Power-law distributions in empirical data
Clauset, A.; Shalizi, C. R. & Newman, M. E. J.
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.
Power-Law Distributions in Empirical Data
Clauset, A.; Shalizi, C. R. & Newman, M. E. J.
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.
Power laws, Pareto distributions and Zipf's law
Newman, M. E. J.
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.
A Brief History of Generative Models for Power Law and Lognormal Distributions
Mitzenmacher, M.
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.