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Universality of citation distributions: Toward an objective measure of scientific impact
Open accessAbstract
We study the distributions of citations received by a single publication within several disciplines, spanning broad areas of science. We show that the probability that an article is cited c times has large variations between different disciplines, but all distributions are rescaled on a universal curve when the relative indicator c f = c / c 0 is considered, where c 0 is the average number of citations per article for the discipline. In addition we show that the same universal behavior occurs when citation distributions of articles published in the same field, but in different years, are compared. These findings provide a strong validation of c f as an unbiased indicator for citation performance across disciplines and years. Based on this indicator, we introduce a generalization of the h index suitable for comparing scientists working in different fields.
Overview
AI snapshot
Written from the paper’s full text.
TL;DR
The paper shows that citation distributions across scientific disciplines and publication years collapse onto a universal curve when citations are rescaled by the field-year average \(c_0\), making \(c_f=c/c_0\) an unbiased relative indicator. It introduces a generalized h-index \(h_f\) based on \(c_f\) and normalized publication rates.
Key findings
- Bare citation distributions vary strongly by discipline: a publication with 100 citations is approximately 50 times more common in Developmental Biology than in Engineering, Aerospace. After rescaling by \(c_0\), all distributions collapse onto a universal curve fitted by a lognormal with \(\sigma^2=1.3\).
- Universality also holds across years within the same field: for Hematology, Neuroimaging, and Physics, Nuclear, articles from 1990, 1999, and 2004 show the same rescaled distribution despite \(c_0\) growing with elapsed time.
- Ranking articles by \(c_f\) reduces field bias: for the top \(z=20\%\), the standard deviation of discipline bin heights is \(\sigma_z=1.15\%\) for \(c_f\)-based ranking versus \(12.37\%\) for bare \(c\), compared with the theoretical unbiased value \(\sigma_z=1.09\%\).
- The generalized h-index \(h_f\) orders articles by \(c_f\) and uses reduced rank \(r/N_0\); an example with \(c_f\) values 4.1, 2.8, 2.2, 1.6, 0.8, 0.4 and \(N_0=2.0\) gives \(h_f=1.5\). Author publication counts also rescale, fitted by a power law with exponent \(-3.5(5)\).
Methods
- Data come from Thomson Scientific’s Web of Science; journals are divided into 172 JCR categories, and only records classified as “article” and “letter” are considered. Citations are counted as the total number of times an article appears as a reference in a more recent published article.
- For each discipline and year, the average citations per article \(c_0\) is computed; distributions \(P(c,c_0)\) are rescaled as \(c_0P(c,c_0)\) versus \(c_f=c/c_0\). Curves are fitted with a lognormal constrained by \(\sigma^2=-2\mu\) because the expected value of \(c_f\) is 1, using exponentially growing bins.
- Ranking analysis pools disciplines, ranks articles by \(c\) and by \(c_f\), and compares the percentage of each discipline in the top \(z\%\) against an unbiased-ranking expectation. The generalized h-index is defined using \(c_f\) and \(r/N_0\).
Limitations
- Stated: uncited articles are neglected, though the authors verified their inclusion only produces a small shift in \(c_0\); “Multidisciplinary sciences” does not fit perfectly because its citation distribution is a convolution of single disciplines, but restricting to Nature, Science, and PNAS fits well.
- Stated: the analysis addresses single publications, not individuals or research groups, and other biases such as number of authors are left for future work; the origin of the universality is also not understood.
- Evident from the text: JCR categories are not mutually exclusive, and the generalized h-index’s practical application is deferred to a forthcoming publication.
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Claims the paper makes
13 Own result · 4 Not checked yet · 4 Own interpretation · 2 No basis given · 1 Own method
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For example a publication with 100 citations is approximately 50 times more common in 'Developmental Biology' than in 'Engineering, Aerospace'.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Fitting a single curve for all categories, a lognormal distribution with σ 2 = 1 . 3 is found, that is reported in Figure 2.”
- Also stated in: § II. VARIABILITY OF CITATION STATISTICS IN DIFFERENT DISCIPLINES · p. 2 · plain
“The dashed line is a lognormal fit with σ 2 = 1 . 3.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § II. VARIABILITY OF CITATION STATISTICS IN DIFFERENT DISCIPLINES · p. 2 · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Values of χ 2 per degree of freedom, also reported in Table I, indicate that the fit is good.”
