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.

@article{clauset2009powerlaw, author = {Clauset, Aaron and Shalizi, Cosma Rohilla and Newman, M. E. J.}, title = {Power-Law Distributions in Empirical Data}, journal = {SIAM Review}, publisher = {SIAM}, year = {2009}, volume = {51}, number = {4}, pages = {661--703}, url = {http://link.aip.org/link/?SIR/51/661/1}, doi = {10.1137/070710111}, issn = {0036-1445}, keywords = {law, power, powerlaw}, abstract = {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.} }

%0 = article %A = Clauset, Aaron and Shalizi, Cosma Rohilla and Newman, M. E. J. %D = 2009 %I = SIAM %T = Power-Law Distributions in Empirical Data %U = http://link.aip.org/link/?SIR/51/661/1

Power-law distributions occur in many situations of scientific interest and
ve significant consequences for our understanding of natural and man-made
enomena. Unfortunately, the detection and characterization of power laws is
mplicated by the large fluctuations that occur in the tail of the
stribution - the part of the distribution representing large but rare events
and by the difficulty of identifying the range over which power-law behavior
lds. Commonly used methods for analyzing power-law data, such as
ast-squares fitting, can produce substantially inaccurate estimates of
rameters for power-law distributions, and even in cases where such methods
turn accurate answers they are still unsatisfactory because they give no
dication of whether the data obey a power law at all. Here we present a
incipled statistical framework for discerning and quantifying power-law
havior in empirical data. Our approach combines maximum-likelihood fitting
thods with goodness-of-fit tests based on the Kolmogorov-Smirnov statistic
d likelihood ratios. We evaluate the effectiveness of the approach with tests
synthetic data and give critical comparisons to previous approaches. We also
ply the proposed methods to twenty-four real-world data sets from a range of
fferent disciplines, each of which has been conjectured to follow a power-law
stribution. In some cases we find these conjectures to be consistent with the
ta while in others the power law is ruled out.

@misc{clauset2007powerlaw, author = {Clauset, Aaron and Shalizi, Cosma Rohilla and Newman, M. E. J.}, title = {Power-law distributions in empirical data}, year = {2007}, note = {cite arxiv:0706.1062Comment: 43 pages, 11 figures, 7 tables, 4 appendices; code available at http://www.santafe.edu/~aaronc/powerlaws/}, url = {http://arxiv.org/abs/0706.1062}, doi = {10.1137/070710111}, keywords = {data, distribution, distributions, empirical, law, power, powerlaw}, abstract = {Power-law distributions occur in many situations of scientific interest andhave significant consequences for our understanding of natural and man-madephenomena. Unfortunately, the detection and characterization of power laws iscomplicated by the large fluctuations that occur in the tail of thedistribution -- the part of the distribution representing large but rare events-- and by the difficulty of identifying the range over which power-law behaviorholds. Commonly used methods for analyzing power-law data, such asleast-squares fitting, can produce substantially inaccurate estimates ofparameters for power-law distributions, and even in cases where such methodsreturn accurate answers they are still unsatisfactory because they give noindication of whether the data obey a power law at all. Here we present aprincipled statistical framework for discerning and quantifying power-lawbehavior in empirical data. Our approach combines maximum-likelihood fittingmethods with goodness-of-fit tests based on the Kolmogorov-Smirnov statisticand likelihood ratios. We evaluate the effectiveness of the approach with testson synthetic data and give critical comparisons to previous approaches. We alsoapply the proposed methods to twenty-four real-world data sets from a range ofdifferent disciplines, each of which has been conjectured to follow a power-lawdistribution. In some cases we find these conjectures to be consistent with thedata while in others the power law is ruled out.} }

%0 = misc %A = Clauset, Aaron and Shalizi, Cosma Rohilla and Newman, M. E. J. %B = } %C = %D = 2007 %I = %T = Power-law distributions in empirical data} %U = http://arxiv.org/abs/0706.1062

@article{newman05power, author = {Newman, M. E. J.}, title = {Power laws, Pareto distributions and Zipf's law}, journal = {Contemporary Physics}, year = {2005}, volume = {46}, pages = {323}, url = {doi:10.1080/00107510500052444}, keywords = {distribution, free, law, long, power, scale, tail, zipf} }

%0 = article %A = Newman, M. E. J. %D = 2005 %T = Power laws, Pareto distributions and Zipf's law %U = doi:10.1080/00107510500052444

Version 1 of Goldstein 04 power law fit containing also the chi 2 test

@article{Goldstein04powerlawfitV1, author = {Goldstein, M. L. and Morris, S. A. and Yen, G. G.}, title = {Fitting to the power-law distribution}, journal = {The European Physical Journal B - Condensed Matter and Complex Systems}, year = {2004}, volume = {41}, number = {2}, pages = {255-258}, url = {http://arxiv.org/abs/cond-mat/0402322v1}, keywords = {distribution, fitting, law, power, powerlaw}, abstract = {Version 1 of Goldstein 04 power law fit containing also the chi 2 test} }

%0 = article %A = Goldstein, M. L. and Morris, S. A. and Yen, G. G. %D = 2004 %T = Fitting to the power-law distribution %U = http://arxiv.org/abs/cond-mat/0402322v1

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
ad 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.

