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

Abstract

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.

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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.

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Acknowledgements

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.

Affiliations

  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

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Contributions

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

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  1. 1.

    Supplementary Information

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

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DOI

https://doi.org/10.1038/nature16948

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