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