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Rev. E"],"accepted":{"date-parts":[[2026,7,23]]},"abstract":"<jats:p>Oscillatory dynamics are common features of complex networks, often playing essential roles in regulating function. Across scales from gene regulatory networks to ecosystems, delayed feedback mechanisms are key drivers of system-scale oscillations. The analysis and prediction of such dynamics are highly challenging, however, due to the combination of high-dimensionality, nonlinearity and delay. Here, we investigate how structural complexity and delayed feedback are associated with oscillatory dynamics in complex systems, and provide an analytical approach combining theoretical dimension reduction and data-driven prediction. We reveal that oscillations emerge from the interplay of structural complexity and delay, with reduced models uncovering their critical thresholds and showing that greater connectivity lowers the delay required for their onset. Our theory is empirically tested in an experiment on a programmable electronic circuit, where oscillations are observed once structural complexity and feedback delay exceeded the critical thresholds predicted by our theory. Finally, we deploy a reservoir computing pipeline to accurately predict the onset of oscillations directly from time-series data. Our findings deepen understanding of oscillatory regulation and provide useful insights into predicting dynamics in complex networks.<\/jats:p>","DOI":"10.1103\/qvlk-6df6","type":"journal-article","created":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T13:27:52Z","timestamp":1784813272000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Predicting oscillations in complex networks with delayed feedback"],"prefix":"10.1103","volume":"114","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-7308-2656","authenticated-orcid":true,"given":"Shijie","family":"Liu","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01xt2dr21","id-type":"ROR","asserted-by":"publisher"}],"name":"China University of Mining and Technology"},{"id":[{"id":"https:\/\/ror.org\/01xt2dr21","id-type":"ROR","asserted-by":"publisher"}],"name":"China University of Mining and Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4526-4520","authenticated-orcid":true,"given":"Jinliang","family":"Han","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01xt2dr21","id-type":"ROR","asserted-by":"publisher"}],"name":"China University of Mining and Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-5830-8737","authenticated-orcid":true,"given":"Jianming","family":"Liu","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01xt2dr21","id-type":"ROR","asserted-by":"publisher"}],"name":"China University of Mining and Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5733-1658","authenticated-orcid":true,"given":"Tim","family":"Rogers","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/002h8g185","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Bath"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9504-5663","authenticated-orcid":true,"given":"Yongzheng","family":"Sun","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01xt2dr21","id-type":"ROR","asserted-by":"publisher"}],"name":"China University of Mining and Technology"},{"id":[{"id":"https:\/\/ror.org\/01xt2dr21","id-type":"ROR","asserted-by":"publisher"}],"name":"China University of Mining and Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2026,8,17]]},"reference":[{"key":"qvlk-6df6Cc1R1","doi-asserted-by":"publisher","DOI":"10.1038\/238413a0","type":"journal-article"},{"key":"qvlk-6df6Cc2R1","doi-asserted-by":"crossref","DOI":"10.2307\/j.ctvs32rq4","type":"book","volume-title":"Stability and Complexity in Model Ecosystems","author":"R. 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All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright_statement","label":"Copyright statement","group":{"name":"copyright","label":"copyright"}},{"value":"2026","name":"copyright_year","label":"Copyright year","group":{"name":"copyright","label":"copyright"}},{"value":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/","name":"creative_commons_license","label":"Creative Commons license","group":{"name":"licenses","label":"Licenses"}},{"value":"\u00a92026 The authors","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"024214"}}