%0 
%0 Journal Article
%A Brzezinski, Michal
%D 2015
%T Power laws in citation distributions: evidence from Scopus
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%B Scientometrics
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%I Springer Netherlands
%V 103
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%N 1
%P 213-228
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%2 Power laws in citation distributions: evidence from Scopus - Springer
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%F brzezinski2015power
%K citation, distribution, fit, powerLaw
%X Modeling 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.
%Z 
%U http://dx.doi.org/10.1007/s11192-014-1524-z
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%0 
%0 Generic
%A Alstott, Jeff; Bullmore, Ed & Plenz, Dietmar
%D 2013
%T Powerlaw: a Python package for analysis of heavy-tailed distributions
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%2 Powerlaw: a Python package for analysis of heavy-tailed distributions
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%F alstott2013powerlaw
%K distribution, fit, powerlaw, python
%X Power laws are theoretically interesting probability distributions that are
also frequently used to describe empirical data. In recent years effective
statistical methods for fitting power laws have been developed, but appropriate
use of these techniques requires significant programming and statistical
insight. In order to greatly decrease the barriers to using good statistical
methods for fitting power law distributions, we developed the powerlaw Python
package. This software package provides easy commands for basic fitting and
statistical analysis of distributions. Notably, it also seeks to support a
variety of user needs by being exhaustive in the options available to the user.
The source code is publicly available and easily extensible.
%Z cite arxiv:1305.0215Comment: 18 pages, 6 figures, code and supporting information at  https://github.com/jeffalstott/powerlaw and  https://pypi.python.org/pypi/powerlaw
%U http://arxiv.org/abs/1305.0215
%+
%^

%0 
%0 Journal Article
%A Clauset, Aaron; Shalizi, Cosma Rohilla & Newman, M. E. J.
%D 2009
%T Power-Law Distributions in Empirical Data
%E 
%B SIAM Review
%C 
%I 
%V 51
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%N 4
%P 661-703
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%F clauset2009powerlaw
%K clauset, empirical, fit, powerLaw
%X 
%Z 
%U /brokenurl#         http://dx.doi.org/10.1137/070710111    
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%^

%0 
%0 Journal Article
%A Clauset, Aaron; Shalizi, Cosma Rohilla & Newman, M. E. J.
%D 2009
%T Power-Law Distributions in Empirical Data
%E 
%B SIAM Review
%C 
%I SIAM
%V 51
%6
%N 4
%P 661--703
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%F clauset2009powerlaw
%K law, power, powerlaw
%X Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution—the part of the distribution representing large but rare events—and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov–Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out.
%Z 
%U http://link.aip.org/link/?SIR/51/661/1
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%^

%0 
%0 Generic
%A Clauset, Aaron; Shalizi, Cosma Rohilla & Newman, M. E. J.
%D 2007
%T Power-law distributions in empirical data
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%2 [0706.1062] Power-law distributions in empirical data
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%F clauset2007powerlaw
%K data, distribution, distributions, empirical, law, power, powerlaw
%X Power-law distributions occur in many situations of scientific interest and
have significant consequences for our understanding of natural and man-made
phenomena. Unfortunately, the detection and characterization of power laws is
complicated by the large fluctuations that occur in the tail of the
distribution -- the part of the distribution representing large but rare events
-- and by the difficulty of identifying the range over which power-law behavior
holds. Commonly used methods for analyzing power-law data, such as
least-squares fitting, can produce substantially inaccurate estimates of
parameters for power-law distributions, and even in cases where such methods
return accurate answers they are still unsatisfactory because they give no
indication of whether the data obey a power law at all. Here we present a
principled statistical framework for discerning and quantifying power-law
behavior in empirical data. Our approach combines maximum-likelihood fitting
methods with goodness-of-fit tests based on the Kolmogorov-Smirnov statistic
and likelihood ratios. We evaluate the effectiveness of the approach with tests
on synthetic data and give critical comparisons to previous approaches. We also
apply the proposed methods to twenty-four real-world data sets from a range of
different disciplines, each of which has been conjectured to follow a power-law
distribution. In some cases we find these conjectures to be consistent with the
data while in others the power law is ruled out.
%Z cite arxiv:0706.1062Comment: 43 pages, 11 figures, 7 tables, 4 appendices; code available at  http://www.santafe.edu/~aaronc/powerlaws/
%U http://arxiv.org/abs/0706.1062
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%^

