The DOI matches its Crossref record; 19 of the 19 references it cites were found in a registry or an index.
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.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The cited source says what the claim attributes to it.
In the citing paper
“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]).”
From Gene co-expression analysis for functional classification and gene–disease predictions (2017) · cites this paper as [69]
The cited source says what the claim attributes to it.
In the citing paper
“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]
The cited source says what the claim attributes to it.
In the citing paper
“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
In the citing paper
“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]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
Collective dynamics of ‘small-world’ networks
Watts, D. J., & Strogatz, S. H. (1998) · Nature · doi:10.1038/30918 · Cited by 43,060
Emergence of Scaling in Random Networks
Barabási, A.-L., & Albert, R. (1999) · Science · doi:10.1126/science.286.5439.509 · Cited by 36,135
Self-citationSocial Network Analysis: Methods and Applications
Wasserman, S., & Faust, K. (1994) · doi:10.1017/cbo9780511815478 · Cited by 17,557
Diameter of the World-Wide Web
Albert, R., Jeong, H., & Barabási, A.-Ĺ. (1999) · Nature · doi:10.1038/43601 · Cited by 4,240
Self-citationFrom molecular to modular cell biology.
Hartwell, L. H., Hopfield, J. J., Leibler, S., & Murray, A. W. (1999) · Nature · doi:10.1038/35011540 · Cited by 3,632 · cited 1×
“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 [1].”
BackgroundOn power-law relationships of the Internet topology
Faloutsos, M., Faloutsos, P., & Faloutsos, C. (1999) · ACM SIGCOMM Computer Communication Review · doi:10.1145/316194.316229 · Cited by 2,643
Mean-field theory for scale-free random networks
Barabási, A.-Ĺ., Albert, R., & Jeong, H. (1999) · Physica A Statistical Mechanics and its Applications · doi:10.1016/s0378-4371(99)00291-5 · Cited by 2,261 · cited 1×
“The inhomogeneous connectivity distribution of many real networks is reproduced by the scale-free model [17,18] that incorporates two ingredients common to real networks: growth and preferential attachment.”
BackgroundSelf-citationHow popular is your paper? An empirical study of the citation distribution
Redner, S. (1998) · The European Physical Journal B · doi:10.1007/s100510050359 · Cited by 1,509 · cited 1×
“In contrast, results on the world-wide web (www) [3–5], Internet [6] and other large networks [17–19] indicate that many systems belong to a class of inhomogeneous networks, referred to as scale-free networks , for which P ( k ) decays as a power-law, i.e.”
ResultSimple rules yield complex food webs
Martinez, N. D., Williams, R. J., & Martinez, N. D. (2000) · Nature · doi:10.1038/35004572 · Cited by 1,453
Accessibility of information on the web
Lawrence, S., & Giles, C. L. (1999) · Nature · doi:10.1038/21987 · Cited by 1,371 · cited 1×
“Networks with a very large number of nodes can have a rather small diameter; for example the diameter of the www, with over 800 million nodes [20], is around 19 [3], while social networks with over six billion individuals are believed to have a diameter of around six [21].”
Background
Flagged first, then by citations
Research works printed but not linked: 5
- Kumar, R., Raghavan, P., Rajalopagan, S. & Tomkins, A. in Proc. 9th ACM Symp. on Principles of Database Systems 1–10 (Association for Computing Machinery, New York, 2000).
- Jeong, H., Tombor, B., Albert, R., Oltvai, Z. & Barabási, A.-L. The large-scale organization of metabolic networks. Nature (in the press).
- Erdös, P. & Rényi, A. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–60 (1960).
- Bollobás, B. Random Graphs (Academic, London, 1985).
- Milgram, S. The small-world problem. Psychol. Today 2, 60–67 (1967).
