules that do not reference connectivity, we find that the signal departs from an initial low-activity baseline well before GCC collapse under random failure, a pattern confirmed on an empirical CAIDA AS-level Internet topology. Under targeted removal, the signal exhibits immediate disruption rather than advance warning. A degree-matched configuration-model experiment reveals that the same rule fails due to ensemble-induced threshold inflation, a mechanistically interpretable limitation on transferability that motivates ensemble-aware deployment of structural monitoring rules. power-law networks network robustness KL divergence early warning signals percolation complex networks information theory degree distribution 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-9204974","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":611654330,"identity":"85b55c15-17b0-49aa-908a-b76155754d11","order_by":0,"name":"Zachary Alexander Kraehling","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5ElEQVRIie3PsQrCMBCA4StCJqHriYKvkCLoIu2rRAJxKeio4FAR6iK6+jpSiIsP4OCQqY7qUjqI2KroIES7ieSfbsjHXQBMpp+MWAqA0XwCRb8jJfok7E6s4AMh+CLwBalPpRimI9FoTSOpWH/nBtXJWumINRNyi9Jv1jaiSxmNeVCTHe2WEnTDLSWDNoLfREYjDug7WkLsfdhnl4zYh+RBeictKaOQsAqzDeiTnLjZFv33EWNeGc9FIxtuhzGCwlnqSH0hnFOacGdp8xiP58izkaujjrzVCQs9z/MKC5PJZPr7rp22R+MN818QAAAAAElFTkSuQmCC","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Zachary","middleName":"Alexander","lastName":"Kraehling","suffix":""}],"badges":[],"createdAt":"2026-03-23 23:38:33","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9204974/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9204974/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109026058,"identity":"3c1e7d3e-caa9-40af-8494-a7093fbe1960","added_by":"auto","created_at":"2026-05-11 21:10:34","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":244546,"visible":true,"origin":"","legend":"","description":"","filename":"submission.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9204974/v1_covered_eabb74a7-edb9-4b8e-af83-3fae095daa24.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Entropy-Based Indicators of Critical Transitions in Power-Law Networks Under Progressive Node Removal","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":"power-law networks, network robustness, KL divergence, early warning signals, percolation, complex networks, information theory, degree distribution","lastPublishedDoi":"10.21203/rs.3.rs-9204974/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9204974/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Power-law networks tolerate extensive random failures yet can collapse abruptly under targeted attacks, often with little warning from connectivity measures alone. We ask whether an information-theoretic rate of structural change can provide early warning of impending connectivity collapse in such networks, and under what conditions any such signal transfers across generative ensembles.\n\nWe track EWMA-smoothed successive Kullback–Leibler divergence between adjacent empirical degree distributions in Chung–Lu scale-free graphs with 2 \u003c γ \u003c 3 under both random and targeted node removal. Using fixed, pre-specified detection rules that do not reference connectivity, we find that the signal departs from an initial low-activity baseline well before GCC collapse under random failure, a pattern confirmed on an empirical CAIDA AS-level Internet topology. Under targeted removal, the signal exhibits immediate disruption rather than advance warning. A degree-matched configuration-model experiment reveals that the same rule fails due to ensemble-induced threshold inflation, a mechanistically interpretable limitation on transferability that motivates ensemble-aware deployment of structural monitoring rules.","manuscriptTitle":"Entropy-Based Indicators of Critical Transitions in Power-Law Networks Under Progressive Node Removal","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-31 03:01:52","doi":"10.21203/rs.3.rs-9204974/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":"8c052912-93f6-459b-b484-c03c4801dced","owner":[],"postedDate":"March 31st, 2026","published":true,"recentEditorialEvents":[{"type":"decision","content":"Rejected","date":"2026-05-11T21:04:03+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-08T08:31:24+00:00","index":40,"fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-04T05:59:43+00:00","index":39,"fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-05-11T21:10:27+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-31 03:01:52","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9204974","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9204974","identity":"rs-9204974","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}