Power laws in citation distributions: evidence from Scopus
distributions of citations to scientific papers is crucial for
understanding how science develops. However, there is a considerable
empirical controversy on which statistical model fits the citation
distributions best. This paper is concerned with rigorous empirical
detection of power-law behaviour in the distribution of citations
received by the most highly cited scientific papers. We have used a
large, novel data set on citations to scientific papers published
between 1998 and 2002 drawn from Scopus. The power-law model is compared
with a number of alternative models using a likelihood ratio test. We
have found that the power-law hypothesis is rejected for around half of
the Scopus fields of science. For these fields of science, the Yule,
power-law with exponential cut-off and log-normal distributions seem to
fit the data better than the pure power-law model. On the other hand,
when the power-law hypothesis is not rejected, it is usually empirically
indistinguishable from most of the alternative models. The pure
power-law model seems to be the best model only for the most highly
cited papers in “Physics and Astronomy”. Overall, our results seem to
support theories implying that the most highly cited scientific papers
follow the Yule, power-law with exponential cut-off or log-normal
distribution. Our findings suggest also that power laws in citation
distributions, when present, account only for a very small fraction of
the published papers (less than 1 % for most of science fields) and that
the power-law scaling parameter (exponent) is substantially higher
(from around 3.2 to around 4.7) than found in the older literature.
is often argued in scientometrics, social physics and other sciences
that distributions of some scientific “items” (e.g., articles,
citations) produced by some scientific sources (e.g., authors, journals)
have heavy tails that can be modelled using a power-law model. These
distributions are then said to conform to the Lotka’s law (Lotka 1926).
Examples of such distributions include author productivity, occurrence
of words, citations received by papers, nodes of social networks, number
of authors per paper, scattering of scientific literature in journals,
and many others (Egghe 2005).
In fact, power-law models are widely used in many sciences as physics,
biology, earth and planetary sciences, economics, finance, computer
science, and others (Newman 2005; Clauset et al. 2009). Models equivalent to Lotka’s law are known as Pareto’s law in economics (Gabaix 2009) and as Zipf’s law in linguistics (Baayen 2001).
Appropriate measuring and providing scientific explanations for power
laws plays an important role in understanding the behaviour of various
natural and social phenomena.
concerned with empirical detection of power-law behaviour in the
distribution of citations received by scientific papers. The power-law
distribution of citations for the highly cited papers was first
suggested by SollaPrice (1965), who also proposed a “cumulative advantage” mechanism that could generate the power-law distribution (SollaPrice 1976).
More recently, a growing literature has developed that aims at
measuring power laws in the right tails of citation distributions. In
particular, Redner (1998), Redner (2005)
found that the right tails of citation distributions for articles
published in Physical Review over a century and of articles published in
1981 in journals covered by Thomson Scientific’s Web of Science (WoS)
follow power laws. The latter data set was also modelled with power-law
techniques by Clauset et al. (2009) and Peterson et al. (2010).
The latter study also used data from 2007 list of the living highest
h-index chemists and from Physical Review D between 1975 and 1994.
observed that the top of the distribution of around 18,000 papers
published between 1991 and 1998 in the field of chemistry in Netherlands
follows a power law distribution. Power-law models were also fitted to
data from high energy physics (Lehmann et al. 2003), data for most cited physicists (Laherrère and Sornette 1998), data for all papers published in journals of the American Physical Society from 1983 to 2008 (Eom and Fortunato 2011), and to data for all physics papers published between 1980 and 1989 (Golosovsky and Solomon 2012).
tested for the power-law behavior using a large WoS dataset of 3.9
million articles published between 1998 and 2002 categorized in 22 WoS
research fields. The same dataset was also used to search for the power
laws in the right tail of citation distributions categorized in 219 WoS
scientific sub-fields (Albarrán et al. 2011a, b).
These studies offer the largest existing body of evidence on the
power-law behaviour of citation distributions. Three major conclusions
appear from them. First, the power-law behavior is not universal. The
existence of power law cannot be rejected in the WoS data for 17 out of
22 and for 140 out of 219 sub-fields studied in Albarrán and
Ruiz-Castillo (2011) and in Albarrán et al. (2011a, b),
respectively. Secondly, in opposition to previous studies, these papers
found that the scaling parameter (exponent) of the power-law
distribution is above 3.5 in most of the cases, while the older
literature suggested that the parameter value is between 2 and 3
(Albarrán et al. 2011).
