- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Origins of instability in dynamical systems on undirected networks

Shraosi Dawn1, Subrata Ghosh2, Chandrakala Meena3, Tim Rogers4, and Chittaranjan Hens1

PDF

Phys. Rev. E 113, 014303 – Published 6 January, 2026

DOI: https://doi.org/10.1103/g2mp-bkhf

Abstract

Robustness to perturbation is a key topic in the study of complex systems occurring across a wide variety of applications from epidemiology to biochemistry. Here, we analyze the eigenspectrum of the Jacobian matrices associated to a general class of networked dynamical systems, which contains information on how perturbations to a stationary state develop over time. We find that stability is always determined by a spectral outlier, but with pronounced differences to the corresponding eigenvector in different regimes. We show that, depending on model details, instability may originate in nodes of anomalously low or high degrees, or may occur everywhere in the network at once. Importantly, the dependence on extremal degrees results in considerable finite-size effects with different scaling depending on the ensemble degree distribution. Our results have potentially useful applications in network monitoring to predict or prevent catastrophic failures, and we validate our analytical findings through applications to epidemic dynamics and gene regulatory systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (43)

  1. R. M. May, Nature (London) 238, 413 (1972).
  2. J. Grilli, T. Rogers, and S. Allesina, Nat. Commun. 7, 12031 (2016).
  3. M. Potters, J.-P. Bouchaud, and L. Laloux, Acta Phys. Pol. Ser. B 35, 2767 (2005).
  4. T. Rogers, Europhys. Lett. 109, 28005 (2015).
  5. S. Moore and T. Rogers, Phys. Rev. Lett. 124, 068301 (2020).
  6. S. Sinha and S. Sinha, Phys. Rev. E 71, 020902(R) (2005).
  7. T. Gibbs, J. Grilli, T. Rogers, and S. Allesina, Phys. Rev. E 98, 022410 (2018).
  8. C. Meena, P. D. Rungta, and S. Sinha, Plos One 12, e0183251 (2017).
  9. T. Gross and U. Feudel, Phys. Rev. E 73, 016205 (2006).
  10. J. W. Baron, T. J. Jewell, C. Ryder, and T. Galla, Phys. Rev. Lett. 128, 120601 (2022).
  11. J. W. Baron, T. J. Jewell, C. Ryder, and T. Galla, Phys. Rev. Lett. 130, 137401 (2023).
  12. S. Azaele and A. Maritan, Phys. Rev. Lett. 133, 127401 (2024).
  13. F. L. Metz, Phys. Rev. Lett. 134, 037401 (2025).
  14. C. Meena, C. Hens, S. Acharyya, S. Haber, S. Boccaletti, and B. Barzel, Nat. Phys. 19, 1033 (2023).
  15. F. L. Metz, I. Neri, and T. Rogers, J. Phys. A: Math. Theor. 52, 434003 (2019).
  16. J. Gao, S. V. Buldyrev, H. E. Stanley, and S. Havlin, Nat. Phys. 8, 40 (2012).
  17. P. Ji, W. Lin, and J. Kurths, Nat. Phys. 16, 1082 (2020).
  18. P. Kundu, H. Kori, and N. Masuda, Phys. Rev. E 105, 024305 (2022).
  19. N. Masuda and P. Kundu, Phys. Rev. Res. 4, 023257 (2022).
  20. P. Ji, J. Ye, Y. Mu, W. Lin, Y. Tian, C. Hens, M. Perc, Y. Tang, J. Sun, and J. Kurths, Phys. Rep. 1017, 1 (2023).
  21. R.-J. Wu, Y.-X. Kong, Z. Di, J. Bascompte, and G.-Y. Shi, Phys. Rev. Lett. 130, 097401 (2023).
  22. I. A. Hatton, O. Mazzarisi, A. Altieri, and M. Smerlak, Science 383, eadg8488 (2024).
  23. S. Ghosh, L. Xue, A. Mishra, S. Saha, D. Dudkowski, S. K. Dana, T. Kapitaniak, J. Kurths, P. Ji, and C. Hens, Phys. Rep. 1147, 1 (2025).
  24. A. Roy, U. Shekhar, A. Bose, S. Ghosh, S. Nannuru, S. Kumar Dana, and C. Hens, Chaos 34, 103120 (2024).
  25. S. Saha, A. Mishra, S. K. Dana, C. Hens, and N. Bairagi, Phys. Rev. E 102, 052307 (2020).
  26. T. Rogers, I. P. Castillo, R. Kühn, and K. Takeda, Phys. Rev. E 78, 031116 (2008).
  27. T. Rogers and I. P. Castillo, Phys. Rev. E 79, 012101 (2009).
  28. This can be shown by introducing Sji=δij|ci/bi|1/2, so that J̃=SJS−1 is real and symmetric.
  29. M. Mézard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond: An Introduction to the Replica Method and its Applications, Vol. 9 (World Scientific, Singapore, 1987).
  30. H. J. Sommers, A. Crisanti, H. Sompolinsky, and Y. Stein, Phys. Rev. Lett. 60, 1895 (1988).
  31. S. Pettersson, V. M. Savage, and M. Nilsson Jacobi, J. R. Soc. Interface. 17, 20190391 (2020).
  32. W. Tarnowski, I. Neri, and P. Vivo, Phys. Rev. Res. 2, 023333 (2020).
  33. J. Grela, Phys. Rev. E 96, 022316 (2017).
  34. I. Neri and F. L. Metz, Phys. Rev. Lett. 117, 224101 (2016).
  35. D. J. Thouless, J. Phys. C 5, 77 (1972).
  36. A.-L. Barabási and R. Albert, Science 286, 509 (1999).
  37. A. L. Krause, L. Kurowski, K. Yawar, and R. A. Van Gorder, J. Theor. Biol. 449, 35 (2018).
  38. C. Hens, U. Harush, S. Haber, R. Cohen, and B. Barzel, Nat. Phys. 15, 403 (2019).
  39. G. Karlebach and R. Shamir, Nat. Rev. Mol. Cell Biol. 9, 770 (2008).
  40. J. Gao, B. Barzel, and A.-L. Barabási, Nature (London) 530, 307 (2016).
  41. X. Bao, Q. Hu, P. Ji, W. Lin, J. Kurths, and J. Nagler, Nat. Commun. 13, 5301 (2022).
  42. S. Kauffman, Physica A 340, 733 (2004).
  43. https://github.com/Rahi-dotcom/Origins-of-Instability-in-Dynamical-Systems-on-Undirected-Networks.git.

Outline

Information

Phys. Rev. E 113, 014303– Published 6 January, 2026
  • Received 8 May 2025
  • Accepted 4 December 2025
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation