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  • Review Article
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Robustness and resilience of complex networks

Nature Reviews Physics volume 6, pages 114–131 (2024)Cite this article

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Abstract

Complex networks are ubiquitous: a cell, the human brain, a group of people and the Internet are all examples of interconnected many-body systems characterized by macroscopic properties that cannot be trivially deduced from those of their microscopic constituents. Such systems are exposed to both internal, localized, failures and external disturbances or perturbations. Owing to their interconnected structure, complex systems might be severely degraded, to the point of disintegration or systemic dysfunction. Examples include cascading failures, triggered by an initially localized overload in power systems, and the critical slowing downs of ecosystems which can be driven towards extinction. In recent years, this general phenomenon has been investigated by framing localized and systemic failures in terms of perturbations that can alter the function of a system. We capitalize on this mathematical framework to review theoretical and computational approaches to characterize robustness and resilience of complex networks. We discuss recent approaches to mitigate the impact of perturbations in terms of designing robustness, identifying early-warning signals and adapting responses. In terms of applications, we compare the performance of the state-of-the-art dismantling techniques, highlighting their optimal range of applicability for practical problems, and provide a repository with ready-to-use scripts, a much-needed tool set.

Key points

  • A variety of biological, social and engineering complex systems can be defined in terms of units that exchange information through interaction networks, exhibiting diverse structural patterns such as heterogeneity, modularity and hierarchy.

  • Owing to their interconnected nature, complex networks can amplify minor disruptions to a system-wide level, making it essential to understand their robustness against both external perturbations and internal failures.

  • The study of complex networks’ robustness and resilience involves investigating phase transitions that usually depend on features such as degree connectivity, spatial embedding, interdependence and coupled dynamics.

  • Network science offers a wide range of theoretical and computational methods for quantifying system robustness against perturbations, as well as grounded approaches to design robustness, identify early-warning signals and devise adaptive responses.

  • These methods find application across a multitude of disciplines, including systems biology, systems neuroscience, engineering, and social and behavioural sciences.

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Fig. 1: Percolation as static approach to network robustness.
Fig. 2: Comparison of state-of-the-art dismantling methods.
Fig. 3: Evolving failures on networks.
Fig. 4: Preventing and reacting to network collapse.

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Code availability

The code used for the performance comparisons can be found in the repository https://github.com/NetworkDismantling/review.

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Acknowledgements

O.A. acknowledges financial support from the Spanish Ministry of Universities through the Recovery, Transformation and Resilience Plan funded by the European Union (Next Generation EU) and the University of the Balearic Island. M.D.D. acknowledges partial financial support from the University of Padua (PRD-BIRD 2022), from the INFN grant ‘LINCOLN’, from the EU funding within the MUR PNRR ‘National Center for HPC, Big Data and Quantum Computing’ (project number CN00000013 CN1) and from the University of Padua (PRD-BIRD 2022). J.P.G. is partly funded by Science Foundation Ireland grant numbers 16/IA/4470 and 12/RC/2289 P2. H.A.M. was supported by NSF-HNDS Award 2214217. G.M. acknowledges financial support from PNRR MUR project PE0000013-FAIR. M.P. was supported by the Slovenian Research and Innovation Agency (grant numbers P1-0403, J1-2457 and N1-0232). F.R. acknowledges support from the Army Research Office, award number W911NF-21-1-0194, and the Air Force Office of Scientific Research, award number FA9550-21-1-0446.

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  1. These authors contributed equally: Oriol Artime and Marco Grassia.

Authors and Affiliations

  1. Departament de Física de la Matèria Condensada, University of Barcelona, Barcelona, Spain

    Oriol Artime

  2. Universitat de Barcelona Institute of Complex Systems (UBICS), University of Barcelona, Barcelona, Spain

    Oriol Artime

  3. Universitat de les Illes Balears, Palma, Spain

    Oriol Artime

  4. Department of Electric, Electronic and Computer Engineering, University of Catania, Catania, Italy

    Marco Grassia & Giuseppe Mangioni

  5. Department of Physics and Astronomy, University of Padua, Padova, Italy

    Manlio De Domenico

  6. Padua Center for Network Medicine, University of Padua, Padova, Italy

    Manlio De Domenico

  7. Istituto Nazionale di Fisica Nucleare, Padova, Italy

    Manlio De Domenico

  8. MACSI, Department of Mathematics & Statistics, University of Limerick, Limerick, Ireland

    James P. Gleeson

  9. Levich Institute and Physics Dept., City College of New York, New York, NY, USA

    Hernán A. Makse

  10. Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia

    Matjaž Perc

  11. Department of Physics, Kyung Hee University, Seoul, Republic of Korea

    Matjaž Perc

  12. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

    Matjaž Perc

  13. Complexity Science Hub Vienna, Vienna, Austria

    Matjaž Perc

  14. Center for Complex Networks and Systems Research, Luddy School of Informatics, Computing, and Engineering, Indiana University, Luddy Center for Artificial Intelligence, Bloomington, IN, USA

    Filippo Radicchi

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Contributions

M.D.D. and G.M. decided the scope of the Review. M.G. performed benchmarking. All authors contributed to the writing of the manuscript.

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Artime, O., Grassia, M., De Domenico, M. et al. Robustness and resilience of complex networks. Nat Rev Phys 6, 114–131 (2024). https://doi.org/10.1038/s42254-023-00676-y

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