Third, power laws in citation distributions are rather small—on average
they cover just about 2 % of the most highly cited articles in a given
WoS field of science and account for about 13.5 % of all citations in
statistically rigorous approach to answer the empirical question of
whether the power-law model describes best the observed distribution of
highly cited papers. We use the statistical toolbox for detecting
power-law behaviour introduced by Clauset et al. (2009).
There are two major contributions of the present paper. First, we use a
very large, previously unused data set on the citation distributions of
the most highly cited papers in several fields of science. This data
set comes from Scopus, a bibliographic database introduced in 2004 by
Elsevier, and contains 2.2 million articles published between 1998 and
2002 and categorized in 27 Scopus major subject areas of science. Most
of the previous studies used rather small data sets, which were not
suitable for rigorous statistical detecting of the power-law behaviour.
In contrast, our sample is even bigger with respect to the most highly
cited papers than the large sample used in the recent contributions
based on WoS data (Albarrán and Ruiz-Castillo 2011; Albarrán et al. 2011a, b). This results from the fact that Scopus indexes about 70 % more sources compared to the WoS (López-Illescas et al. 2008; Chadegani et al. 2013) and therefore gives a more comprehensive coverage of citation distributions.1
second major contribution of the paper is to provide a rigorous
statistical comparison of the power-law model and a number of
alternative models with respect to the problem which theoretical
distribution fits better empirical data on citations. This problem of
model selection has been previously studied in some contributions to the
literature. It has been argued that models like stretched exponential
(Laherrère and Sornette 1998), Yule (SollaPrice 1976), log-normal (Redner 2005; Stringer et al. 2008; Radicchi et al. 2008), Tsallis (Tsallis and deAlbuquerque 2000; Anastasiadis et al. 2010; Wallace et al. 2009) or shifted power law (Eom and Fortunato 2011)
fit citation distributions equally well or better than the pure
power-law model. However, previous papers have either focused on a
single alternative distribution or used only visual methods to choose
between the competing models. The present paper fills the gap by
providing a systematic and statistically rigorous comparison of the
power-law distribution with such alternative models as the log-normal,
exponential, stretched exponential (Weibull), Tsallis, Yule and
power-law with exponential cut-off. The comparison between models was
performed using a likelihood ratio test (Vuong 1989; Clauset et al. 2009).
Materials and methods
Fitting power-law model to citation data
in choosing methods for fitting power laws to citation distributions.
These authors carefully show that, in general, the appropriate methods
depend on whether the data are continuous or discrete. In our case, the
latter is true as citations are non-negative integers. Let x
be the number of citations received by an article in a given field of
science. The probability density function (pdf) of the discrete
power-law model is defined as
is a shape parameter of the power-law distribution, known as the
power-law exponent or scaling parameter. The power-law behaviour is
usually found only for values greater than some minimum, denoted by x0.
In case of citation distributions, the power-law behaviour has been
found on average only in the top 2 % of all articles published in a
field of science (Albarrán et al. 2011a, b).
should be therefore estimated if we want to measure precisely in which
part of a citation distribution the model applies. Moreover, we need an
estimate of x0 if we want to obtain an unbiased estimate of the power-law exponent, α.
and then we compute the well-known Kolmogorov–Smirnov (KS) statistic
for the data and the fitted model. The KS statistic is defined as
Goodness-of-fit and model selection tests
next step in measuring power laws involves testing goodness of fit. A
positive result of such a test allows to conclude that a power-law model
is consistent with data. Following Clauset et al. (2009) again, we use a test based on a semi-parametric bootstrap approach.3 The procedure starts with fitting a power-law model to data and calculating a KS statistic (see Eq. 3) for this fit, denoted by k.
Next, a large number of synthetic data sets is generated that follow
the originally fitted power-law model above the estimated x0 and have the same non-power-law distribution as the original data set below
Then, a power-law model is fitted to each of the generated data sets
using the same methods as for the original data set, and the KS
statistics are calculated. The fraction of data sets for which their own
KS statistic is larger than k is the p
value of the test. It represents a probability that the KS statistics
computed for data drawn from the power-law model fitted to the original
data is at least as large as k. The power-law hypothesis is rejected if the p value is smaller than some chosen threshold.4 Following Clauset et al. (2009), we rule out the power-law model if the estimated p value for this test is smaller than 0.1. In the present paper, we use 1,000 generated data sets.
