Multivariate Recovery Dependency in Networks with Time Delays: Effects on Resilience and Scaling Laws

Multivariate Recovery Dependency in Networks with Time Delays: Effects on Resilience and Scaling Laws | 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 Multivariate Recovery Dependency in Networks with Time Delays: Effects on Resilience and Scaling Laws Gaogao Dong, Xun Zhou, Fan Wang, Michael Danziger, ruijin Du, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7107176/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 recovery process in real-world systems needs to consider the combined effects of internal and external failures and is rarely instantaneous, typically involving inherent time delays.Here, we investigate the impact of a time delay, τ , on system recovery in dependency networks, that can be homogeneous or heterogeneous, with both internal and external recovery mechanisms. Through theoretical analysis and numerical simulations, our findings reveal that the system exhibits two critical thresholds, p c1 and p c2 , associated to complete recovery and irreversible failure, respectively. These thresholds divide the network recovery states into three regions: fully recoverable , potentially recoverable and unrecoverable , where the first region exhibits the optimal recovery time cost, which is reflected as the critical window in the rescue measures. Additionally, we observe that the network size N and time delay τ have a significant impact on p c1 , but no effect on p c2 , and exhibit scaling and linear relationships, respectively, with the value | p c1 - p c2 |. However, when the system topology changes, the stability of p c2 is disrupted, exhibiting characteristics of a first-order phase transition. These findings highlight the importance of minimizing delays to enhance network resilience, providing valuable insights for designing robust recovery strategies in critical infrastructures. Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Complex networks Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Phase transitions and critical phenomena Full Text Additional Declarations There is NO Competing Interest. 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. 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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-7107176","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":497804506,"identity":"745957e6-a9d5-4940-a77f-dcf3832a4503","order_by":0,"name":"Gaogao Dong","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwUlEQVRIiWNgGAWjYLCCDxU2cmzszQeI18E440yaMR/PsQTitTDzth1OnCeRo0Cccnn34xcfA7WktzHkMDD8qNhGWIvhmZxiwznn0nPbGM4eYOw5c5sILQ05aRJvyqxz2xj7EpgZ24jR0v8mTYKHjTmdjZnHgDgt8hLpxyR52pwT2NiI1WIg8YbZEBjIhm08bAkHifKLfH/6wwfAqJSXn//44IMfFcTYcoDHAM45QFg9yJYG9gdEKRwFo2AUjIIRDAAswT2qBDVDsQAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0003-4425-7724","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Gaogao","middleName":"","lastName":"Dong","suffix":""},{"id":497804507,"identity":"98e372d1-fecd-4ed0-8f42-00ed5a36f7c1","order_by":1,"name":"Xun Zhou","email":"","orcid":"","institution":"Jiangsu University","correspondingAuthor":false,"prefix":"","firstName":"Xun","middleName":"","lastName":"Zhou","suffix":""},{"id":497804508,"identity":"7833c045-c983-4622-b44d-24b07c0bfb25","order_by":2,"name":"Fan Wang","email":"","orcid":"","institution":"Bar Ilan University","correspondingAuthor":false,"prefix":"","firstName":"Fan","middleName":"","lastName":"Wang","suffix":""},{"id":497804509,"identity":"7cffc04a-b620-4787-97f6-f9aff0631372","order_by":3,"name":"Michael Danziger","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Michael","middleName":"","lastName":"Danziger","suffix":""},{"id":497804510,"identity":"27c3bf1f-7dc2-4fd7-9944-69e31333361a","order_by":4,"name":"ruijin Du","email":"","orcid":"","institution":"University of Oxford","correspondingAuthor":false,"prefix":"","firstName":"ruijin","middleName":"","lastName":"Du","suffix":""},{"id":497804511,"identity":"3aac550a-5340-4d16-ae5d-fc7f623985f8","order_by":5,"name":"Renaud Lambiotte","email":"","orcid":"https://orcid.org/0000-0002-0583-4595","institution":"University of Oxford","correspondingAuthor":false,"prefix":"","firstName":"Renaud","middleName":"","lastName":"Lambiotte","suffix":""}],"badges":[],"createdAt":"2025-07-12 09:30:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7107176/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7107176/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":91374760,"identity":"6541fb14-b01e-41bd-8b23-7c430335c088","added_by":"auto","created_at":"2025-09-15 19:48:39","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":864618,"visible":true,"origin":"","legend":"Article File","description":"","filename":"Manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7107176/v1_covered_a1c864b7-25de-4c4f-b822-d95b88212eb7.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Multivariate Recovery Dependency in Networks with Time Delays: Effects on Resilience and Scaling Laws","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":"","lastPublishedDoi":"10.21203/rs.3.rs-7107176/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7107176/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The recovery process in real-world systems needs to consider the combined effects of internal and external failures and is rarely instantaneous, typically involving inherent time delays.Here, we investigate the impact of a time delay, \u003ci\u003eτ\u003c/i\u003e, on system recovery in dependency networks, that can be homogeneous or heterogeneous, with both internal and external recovery mechanisms. Through theoretical analysis and numerical simulations, our findings reveal that the system exhibits two critical thresholds, \u003ci\u003ep\u003c/i\u003e\u003csub\u003ec1\u003c/sub\u003e and \u003ci\u003ep\u003c/i\u003e\u003csub\u003ec2\u003c/sub\u003e, associated to complete recovery and irreversible failure, respectively. These thresholds divide the network recovery states into three regions: \u003ci\u003efully recoverable\u003c/i\u003e, \u003ci\u003epotentially recoverable\u003c/i\u003e and \u003ci\u003eunrecoverable\u003c/i\u003e, where the first region exhibits the optimal recovery time cost, which is reflected as the critical window in the rescue measures. Additionally, we observe that the network size \u003ci\u003eN\u003c/i\u003e and time delay \u003ci\u003eτ\u003c/i\u003e have a significant impact on \u003ci\u003ep\u003c/i\u003e\u003csub\u003ec1\u003c/sub\u003e, but no effect on \u003ci\u003ep\u003c/i\u003e\u003csub\u003ec2\u003c/sub\u003e, and exhibit scaling and linear relationships, respectively, with the value |\u003ci\u003ep\u003c/i\u003e\u003csub\u003ec1\u003c/sub\u003e-\u003ci\u003ep\u003c/i\u003e\u003csub\u003ec2\u003c/sub\u003e|. However, when the system topology changes, the stability of \u003ci\u003ep\u003c/i\u003e\u003csub\u003ec2\u003c/sub\u003e is disrupted, exhibiting characteristics of a first-order phase transition. These findings highlight the importance of minimizing delays to enhance network resilience, providing valuable insights for designing robust recovery strategies in critical infrastructures.","manuscriptTitle":"Multivariate Recovery Dependency in Networks with Time Delays: Effects on Resilience and Scaling Laws","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-08-20 08:02:25","doi":"10.21203/rs.3.rs-7107176/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":"4bd397ec-3260-49db-a2f7-2df44941612c","owner":[],"postedDate":"August 20th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":52884163,"name":"Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Complex networks"},{"id":52884164,"name":"Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Phase transitions and critical phenomena"}],"tags":[],"updatedAt":"2025-09-15T19:40:32+00:00","versionOfRecord":[],"versionCreatedAt":"2025-08-20 08:02:25","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7107176","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7107176","identity":"rs-7107176","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}