Power Laws Otherwise known as any semi- straight line on a log-log plot

Self Similar The distribution maintains its shape This is the only distribution with this property

Fitting a line Assumptions of linear Regression do not hold: noise is not Gaussian Many distributions approximate power laws, leading to high R 2 indepent of the quality of the fit Regressions will not be properly normalized

Maximum Likelihood Estimator for the continuous case α is greater than 1 – necessary for convergence There is some x min below which power law behavior does not occur – necessary for convergence Converges as n→∞ This will give the best power law, but does not test if a power law is a good distribution!!!

How Does it do? Actual Value: 2.5 Continuous Discreet

Error as a function of X min and n For Discreet DataFor Continous Data

Setting X min Too low: we include non power-law data Too high: we lose a lot of data Clauset suggests “the value x min that makes the probability distributions between the measured data and the best- fit power-law model as similar as possible above x min ” Use KS statistic

How does it perform?

But How Do We Know it’s a Power Law? Calculate KS Statistic between data and best fitting power law Find p-value – theoretically, there exists a function p=f(KS value) But, the best fit distribution is not the “true” distribution due to statistical fluctuations Do a numerical approach: create distributions and find their KS value Compare D value to best fit value for each data set We can now rule out a power law, but can we conclude that it is a power law?

Comparison of Models Which of two fits is least bad Compute likelihood (R) of two distributions, higher likelihood = better fit But, we need to know how large statistical fluctuations will be Using central limit theroem, R will be normally distributed – we can calculate p values from the standard deviation

How does real world data stack up?

Mechanisms Summation of exponentials Random walk – often first return The Yule process, whereby probabilities are related to the number that are already present Self-organized criticality – the burning forest

Conclusions It’s really hard to show something is a power law With high noise or few points, it’s hard to show something isn’t a power law