[0706.1062] Power-law distributions in empirical data

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.