Error and attack tolerance of complex networks
Open accessAbstract
Many complex systems display a surprising degree of tolerance against errors. For example, relatively simple organisms grow, persist and reproduce despite drastic pharmaceutical or environmental interventions, an error tolerance attributed to the robustness of the underlying metabolic network. Complex communication networks display a surprising degree of robustness: although key components regularly malfunction, local failures rarely lead to the loss of the global information-carrying ability of the network. The stability of these and other complex systems is often attributed to the redundant wiring of the functional web defined by the systems' components. Here we demonstrate that error tolerance is not shared by all redundant systems: it is displayed only by a class of inhomogeneously wired networks, called scale-free networks, which include the World-Wide Web, the Internet, social networks and cells. We find that such networks display an unexpected degree of robustness, the ability of their nodes to communicate being unaffected even by unrealistically high failure rates. However, error tolerance comes at a high price in that these networks are extremely vulnerable to attacks (that is, to the selection and removal of a few nodes that play a vital role in maintaining the network's connectivity). Such error tolerance and attack vulnerability are generic properties of communication networks.
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Hubs are frequently more relevant to the functionality of networks than other nodes.
“Hubs are frequently more relevant to the functionality of networks than other nodes [69]. This is also the case in biological networks [32], although mathematical derivations show that this is only the case for intra-modular hub genes (as opposed to inter-modular hub genes [64, 65]).”
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Other types of perturbations natural for classical systems, such as targeted attacks of network hubs or cascade-based attacks, still have no clear quantum counterpart.
“Other types of perturbations that are natural for classical systems, such as targeted attacks of network hubs 26 or cascade-based attacks 105 , still have no clear quantum counterpart”
From Complex networks from classical to quantum (2019) · cites this paper as [26]
A different class of problems concerns changes in the network structure itself.
“A different class of problems concerns not changes in the content transmitted through the network but changes in the network structure itself (Albert et al., 2000).”
From Multi-Agent Risks from Advanced AI (2025) · cites this paper as Albert et al., 2000
Results were shown for the Gave thresholded at R = 0.31; at this threshold the group averaged connectomes is sparse (edge density = 10%) and fully connected.
“Results were shown for the Gave thresholded at R = 0.3159; at this threshold the group averaged connectomes is sparse (edge density = 10%) and fully connected.”
From Functional properties of resting state networks in healthy full-term newborns (2015) · cites this paper as [59]
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- Authors
- Albert, R., Jeong, H., & Barabási, A.-L.
- Affiliations
- University of Notre Dame (US)
- Venue
- Nature
- Published
- 2000-07-27
- Volume
- 406
- Issue
- 6794
- Pages
- 378-382
- ISSN
- 0028-0836 (ISSN-L)
- 1476-4687
- OpenAlex
- W2065769502
- Cited by
- 7,534
- References
- 19
- Citations by year
- 2026: 407 · 2025: 315 · 2024: 304 · 2023: 287 · 2022: 331 · 2021: 353 · 2020: 334 · 2019: 369 · 2018: 402 · 2017: 384 · 2016: 398 · 2015: 402 · 2014: 376 · 2013: 315 · 2012: 318
- FWCI
- 63.07
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Nature (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Complex Network Analysis Techniques
- Physical Sciences › Engineering › Civil and Structural Engineering › Infrastructure Resilience and Vulnerability Analysis
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Complex Systems and Time Series Analysis
- Keywords
- error tolerance, attack vulnerability, scale-free networks, network robustness, complex networks, node removal, network connectivity, redundancy, communication networks, metabolic networks, social networks, World Wide Web
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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