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.