@article{mitzenmacher2004history, author = {Mitzenmacher, M.}, title = {A Brief History of Generative Models for Power Law and Lognormal Distributions
}, journal = {Internet Mathematics}, year = {2004}, volume = {1}, number = {2}, pages = {226--251}, url = {http://www.eecs.harvard.edu/~michaelm/CS223/powerlaw.pdf}, keywords = {generative, law, model, power, powerlaw}, abstract = {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 thatlead 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.} }

%0 = article %A = Mitzenmacher, M. %D = 2004 %T = A Brief History of Generative Models for Power Law and Lognormal Distributions
%U = http://www.eecs.harvard.edu/~michaelm/CS223/powerlaw.pdf

When the probability of measuring a particular value of some quantity varies
versely as a power of that value, the quantity is said to follow a power law,
so known variously as Zipf's law or the Pareto distribution. Power laws
pear widely in physics, biology, earth and planetary sciences, economics and
nance, computer science, demography and the social sciences. For instance,
e distributions of the sizes of cities, earthquakes, solar flares, moon
aters, wars and people's personal fortunes all appear to follow power laws.
e origin of power-law behaviour has been a topic of debate in the scientific
mmunity for more than a century. Here we review some of the empirical
idence for the existence of power-law forms and the theories proposed to
plain them.

@misc{newman2004power, author = {Newman, M. E. J.}, title = {Power laws, Pareto distributions and Zipf's law}, year = {2004}, url = {http://arxiv.org/abs/cond-mat/0412004}, keywords = {law, power, powerlaw}, abstract = {When the probability of measuring a particular value of some quantity variesinversely 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 lawsappear widely in physics, biology, earth and planetary sciences, economics andfinance, computer science, demography and the social sciences. For instance,the distributions of the sizes of cities, earthquakes, solar flares, mooncraters, 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 scientificcommunity for more than a century. Here we review some of the empiricalevidence for the existence of power-law forms and the theories proposed toexplain them.} }

%0 = misc %A = Newman, M. E. J. %B = } %C = %D = 2004 %I = %T = Power laws, Pareto distributions and Zipf's law} %U = http://arxiv.org/abs/cond-mat/0412004

@inproceedings{stumme2004iceberg, author = {Stumme, Gerd}, title = {Iceberg Query Lattices for Datalog}, editor = {Wolff, Karl Erich and Pfeiffer, Heather D. and Delugach, Harry S.}, booktitle = {Conceptual Structures at Work: 12th International Conference on Conceptual Structures (ICCS 2004)}, series = {LNCS}, publisher = {Springer}, address = {Heidelberg}, year = {2004}, volume = {3127}, pages = {109-125}, url = {http://www.kde.cs.uni-kassel.de/stumme/papers/2004/stumme2004iceberg.pdf}, keywords = {2004, analysis, concept, context, datalog, families, family, fca, formal, iceberg, itegpub, l3s, lattices, myown, pcf, power, queries, query} }

%0 = inproceedings %A = Stumme, Gerd %B = Conceptual Structures at Work: 12th International Conference on Conceptual Structures (ICCS 2004) %C = Heidelberg %D = 2004 %I = Springer %T = Iceberg Query Lattices for Datalog %U = http://www.kde.cs.uni-kassel.de/stumme/papers/2004/stumme2004iceberg.pdf

@article{newman03structure, author = {Newman, M. E. J.}, title = {The structure and function of complex networks}, journal = {SIAM Review}, year = {2003}, volume = {45}, pages = {167}, url = {http://www.citebase.org/abstract?id=oai:arXiv.org:cond-mat/0303516}, keywords = {complex, free, law, long, networks, power, scale, tail} }

%0 = article %A = Newman, M. E. J. %D = 2003 %T = The structure and function of complex networks %U = http://www.citebase.org/abstract?id=oai:arXiv.org:cond-mat/0303516

@article{adamic02zipf, author = {Adamic, L. A. and Huberman, B. A.}, title = {Zipf's Law and the Internet}, journal = {Glottometrics}, year = {2002}, volume = {3}, pages = {143-150}, keywords = {Zipf, flat, law, long, pareto, power, powerlaw, tail} }

%0 = article %A = Adamic, L. A. and Huberman, B. A. %D = 2002 %T = Zipf's Law and the Internet

@misc{adamic02tutorial, author = {Adamic, Lada}, title = {Zipf, Power-laws, and Pareto -- a ranking tutorial }, year = {2002}, url = {http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html}, keywords = {Pareto, Power, Zipf, law, long, powerlaw, tail} }

%0 = misc %A = Adamic, Lada %B = } %C = %D = 2002 %I = %T = Zipf, Power-laws, and Pareto -- a ranking tutorial } %U = http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html

@article{barabasi99emergence, author = {Barabási, Albert-László and Albert, Réka}, title = {Emergence of scaling in random networks}, journal = {Science}, year = {1999}, volume = {286}, pages = {509--512}, keywords = {attachment, law, power, preferential} }

%0 = article %A = Barabási, Albert-László and Albert, Réka %D = 1999 %T = Emergence of scaling in random networks