the goodness-of-fit test rejects the power-law hypothesis, we may
conclude that the power law has not been found. However, if a data set
is fitted well by a power law, the question remains if there is an
alternative distribution, which is an equally good or better fit to this
data set. We need, therefore, to fit some rival distributions and
evaluate which distribution gives a better fit. To this aim, we use the
likelihood ratio test, which tests if the compared models are equally
close to the true model against the alternative that one is closer. The
test computes the logarithm of the ratio of the likelihoods of the data
under two competing distributions, LR, which is negative or positive
depending on which model fits data better. Specifically, let us consider
two distributions with pdfs denoted by p1(x) and p2(x). The LR is defined as:
fits the data better. However, the sign of the LR can be used to
determine which model should be favored only if the LR is significantly
different from zero. Vuong (1989) showed that in the case of non-nested models the normalized log-likelihood ratio NLR = n-1/2LR/σ, where σ is the estimated standard deviation of LR, has a limit standard normal distribution.5 This result can be used to compute a p value for the test discriminating between the competing models. If the p
value is small (for example, smaller than 0.1), then the sign of the LR
can probably be trusted as an indicator of which model is preferred.
However, if the p value is large, then the test is unable to choose between the compared distributions.
in choosing the following alternative discrete distributions:
exponential, stretched exponential (Weibull), log-normal, Yule and the
power law with exponential cut-off.6
Most of these models have been considered in previous literature on
modeling citation distribution. As another alternative, we also use the
Tsallis distribution, which has been also proposed as a model for
citation distributions (Wallace et al. 2009; Anastasiadis et al. 2010).
Finally, we also consider a “digamma” model using exponential functions
of a digamma function, which was recently introduced for distributions
with heavy tails in a statistical physics framework based on the
principle of maximum entropy (Peterson et al. 2013).7
use citation data from Scopus, a bibliographic database introduced in
2004 by Elsevier. Scopus is a major competitor to the most-widely used
data source in the literature on modeling citation distributions—Web of
Science (WoS) from Thomson Reuters. Scopus covers 29 million records
with references going back to 1996 and 21 million pre-1996 records going
back as far as 1823. An important limitation of the database is that it
does not cover cited references for pre-1996 articles. Scopus contains
21,000 peer-reviewed journals from more than 5,000 international
publishers. It covers about 70 % more sources compared to the WoS
(López-Illescas et al. 2008),
but a large part of the additional sources are low-impact journals. A
recent literature review has found that the quite extensive literature
that compares WoS and Scopus from the perspective of citation analysis
offers mixed results (Chadegani et al. 2013).
However, most of the studies suggest that, at least for the period from
1996 on, the number of citations in both databases is either roughly
similar or higher in Scopus than in WoS. Therefore, is seems that Scopus
constitutes a useful alternative to WoS from the perspective of
modeling citation distributions.
Scopus are classified under four main subject areas: life sciences
(4,200 journals), health sciences (6,500 journals), physical sciences
(7,100 journals) and social sciences including arts and humanities
(7,000 journals). The four main subject areas are further divided into
27 major subject areas and more than 300 minor subject areas. Journals
may be classified under more than one subject area.
From the various document types contained in Scopus, we have selected
only articles. For the purpose of comparability with the recent
WoS-based studies (Albarrán and Ruiz-Castillo 2011; Albarrán et al. 2011a),
only the articles published between 1998 and 2002 were considered.
Following previous literature, we have chosen a common 5-year citation
window for all articles published in 1998–2002.9 See Albarrán and Ruiz-Castillo (2011) for a justification of choosing the 5-year citation window common for all fields of science.
order to measure the power-law behaviour of citations, we need data on
the right tails of citation distributions. To this end, we have used the
Scopus Citation Tracker to collect citations for
is the actual number of articles published in a given field of science
during 1998–2002. This analysis was performed separately for each of the
27 science fields categorized by Scopus.
some cases, there was less than 100,000 articles published in a field
of science during 1998–2002 and we were able to obtain complete or
almost complete distributions of citations (see columns 2–4 of Table 2).10
In other cases, we have obtained only a part of the relevant
distribution encompassing the right tail and some part of the middle of
the distribution. The smallest portions of citation distributions were
obtained for Medicine (8.4 % of total papers), Biochemistry, Genetics
and Molecular Biology (15.7 %) and Physics and Astronomy (18.4 %).
