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.
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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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- Findingown data§ II. VARIABILITY OF CITATION STATISTICS IN DIFFERENT DISCIPLINES · p. 2
A publication with 100 citations is approximately 50 times more common in Developmental Biology than in Engineering, Aerospace.
“For example a publication with 100 citations is approximately 50 times more common in 'Developmental Biology' than in 'Engineering, Aerospace'.”
Quote in the paperSubstantial claim - Findingown data§ III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 2
The universal rescaled citation distribution resembles a lognormal distribution with σ^2 = 1.3 when a single curve is fitted for all categories.
“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.”
Quote in the paperSubstantial claim - Also stated in: § II. VARIABILITY OF CITATION STATISTICS IN DIFFERENT DISCIPLINES · p. 2 · plain
- Findingown data§ III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 2
Values of χ2 per degree of freedom indicate that the lognormal fit is good.
“Values of χ 2 per degree of freedom, also reported in Table I, indicate that the fit is good.”
Quote in the paperSubstantial claim - Findingown data§ Universality of citation distributions: towards an objective measure of scientific impact. · p. 1
The same universal behavior occurs when citation distributions of articles published in the same field but in different years are compared.
“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.”
Quote in the paperSubstantial claim - Also stated in: § III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 4 · plain
- Findingown data§ IV. A GENERALIZED H-INDEX · p. 4
The average number of articles published by an author in a year depends on the discipline.
“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.”
Quote in the paperSubstantial claim
- Findingown data§ IV. A GENERALIZED H-INDEX · p. 4
The universal curve for rescaled number of publications per author is fitted reasonably well over almost two decades by a power law P(N,N0) ≈ (N/N0)^-δ with δ = 3.5(5).
“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).”
Quote in the paperSubstantial claim - Findingcitation§ III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 3
c0 shows very little correlation with the overall size of the field, as measured by the total number of articles.
“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.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [11] Althouse BM, West JD, Bergstrom T, Bergstrom CT (2008) Differences in Impact Factor Across Fields and Over Time. arxiv 0804.3116 . source
Not checked in the free run.
- Findingreasoning§ VI. METHODS · p. 6
The category 'Multidisciplinary sciences' does not fit perfectly into the universal picture found for other categories because its citation distribution is a convolution of the distributions of the single disciplines represented in its journals.
“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.”
Quote in the paperSubstantial claim - Findingreasoning§ VI. METHODS · p. 6
If one restricts to Nature, Science, and PNAS, the 'Multidisciplinary sciences' category fits very well into the global universal picture.
“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.”
Quote in the paperNot a substantial claim - Findingown data§ Universality of citation distributions: towards an objective measure of scientific impact. · p. 1
When the relative indicator c_f = c/c0 is used, citation distributions for different disciplines are rescaled onto a universal curve, where c0 is the average number of citations per article for the discipline.
“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.”
Quote in the paperSubstantial claim - Also stated in: § III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 2 · boosted
- Findingown data§ Universality of citation distributions: towards an objective measure of scientific impact. · p. 1
The probability that an article is cited c times varies largely between different scientific disciplines.
“We show that the probability that an article is cited c times has large variations between different disciplines”
- Also stated in: § II. VARIABILITY OF CITATION STATISTICS IN DIFFERENT DISCIPLINES · p. 2 · boosted
“From this figure it is apparent that the chance of a publication to be cited strongly depends on the category the article belongs to.”
Quote in the paperSubstantial claim - Also stated in: § II. VARIABILITY OF CITATION STATISTICS IN DIFFERENT DISCIPLINES · p. 2 · boosted
- Findingown data§ IV. A GENERALIZED H-INDEX · p. 4
The variability in the number of publications per author across disciplines is rescaled away when N is divided by the average value N0 in the discipline.
“But also in this case this variability is rescaled away if the number N of publications in a year by an author is divided by the average value in the discipline N 0 .”
Quote in the paperSubstantial claim - Interpretationreasoning§ IV. A GENERALIZED H-INDEX · p. 5
The generalized h-index h_f factors out the additional bias due to different publication rates, allowing comparisons among scientists working in different fields.
“This universality allows one to define a generalized hindex, h f , that factors out also the additional bias due to different publication rates, thus allowing comparisons among scientists working in different fields.”
Quote in the paperSubstantial claim - Interpretationown data§ V. CONCLUSIONS · p. 5
An article in Aerospace Engineering with only 20 citations (c_f ≈ 3.54) is more successful than an article in Developmental Biology with 100 citations (c_f ≈ 2.58).