%0 
%0 Journal Article
%A Goldstein, M. L.; Morris, S. A. & Yen, G. G.
%D 2004
%T Fitting to the power-law distribution
%E 
%B The European Physical Journal B - Condensed Matter and Complex Systems
%C 
%I 
%V 41
%6
%N 2
%P 255-258
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%F Goldstein04powerlawfitV1
%K distribution, fitting, law, power, powerlaw
%X Version 1 of Goldstein 04 power law fit containing also the chi 2 test
%Z 
%U http://arxiv.org/abs/cond-mat/0402322v1
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%0 
%0 Journal Article
%A Mitzenmacher, M.
%D 2004
%T A Brief History of Generative Models for Power Law and Lognormal Distributions

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%B Internet Mathematics
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%V 1
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%N 2
%P 226--251
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%F mitzenmacher2004history
%K generative, law, model, power, powerlaw
%X Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a lognormal distribution. In trying to learn enough about these distributions to settle the question, I found a rich and long history, spanning many fields. Indeed, several recently proposed models from the computer science community have antecedents in work from decades ago. Here, I briefly survey some of this history, focusing on underlying generative models that
lead to these distributions. One finding is that lognormal and power law distributions connect quite naturally, and hence, it is not surprising that lognormal distributions have arisen as a possible alternative to power law distributions across many fields.

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%U http://www.eecs.harvard.edu/~michaelm/CS223/powerlaw.pdf
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%0 
%0 Generic
%A Newman, M. E. J.
%D 2004
%T Power laws, Pareto distributions and Zipf's law
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%F newman2004power
%K law, power, powerlaw
%X When the probability of measuring a particular value of some quantity varies
inversely as a power of that value, the quantity is said to follow a power law,
also known variously as Zipf's law or the Pareto distribution. Power laws
appear widely in physics, biology, earth and planetary sciences, economics and
finance, computer science, demography and the social sciences. For instance,
the distributions of the sizes of cities, earthquakes, solar flares, moon
craters, wars and people's personal fortunes all appear to follow power laws.
The origin of power-law behaviour has been a topic of debate in the scientific
community for more than a century. Here we review some of the empirical
evidence for the existence of power-law forms and the theories proposed to
explain them.

%Z 
%U http://arxiv.org/abs/cond-mat/0412004
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%0 
%0 Journal Article
%A Adamic, L. A. & Huberman, B. A.
%D 2002
%T Zipf's Law and the Internet
%E 
%B Glottometrics
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%I 
%V 3
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%P 143-150
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%2 Zipf, Power-law, Pareto
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%F adamic02zipf
%K Zipf, flat, law, long, pareto, power, powerlaw, tail
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%Z 
%U 
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%0 
%0 Generic
%A Adamic, Lada
%D 2002
%T Zipf, Power-laws, and Pareto -- a ranking tutorial 
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%B 
%C
%I http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html
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%F adamic02tutorial
%K Pareto, Power, Zipf, law, long, powerlaw, tail
%X 
%Z 
%U http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html
%+
%^

%0 
%0 Journal Article
%A Pennock, David; Flake, Gary; Lawrence, Steve; Glover, Eric & Giles, C. Lee
%D 2002
%T Winners don't take all: Characterizing the
                  competition for links on the web
%E 
%B Proc.\ National Academy of Sciences
%C 
%I 
%V 99
%6
%N 8
%P 5207--5211
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%F pennock2002winners
%K community, degree, distribution, model, powerlaw, smallworld
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%0 
%0 Journal Article
%A Vuong, Quang H.
%D 1989
%T Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses
%E 
%B Econometrica
%C 
%I The Econometric Society
%V 57
%6
%N 2
%P pp. 307-333
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%2 Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses on JSTOR
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%F vuong1989likelihood
%K comparision, hypothesis, likelihood, powerLaw, testing
%X In this paper, we develop a classical approach to model selection. Using the Kullback-Leibler Information Criterion to measure the closeness of a model to the truth, we propose simple likelihood-ratio based statistics for testing the null hypothesis that the competing models are equally close to the true data generating process against the alternative hypothesis that one model is closer. The tests are directional and are derived successively for the cases where the competing models are non-nested, overlapping, or nested and whether both, one, or neither is misspecified. As a prerequisite, we fully characterize the asymptotic distribution of the likelihood ratio statistic under the most general conditions. We show that it is a weighted sum of chi-square distribution or a normal distribution depending on whether the distributions in the competing models closest to the truth are observationally identical. We also propose a test of this latter condition.
%Z 
%U http://www.jstor.org/stable/1912557
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