However, using the WoS data for 22 science categories, Albarrán and
found that power laws account usually only for less than 2 % of the
highest-cited articles. Therefore, it seems that the coverage of the
right tails of citation distributions in our samples is satisfactory for
Results and discussion
presents results of fitting the discrete power-law model to our data
sets consisting of citations to scientific articles published over
1998–2002 (with a common 5-year citation window), separately for each of
the 27 Scopus major subject areas of science. The last row gives also
results for all subject areas combined (“All sciences”). Beside
estimates of the power-law exponent
with respect to the goodness-of-fit suggest that the power-law
hypothesis cannot be rejected for the following 14 Scopus science
fields: “Agricultural and Biological Sciences”, “Biochemistry, Genetics
and Molecular Biology”, “Chemical Engineering”, “Chemistry”, “Energy”,
“Environmental Science”, “Materials Science”, “Neuroscience”, “Nursing”,
“Pharmacology, Toxicology and Pharmaceutics”, “Physics and Astronomy”,
“Psychology”, “Health Professions”, and “Multidisciplinary”. The
remaining 13 Scopus fields of science for which the power-law model is
rejected include humanities and social sciences (“Arts and Humanities”,
“Business, Management and Accounting”, “Economics, Econometrics and
Finance”, “Social Sciences”), but also formal sciences (“Computer
Science”, “Decision Sciences”, “Mathematics”), life sciences
(“Immunology and Microbiology”, “Medicine”, “Veterinary”, “Dentistry”),
as well as “Earth and Planetary Sciences” and “Engineering”. The best
power-law fits for these fields of science are shown on
it can be clearly seen that their right tails decay faster than the
pure power-law model indicates. This suggest that the largest
observations for these distributions should be rather modeled with a
distribution having a lighter tail than the pure power-law model like
the log-normal or power-law with exponential cut-off models.
value for our goodness-of-fit test in case of “All Sciences” is 0.076,
which is below our acceptance threshold of 0.1. However, this p value is non-negligible and significantly higher than p
values for most of the 13 Scopus fields of science for which we reject
the power-law hypothesis. For this reason, we conclude that the evidence
is not conclusive in this case. Our result for “All Sciences” is,
however, in a stark contrast with that of Albarrán and Ruiz-Castillo (2011), who using the WoS data found that the fit for a corresponding data set was very good (with a p value of 0.85).11
estimates of the power-law exponent for the 14 Scopus science fields
for which the power law seems to be a plausible hypothesis range from
3.24 to 4.69. This is in a good agreement with Albarrán and
and confirms their assessment that the true value of this parameter is
substantially higher than found in the earlier literature (Redner 1998; Lehmann et al. 2003; Tsallis and deAlbuquerque 2000), which offered estimates ranging from around 2.3 to around 3. We also confirm the observation of Albarrán and Ruiz-Castillo (2011)
that power laws in citation distributions are rather small—they account
usually for less than 1 % of total articles published in a field of
science. The only two fields in our study with slightly “bigger” power
laws are “Chemistry” (2 %) and “Multidisciplinary” (2.8 %).
It can be observed that the exponential model can be ruled out in most
of the cases. We discuss other results first for the 13 Scopus fields of
science that did not pass our goodness-of-fit test. For all of these
fields, except for “Veterinary”, the Yule and power-law with exponential
cut-off models fit the data better than the pure power-law model in a
statistically significant way. The log-normal model is better than the
pure power-law model in 10 of the discussed fields; the same holds for
the Weibull distribution in case of 5 fields and for the digamma
distribution in case of 4 fields. However, these results do not imply
that the distributions, which give a better fit to the non-power-law
distributed data than the pure power-law model are plausible hypotheses
for these data sets. This issue should be further studied using
appropriate goodness-of-fit tests.
now turn to results for the remaining Scopus fields of science that
were not rejected by our goodness-of-fit test. The power-law hypothesis
seems to be the best model only for “Physics and Astronomy”. In this
case, the test statistics is always non-negative implying that the
power-law model fits the data as good as or better than each of the
alternatives. For the remaining 13 fields of science, the log-normal,
Yule and power-law with exponential cut-off models have always higher
log-likelihoods suggesting that these models may fit the data better
than the pure power-law distribution. However, only in a few cases the
differences between models are statistically significant. For
“Chemistry” and “Multidisciplinary” both the Yule and power-law with
exponential cut-off models are favoured over the pure power-law model.