“For instance, Figure 2 leads to the counterintuitive conclusion that an article in Aerospace Engineering with only 20 citations ( c f ≈ 3 . 54) is more successful than an article in Developmental Biology with 100 citations ( c f ≈ 2 . 58).”
Quote in the paperSubstantial claim - Study design and conductown data§ VI. METHODS · p. 6
Neglecting uncited articles produces only a small shift in c0 that does not affect the results of the analysis.
“Our calculations neglect uncited articles; we have verified however that their inclusion just produces a small shift in c 0 , which does not affect the results of our analysis.”
Quote in the paperNot a substantial claim - Premisecitation§ I. INTRODUCTION · p. 1
Empirical studies have shown that distributions of article citations are very skewed, even within single disciplines.
“The use of relative indicators is widespread, but empirical studies [19, 20, 21] have shown that distributions of article citations are very skewed, even within single disciplines.”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [19] source
Not checked in the free run.
- cites [20] Naranan S (1971) Power law relations in science bibliography: a self-consistent interpretation. J Docum 27:8397. source
Not checked in the free run.
- cites [21] Seglen PO (1992) The skewness of science. J Am Soc Inf Sci 43:628-638.
Not checked in the free run.
- Premisecitation§ IV. A GENERALIZED H-INDEX · p. 4
The h-index has the shortcoming of difficulty in comparing authors in different disciplines.
“Despite its success, as all other performance metrics the h-index has some shortcomings, as already pointed out by Hirsch himself. One of them is the difficulty to compare authors in different disciplines.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [1] source
Not checked in the free run.
- Interpretationown data§ V. CONCLUSIONS · p. 5
The analysis justifies the use of relative indicators to compare fairly the impact of articles across different disciplines and years.
“The analysis presented here justifies the use of relative indicators to compare in a fair manner the impact of articles across different disciplines and years.”
Quote in the paperSubstantial claim- The authors state this as an assumption: Under an unbiased ranking, articles of the various disciplines are scattered uniformly along the rank axis.
- Interpretationown data§ Universality of citation distributions: towards an objective measure of scientific impact. · p. 1
c_f is an unbiased indicator for comparing citation performance across disciplines and years.
“These findings provide a strong validation of c f as an unbiased indicator for citation performance across disciplines and years.”
- Also stated in: § III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 3 · boosted
“Figures 2 and 3 allow to conclude that c f is an unbiased indicator for comparing the scientific impact of publications in different disciplines.”
Quote in the paperSubstantial claim- The authors state this as an assumption: Under an unbiased ranking, articles of the various disciplines are scattered uniformly along the rank axis.
- Also stated in: § III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 3 · boosted
- May combine two claims. 5
- Classified as probably not a substantial claim. 1
May combine two claims.
Findingown data§ III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 3For z = 20%, the standard deviation of bin heights is 1.15% for c_f-based ranking and 12.37% for c-based ranking, compared with 1.09% expected under unbiased ranking.
“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.”
Quote in the paperSubstantial claim- The authors state this as an assumption: Under an unbiased ranking, articles of the various disciplines are scattered uniformly along the rank axis.
May combine two claims.
Findingown data§ III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 2All fitted values of σ^2 for the disciplines are compatible within two standard deviations, except Anesthesiology, which is within three standard deviations of all the others.
“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.”
Quote in the paperSubstantial claimMay combine two claims.
Findingown data§ III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 3When articles from different disciplines are ranked by c_f, the percentage of each discipline in the top z% agrees well with the unbiased-ranking expectation, whereas ranking by bare c shows strong bias.
“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.”
Quote in the paperSubstantial claim- The authors state this as an assumption: Under an unbiased ranking, articles of the various disciplines are scattered uniformly along the rank axis.
May combine two claims.
Findingown data§ III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 4The value of c0 grows as older publications are considered, but the rescaled citation distribution remains the same.
“The value of c 0 obviously grows as older publications are considered, but the rescaled distribution remains conspicuously the same.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim. May combine two claims.
Premisecitation§ III. DISTRIBUTION OF THE RELATIVE INDICATOR c f · p. 4Different temporal patterns of citations exist, with some articles starting soon to receive citations while others ('sleeping beauties') go unnoticed for a long time before attracting many citations.
“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].”
Quote in the paperSubstantial claim0 of 2 sources checked
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?
The number of citations a paper receives is the most commonly used measure of scientific impact.
“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
Variability in citation practices across disciplines prevents direct comparability of citation counts across fields unless appropriate corrective methods are applied.
“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
Recent work has improved theoretical understanding of citation dynamics.
“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]
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SciSciNet: A large-scale open data lake for the science of science research
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- 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 › Decision Sciences › Statistics, Probability and Uncertainty › 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
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.
Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-06.
Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-06.
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 ›
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