Letter | Published:

Universal resilience patterns in complex networks

Nature volume 530, pages 307312 (18 February 2016) | Download Citation

  • An Erratum to this article was published on 04 May 2016
  • An Author Correction to this article was published on 28 March 2019

This article has been updated


Resilience, a system’s ability to adjust its activity to retain its basic functionality when errors, failures and environmental changes occur, is a defining property of many complex systems1. Despite widespread consequences for human health2, the economy3 and the environment4, events leading to loss of resilience—from cascading failures in technological systems5 to mass extinctions in ecological networks6—are rarely predictable and are often irreversible. These limitations are rooted in a theoretical gap: the current analytical framework of resilience is designed to treat low-dimensional models with a few interacting components7, and is unsuitable for multi-dimensional systems consisting of a large number of components that interact through a complex network. Here we bridge this theoretical gap by developing a set of analytical tools with which to identify the natural control and state parameters of a multi-dimensional complex system, helping us derive effective one-dimensional dynamics that accurately predict the system’s resilience. The proposed analytical framework allows us systematically to separate the roles of the system’s dynamics and topology, collapsing the behaviour of different networks onto a single universal resilience function. The analytical results unveil the network characteristics that can enhance or diminish resilience, offering ways to prevent the collapse of ecological, biological or economic systems, and guiding the design of technological systems resilient to both internal failures and environmental changes.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Change history

  • 28 March 2019

    In this Letter, in Fig. 3c and f the Saccharomyces cerevisiae and Escherichia coli networks were subject to both weight loss and node deletion, a combination of two types of perturbation, as opposed to weight loss only (as the labelling incorrectly indicated). The collapse in Fig. 3h was also obtained from this combined perturbation, and therefore the results displayed in Fig. 3h remain fully consistent with the theoretical framework presented in this Letter. Figure 1 in the accompanying Amendment shows the corrected Fig. 3c, f and h, in which Fig. 3c and f have been generated with weight-loss perturbations only, as originally reported, together with the published, incorrect panels, for completeness and transparency. The codes used to generate the original and the corrected Fig. 3 are available at https://github.com/jianxigao/NuRsE. We thank Travis A. Gibson for alerting us to this error. The original Letter has not been corrected.


  1. 1.

    , , & Resilience of the internet to random breakdown. Phys. Rev. Lett. 85, 4626–4628 (2000)

  2. 2.

    et al. Self-organized patchiness in asthma as a prelude to catastrophic shifts. Nature 434, 777–782 (2005)

  3. 3.

    Resilience in the dynamics of economy-environment systems. Environ. Resour. Econ. 11, 503–520 (1998)

  4. 4.

    Thresholds and breakpoints in ecosystems with a multiplicity of stable states. Nature 269, 471–477 (1977)

  5. 5.

    The general problem of the stability of motion. Int. J. Control 55, 531–534 (1992)

  6. 6.

    & Quantifying the connectivity of a network: the network correlation function method. Phys. Rev. E 80, 046104 (2009)

  7. 7.

    & Complexity and fragility in ecological networks. Proc. R. Soc. Lond. B 268, 2039–2045 (2001)

  8. 8.

    , & Complex systems: Ecology for bankers. Nature 451, 893–895 (2008)

  9. 9.

    et al. Anticipating critical transitions. Science 338, 344–348 (2012)

  10. 10.

    & Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)

  11. 11.

    , & Complex Networks Vol. 650 (Springer Science & Business Media, 2004)

  12. 12.

    & Universality in network dynamics. Nature Phys. 9, 673–681 (2013)

  13. 13.

    , & Constructing minimal models for complex system dynamics. Nature Commun. 6, 7186 (2015)

  14. 14.

    An Introduction to Systems Biology: Design Principles of Biological Circuits (CRC Press, 2006)

  15. 15.

    et al. Simple prediction of interaction strengths in complex food webs. Proc. Natl Acad. Sci. USA 106, 187–191 (2009)

  16. 16.

    , & Population dynamics and mutualism: functional responses of benefits and costs. Am. Nat. 159, 231–244 (2002)

  17. 17.

    & Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200 (2001)

  18. 18.

    , & Network structure and biodiversity loss in food webs: robustness increases with connectance. Ecol. Lett. 5, 558–567 (2002)

  19. 19.

    , & Analysis of structural robustness of metabolic networks. Syst. Biol. 1, 114–120 (2004)

  20. 20.

    , , & Coral resilience to ocean acidification and global warming through ph up-regulation. Nature Clim. Change 2, 623–627 (2012)

  21. 21.

    Carrying capacity, population equilibrium, and environment’s maximal load. Ecol. Modell. 192, 317–320 (2006)

  22. 22.

    et al. Principles of Animal Ecology Edn 1 (WB Saundere, 1949)

  23. 23.

    Interaction Web Database (accessed 30 September 2010)

  24. 24.

    , , , & Principles of combinatorial regulation in the transcriptional regulatory network of yeast. J. Mol. Biol. 360, 213–227 (2006)

  25. 25.

    et al. Regulondb (version 6.0): gene regulation model of Escherichia coli k-12 beyond transcription, active (experimental) annotated promoters and textpresso navigation. Nucleic Acids Res. 36, D120–D124 (2008)

  26. 26.

    & Controlling edge dynamics in complex networks. Nature Phys. 8, 568–573 (2012)

  27. 27.

    , & Realistic control of network dynamics. Nature Commun. 4, 1942 (2013)

  28. 28.

    et al. Spontaneous recovery in dynamical networks. Nature Phys. 10, 34–38 (2013)

  29. 29.

    & Optimal self-organization. New J. Phys. 1, 13 (1999)

  30. 30.

    , , & Breakdown of the internet under intentional attack. Phys. Rev. Lett. 86, 3682–3685 (2001)

Download references


We thank A. Mohan, S. E. Flynn and A. R. Ganguly for discussions. This work was supported by an Army Research Laboratories Network Science Collaborative Technology Alliance grant (ARL NS-CTA W911NF-09-2-0053), by The John Templeton Foundation: Mathematical and Physical Sciences (grant number PFI-777), by The Defense Threat Reduction Agency (basic research grant number HDTRA1-10-1-0100) and by the European Commission (grant numbers FP7317532 (MULTIPLEX) and 641191 (CIMPLEX)).

Author information

Author notes

    • Jianxi Gao
    •  & Baruch Barzel

    These authors contributed equally to this work.


  1. Center for Complex Network Research, Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA

    • Jianxi Gao
    •  & Albert-László Barabási
  2. Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

    • Baruch Barzel
  3. Center for Cancer Systems Biology, Dana-Farber Cancer Institute, Harvard University, Boston, Massachusetts 02215, USA

    • Albert-László Barabási
  4. Department of Medicine, Brigham and Women’s Hospital, Harvard Medical School, Boston, Massachusetts 02115, USA

    • Albert-László Barabási
  5. Center for Network Science, Central European University, Budapest 1051, Hungary

    • Albert-László Barabási


  1. Search for Jianxi Gao in:

  2. Search for Baruch Barzel in:

  3. Search for Albert-László Barabási in:


All authors designed and did the research. J.G. and B.B. did the analytical calculations. J.G. analysed the empirical data and did the numerical calculations. A.-L.B. and B.B. were the lead writers of the manuscript.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Albert-László Barabási.

All code for the reproduction of the reported results can be downloaded from https://github.com/jianxigao/NuRsE.

Supplementary information

PDF files

  1. 1.

    Supplementary Information

    This file contains Supplementary Text sections 1-6, Supplementary References, Supplementary Figures 1-19 and Supplementary Tables 1-5.

About this article

Publication history






Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.