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Predicting oscillations in complex networks with delayed feedback

Shijie Liu1,2, Jinliang Han2, Jianming Liu2, Tim Rogers3,*, and Yongzheng Sun1,2,†

  • *Contact author: t.c.rogers@bath.ac.uk
  • †Contact author: yzsung@gmail.com
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Phys. Rev. E 114, 024214 – Published 17 August, 2026

DOI: https://doi.org/10.1103/qvlk-6df6

Abstract

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.

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Phys. Rev. E 114, 024214– Published 17 August, 2026
  • Received 5 March 2026
  • Accepted 23 July 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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