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In addition we show that the same universal behavior occurs when citation distributions of articles published in the same field, but in different years, are compared.”
- Also stated in: § III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 4 · plain
“The value of c 0 obviously grows as older publications are considered, but the rescaled distribution remains conspicuously the same.”
- Also stated in: § IV. A GENERALIZED H-INDEX · p. 5 · plain
“the universal scaling observed over different disciplines naturally holds also for articles published in different periods of time.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 4 · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As Fig. 5 shows, such a quantity also depends on the discipline considered: in some disciplines the average number of articles published by an author in a year is much larger than in others.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Interestingly, the universal curve is fitted reasonably well over almost two decades by a power-law behavior P ( N,N 0 ) ≈ ( N/N 0 ) -δ with δ = 3 . 5(5).”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
In the paper
“In agreement with the findings of Ref. [11] c 0 shows very little correlation with the overall size of the field, as measured by the total number of articles.”
What the sources say
- Not checked yetcites [11] Althouse BM, West JD, Bergstrom T, Bergstrom CT (2008) Differences in Impact Factor Across Fields and Over Time. arxiv 0804.3116 . source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The category 'Multidisciplinary sciences' does not fit perfectly into the universal picture found for other categories, because the distribution of the number of citations is a convolution of the distributions corresponding to the single disciplines represented in the journals.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“However, if one restricts only to the three most important multidisciplinary journals ( Nature , Science , Proc. Natl. Acad. Sci. USA ) also this category fits very well into the global universal picture.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“all distributions are rescaled on a universal curve when the relative indicator c f = c/c 0 is considered, where c 0 is the average number of citations per article for the discipline.”
- Also stated in: § III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 2 · boosted
“Fig. 2 shows that this procedure leads to a very good collapse of all curves for different values of c 0 onto a single shape.”
- Also stated in: § III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 2 · boosted
“This allows to conclude that, rescaling the distribution of citations for publications in a scientific discipline by their average number, a universal curve is found, independent of the specific discipline.”
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 2 · boosted
- It may state two separate claims as one. 5
- Probably not a substantial claim. 1
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For example, for z = 20%, σ z = 1 . 15% for c f -based ranking, while σ z = 12 . 37% if c is used, as opposed to the value σ z = 1 . 09% in the hypothesis of unbiased ranking.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: Under an unbiased ranking, articles of the various disciplines are scattered uniformly along the rank axis.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“All fitted values of σ 2 , reported in Table I, are compatible within two standard deviations, except for one (Anesthesiology) that is in any case within three standard deviations of all the others.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“When the ranking is performed according to c f = c/c 0 we find (Table II) a very good agreement with the hypothesis that the ranking is unbiased, while strong evidence that the ranking is biased is found when c is used.”
boostedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: Under an unbiased ranking, articles of the various disciplines are scattered uniformly along the rank axis.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The value of c 0 obviously grows as older publications are considered, but the rescaled distribution remains conspicuously the same.”
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“It is known that different temporal patterns of citations exist, with some articles starting soon to receive citations, while others ('sleeping beauties') go unnoticed for a long time, after which they are recognized as seminal and begin to attract a large number of citations [26, 27].”
What the sources say
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may state two separate claims as one.
The authors state this as an assumption.
Under an unbiased ranking, articles of the various disciplines are scattered uniformly along the rank axis.
“assuming that articles of the various disciplines are scattered uniformly along the rank axis, one would expect the average bin height in Fig. 3 to be z % with a standard deviation”
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?
In the citing paper
“The number of citations a paper receives is the most commonly used measure of scienti fi c impact.”
From Unraveling scientific impact: Citation types in marketing journals (2015) · cites this paper as Radicchi, Fortunato, & Castellano, 2008
In the citing paper
“Variability in citation practices across disciplines, for instance, is a well-recognized problem which prevents direct comparability of citation counts across fi elds (Bornmann & Daniel, 2008), unless appropriate corrective methods are applied (Radicchi et al., 2008).”
From Unraveling scientific impact: Citation types in marketing journals (2015) · cites this paper as Radicchi et al., 2008
In the citing paper
“Recent work has improved our theoretical understanding of citation dynamics [27–29].”
