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Reviving a failed network through microscopic interventions


From mass extinction to cell death, complex networked systems often exhibit abrupt dynamic transitions between desirable and undesirable states. These transitions are often caused by topological perturbations (such as node or link removal, or decreasing link strengths). The problem is that reversing the topological damage, namely, retrieving lost nodes or links or reinforcing weakened interactions, does not guarantee spontaneous recovery to the desired functional state. Indeed, many of the relevant systems exhibit a hysteresis phenomenon, remaining in the dysfunctional state, despite reconstructing their damaged topology. To address this challenge, we develop a two-step recovery scheme: first, topological reconstruction to the point where the system can be revived and then dynamic interventions to reignite the system’s lost functionality. By applying this method to a range of nonlinear network dynamics, we identify the recoverable phase of a complex system, a state in which the system can be reignited by microscopic interventions, for instance, controlling just a single node. Mapping the boundaries of this dynamical phase, we obtain guidelines for our two-step recovery.

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Fig. 1: Reviving a failed network.
Fig. 2: Can a system be reignited by a single node?
Fig. 3: Recoverability of cellular dynamics.
Fig. 4: Two-step recovery for reviving a failed cellular system.
Fig. 5: Recoverability of neuronal dynamics.
Fig. 6: Recoverability of microbial dynamics.

Data availability

Empirical data required for constructing the real-world networks (Microbiome, Brain, Yeast PPI, Human PPI) are available at

Code availability

All codes to reproduce, examine and improve our proposed analysis are available at


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H.S. acknowledges the support of the Presidential Fellowship of Bar-Ilan University, Israel, and the Mordecai and Monique Katz Graduate Fellowship Program. This research was supported by the Israel Science Foundation (grant nos. 499/19 and 189/19), the US National Science Foundation CRISP award (grant no. 1735505), the Bar-Ilan University Data Science Institute grant for research on network dynamics, the ISF-NSFC joint research program (grant nos. 3132/19 and 3552/21), the US–Israel NSF–BSF programme (grant no. 2019740), the EU H2020 project RISE (grant no. 821115), the EU H2020 project DIT4TRAM (grant no. 953783), the Defense Threat Reduction Agency (DTRA grant no. HDTRA-1-19-1-0016), the US National Science Foundation (grant no. 2047488) and the Rensselaer-IBM AI Research Collaboration.

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Authors and Affiliations



All the authors designed the research. H.S. and B.B. conducted the mathematical analysis. H.S. performed the numerical simulations and analysed the data. A.B. supervised the microbiome analysis. B.B. was the lead writer of the paper.

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Correspondence to Baruch Barzel.

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The authors declare no competing interests.

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Nature Physics thanks Patrick Desrosiers and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Sections 1–5.

Supplementary Table 1

Dataset 1: reigniting capacity of all the microbial species.

Supplementary Table 2

Dataset 2: adversarial impact of all the nutrients.

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Sanhedrai, H., Gao, J., Bashan, A. et al. Reviving a failed network through microscopic interventions. Nat. Phys. 18, 338–349 (2022).

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