The power-law with exponential cut-off is also favoured in case of
“Health Professions”. In other cases, the p values for the
likelihood ratio test are large, which implies that there is no
conclusive evidence that would allow to distinguish between the pure
power-law, log-normal, Yule and power-law with exponential cut-off
distributions. Comparing the power-law distribution with the Weibull and
Tsallis distributions, we observe that the sign of the test statistic
is positive in roughly half of the cases, but the p values are
always large and neither model can be ruled out. For the considered 13
fields of science, the digamma model is never better than the power law,
judging by the sign of the test statistic. Our likelihood ratio tests
suggest therefore that when the power law is a plausible hypothesis
according to our goodness-of-fit test it is often indistinguishable from
some alternative models.
our results show that the evidence in favour of the power-law behaviour
of the right-tails of citation distributions is rather weak. For
roughly half of the Scopus fields of science studied, the power-law
hypothesis is rejected. Other distributions, especially the Yule,
power-law with exponential cut-off and log-normal seem to fit the data
from these fields of science better than the pure power-law model. On
the other hand, when the power-law hypothesis is not rejected, it is
usually empirically indistinguishable from all alternatives with the
exception of the exponential distribution. The pure power-law model
seems to be favoured over alternative models only for the most highly
cited papers in “Physics and Astronomy”. Our results suggest that
theories implying that the most highly cited scientific papers follow
the Yule, power-law with exponential cut-off or log-normal distribution
may have slightly more support in data than theories predicting the pure
have used a large, novel data set on citations to scientific papers
published between 1998 and 2002 drawn from Scopus to test empirically
for the power-law behaviour of the right-tails of citation
distributions. We have found that the power-law hypothesis is rejected
for around half of the Scopus fields of science. For the remaining
fields of science, the power-law distribution is a plausible model, but
the differences between the power law and alternative models are usually
statistically insignificant. The paper also confirmed recent findings
of Albarrán and Ruiz-Castillo (2011)
that power laws in citation distributions, when they are a plausible,
account only for a very small fraction of the published papers (less
than 1 % for most of science fields) and that the power-law exponent is
substantially higher than found in the older literature.
would like to thank two anonymous referees for helpful comments and
suggestions that improved this paper. The use of Matlab and R software
accompanying the papers by Clauset et al. (2009), Shalizi (2007) and Peterson et al. (2013) is gratefully acknowledged. Any remaining errors are my responsibility.
the perspective of measuring power laws in citation distributions, the
most important part of the distribution is the right tail. It seems that
the database used in this paper has a better coverage of the right tail
of citation distributions. The most highly cited paper in our database
has received 5,187 citations (see Table 2), while the corresponding number for the database based on WoS is 4,461 (Li and Ruiz-Castillo 2013). Our database is further described in “Materials and methods” section.
Clauset et al. (2009) provide also an approximate method of estimating α
for the discrete power-law model by assuming that continuous power-law
distributed reals are rounded to the nearest integers. However, it this
paper we use an exact approach based on maximizing (2).
our data were drawn from a given model, then we could use the KS
statistic in testing goodness of fit, because the distribution of the KS
statistic is known in such a case. However, when the underlying model
is not known or when its parameters are estimated from the data, which
is our case, the distribution of the KS statistic must be obtained by
goodness-of-fit test we are interested in verifying if the power-law
model is a plausible hypothesis for our data sets. Hence, high p
values suggest that the power law is not ruled out. This approach is to
be contrasted with the usual approach, which for a given null
hypothesis interprets low p values as evidence in favor of the alternative hypothesis. See Clauset et al. (2009) for a more detailed discussion of these interpretations of p values.
it behaves like an exponential distribution. The pure power-law model
is nested within the power-law with exponential cut-off, and for this
reason the latter always provides a fit at least as good as the former.
would like to thank an anonymous referee for suggesting the inclusion
of this distribution in our comparison of alternative models.
example, for articles published in 1998 we have analyzed citations
received during 1998–2002, while for articles published in 2002, those
received during 2002–2006.
all fields of science analyzed, there were some articles with missing
information on citations. These articles were removed from our samples.
However, this has usually affected only about 0.1 % of our samples.
the power-law hypothesis is found plausible for 17 out of 22 WoS fields
of science. It is rejected for “Pharmacology and Toxicology”,
“Physics”, “Agricultural Sciences”, “Engineering”, and “Social Sciences,
General”. These results are not directly comparable with those of the
present paper as Scopus and WoS use different classification systems to
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