From Relative Citation Ratio (RCR): A New Metric That Uses Citation Rates to Measure Influence at the Article Level (2016) · cites this paper as [28]
Citations
Cites
An index to quantify an individual’s scientific research output
Hirsch, J. E. (2005) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0507655102 · Cited by 11,669
Theory and practise of the g-index
Egghe, L. (2006) · Scientometrics · doi:10.1007/s11192-006-0144-7 · Cited by 2,354
Citation Indexing-Its Theory and Application in Science, Technology, and Humanities
Goodwin, J., & Garfield, E. (1980) · Technology and Culture · doi:10.2307/3104125 · Cited by 1,648
What do citation counts measure? A review of studies on citing behavior
Bornmann, L., & Daniel, H. (2008) · Journal of Documentation · doi:10.1108/00220410810844150 · Cited by 1,607
How 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
The scientific impact of nations
King, D. A. (2004) · Nature · doi:10.1038/430311a · Cited by 1,245
Does the h index have predictive power?
Hirsch, J. E. (2007) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0707962104 · Cited by 980
van Raan, A. F. J. (2004) · Scientometrics · doi:10.1023/b:scie.0000018543.82441.f1 · Cited by 579
Moed, H. F., Bruin, R. E. D., & van Leeuwen, T. N. (1995) · Scientometrics · doi:10.1007/bf02017338 · Cited by 502
Schubert, A. P., & Braun, T. (1986) · Scientometrics · doi:10.1007/bf02017249 · Cited by 450
Cited by
Sorted by citations
Science Mapping: A Systematic Review of the Literature
Chen, C. (2017) · Journal of Data and Information Science · doi:10.1515/jdis-2017-0006 · Cited by 2,134
What do citation counts measure? A review of studies on citing behavior
Bornmann, L., & Daniel, H. (2008) · Journal of Documentation · doi:10.1108/00220410810844150 · Cited by 1,607
Fortunato, S., Bergstrom, C. T., Börner, K., Evans, J. A., Helbing, D., Milojević, S., Petersen, A. M., Radicchi, F., Sinatra, R., Uzzi, B., et al. (2018) · Science · doi:10.1126/science.aao0185 · Cited by 1,497
Self-citationA review of the literature on citation impact indicators
Waltman, L. (2016) · Journal of Informetrics · doi:10.1016/j.joi.2016.02.007 · Cited by 1,356
Historical comparison of gender inequality in scientific careers across countries and disciplines
Huang, J., Gates, A. J., Sinatra, R., & Barabási, A.-Ĺ. (2020) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1914221117 · Cited by 978
Metadata
- Authors
- Radicchi, F., Fortunato, S., & Castellano, C.
- Affiliations
- Institute for Scientific Interchange (IT)
- National Research Council (IT)
- Corresponding author
- Castellano, Claudio
- Venue
- Proceedings of the National Academy of Sciences
- Publisher
- National Academy of Sciences
- Published
- 2008-11-11
- Volume
- 105
- Issue
- 45
- Pages
- 17268-17272
- ISSN
- 0027-8424 (ISSN-L)
- 1091-6490
- PMC
- PMC2582263
- OpenAlex
- W2151866568
- Semantic Scholar
- 1ead99845d1dc1c0d58e47eda9ba21aa0234878f
- Cited by
- 828
- References
- 31
- Citations by year
- 2026: 20 · 2025: 38 · 2024: 42 · 2023: 40 · 2022: 41 · 2021: 48 · 2020: 53 · 2019: 64 · 2018: 44 · 2017: 75 · 2016: 58 · 2015: 56 · 2014: 57 · 2013: 61 · 2012: 48
- FWCI
- 31.18
- Type
- journal article
- Language
- en
- Open access
- green
- www.ncbi.nlm.nih.gov
- Locations
- Proceedings of the National Academy of Sciences (publishedVersion)
- PubMed Central (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Social Sciences › Library and Information Sciences › scientometrics and bibliometrics research
- Physical Sciences › Computer Science › Information Systems › Web visibility and informetrics
- Social Sciences › Psychology › Experimental and Cognitive Psychology › Psychology Research and Bibliometrics
- Keywords
- citation distribution, scientific impact, h-index, cross-disciplinary comparison, rescaling, citation performance, bibliometric analysis
- MeSH terms
- Models, Theoretical, Selection Bias, Interdisciplinary Communication, Journal Impact Factor
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