Cited by
Sorted by citations
Items 1,001–1,024 of 1,024
Reuter, J. A., Ortiz-Urda, S., Kretz, M., Garcia, J., Scholl, F. A., Pasmooij, A. M. G., Cassarino, D., Chang, H. Y., & Khavari, P. A. (2009) · Cancer Cell · doi:10.1016/j.ccr.2009.04.002 · Cited by 88
Functional properties of resting state networks in healthy full-term newborns
De Asis-Cruz, J., Bouyssi-Kobar, M., Evangelou, I., Vezina, G., & Limperopoulos, C. (2015) · Scientific Reports · doi:10.1038/srep17755 · Cited by 85
Information cascades in complex networks
Jalili, M., & Perc, M. (2017) · Journal of Complex Networks · doi:10.1093/comnet/cnx019 · Cited by 248
The Molecular Roots of Compositional Inheritance
SEGRÉ, D., SHENHAV, B., KAFRI, R., & LANCET, D. (2001) · Journal of Theoretical Biology · doi:10.1006/jtbi.2001.2440 · Cited by 82
Reconstruction of Cell-type-Specific Interactomes at Single-Cell Resolution
Mohammadi, S., Davila-Velderrain, J., & Kellis, M. (2019) · Cell Systems · doi:10.1016/j.cels.2019.10.007 · Cited by 73
Li, Z. (., Fang, X., & Sheng, O. R. L. (2018) · ACM Transactions on Management Information Systems · doi:10.1145/3131782 · Cited by 65
Stevens, A., Hanson, D., Whatmore, A., Destenaves, B., Chatelain, P., & Clayton, P. (2013) · BMC Genomics · doi:10.1186/1471-2164-14-547 · Cited by 65
Sreekumar, V., Nielson, D. M., Smith, T. A., Dennis, S. J., & Sederberg, P. B. (2018) · Scientific Reports · doi:10.1038/s41598-018-32879-0 · Cited by 64
Palva, J. M., & Palva, S. (2011) · Progress in Brain Research · doi:10.1016/b978-0-444-53839-0.00022-3 · Cited by 63
Evans, T., & Fu, F. (2018) · Royal Society Open Science · doi:10.1098/rsos.181122 · Cited by 61
Generative models of rich clubs in Hebbian neuronal networks and large-scale human brain networks
Vértes, P. E., Alexander-Bloch, A., & Bullmore, E. T. (2014) · Philosophical Transactions of the Royal Society B: Biological Sciences · doi:10.1098/rstb.2013.0531 · Cited by 50
Massey, S. (2015) · Life · doi:10.3390/life5021301 · Cited by 38
“In an interesting parallel, many non biological complex systems also show the property of robustness, often associated with network topology [69], an emergent property [70].”
BackgroundStructural and dynamical analysis of biological networks
Klein, C., Marino, A., Sagot, M.-F., Vieira Milreu, P., & Brilli, M. (2012) · Briefings in Functional Genomics · doi:10.1093/bfgp/els030 · Cited by 34
Bret, C., Sueur, C., Ngoubangoye, B., Verrier, D., Deneubourg, J.-L., & Petit, O. (2013) · PLoS ONE · doi:10.1371/journal.pone.0083015 · Cited by 37
Convergent Evolutionary Paths in Biological and Technological Networks
Solé, R. V., Valverde, S., & Rodriguez-Caso, C. (2011) · Evolution: Education and Outreach · doi:10.1007/s12052-011-0346-1 · Cited by 20
Graph theory and stability analysis of protein complex interaction networks
Huang, C., Chen, T., & Ng, K. (2016) · IET Systems Biology · doi:10.1049/iet-syb.2015.0007 · Cited by 18
Quantum Google algorithm: Construction and application to complex networks
Paparo, G. D., Müller, M., Comellas, F., & Martin-Delgado, M. A. (2014) · The European Physical Journal Plus · doi:10.1140/epjp/i2014-14150-y · Cited by 17
Excess reciprocity distorts reputation in online social networks
Livan, G., Caccioli, F., & Aste, T. (2017) · Scientific Reports · doi:10.1038/s41598-017-03481-7 · Cited by 20
Hierarchy in directed random networks
Mones, E. (2013) · Physical Review E · doi:10.1103/physreve.87.022817 · Cited by 16
Multi-Agent Risks from Advanced AI
Hammond, L., Chan, A., Clifton, J., Hoelscher-Obermaier, J., Khan, A., McLean, E., Smith, C., Barfuss, W., Foerster, J., Gavenčiak, T., et al. (2025) · arXiv · doi:10.48550/arxiv.2502.14143 · Cited by 305
Jaemthaworn, T., Kalapanulak, S., & Saithong, T. (2021) · Scientific Reports · doi:10.1038/s41598-021-86806-x · Cited by 7
dos Santos, E. C., Rohan, P., Binato, R., & Abdelhay, E. (2023) · Cancers · doi:10.3390/cancers15225374 · Cited by 3
“This information is central not only to understanding the network structure but also to predicting the magnitude of changes that can occur if a given hub is subjected to management as they are important for network maintenance [39].”
BackgroundColglazier, C., TeBlunthuis, N., & Shaw, A. (2024) · Proceedings of the International AAAI Conference on Web and Social Media · doi:10.1609/icwsm.v18i1.31316 · Cited by 8
When Your AI Agent Succumbs to Peer-Pressure: Studying Opinion-Change Dynamics of LLMs
Mehdizadeh, A., & Hilbert, M. (2025) · arXiv · doi:10.48550/arxiv.2510.19107
“Decentralized networks are characterized by a more homogeneous distribution of centrality and reduced reliance on a few highly connected nodes, making them structurally resilient and less prone to bottlenecks [125].”
Metadata
- 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.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
