**Aaron Clauset, Cosma Rohilla Shalizi, and M. E. J. Newman**

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 empirical detection
and characterization of power laws is made difficult by the large
fluctuations that occur in the tail of the distribution. In
particular, standard methods such as least-squares fitting are known
to produce systematically biased estimates of parameters for power-law
distributions and should not be used in most circumstances. Here we
describe statistical techniques for making accurate parameter
estimates for power-law data, based on maximum likelihood methods and
the Kolmogorov-Smirnov statistic. We also show how to tell whether the
data follow a power-law distribution at at all, defining quantitative
measures that indicate when the power law is a reasonable fit to the
data and when it is not. We demonstrate these methods by applying them
to twenty-four real-world data sets from a range of different
disciplines. Each of the data sets has been conjectured previously 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.

*Keywords:* Power-law distributions; Pareto; Zipf; maximum
likelihood; heavy-tailed distributions; likelihood ratio test; model selection

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