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 R2 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 xmin 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 Xmin and n

For Discreet Data For Continous Data

Setting Xmin

• Too low: we include non power-law data• Too high: we lose a lot of data• Clauset suggests “the value xmin that

makes the probability distributions between the measured data and the best-fit power-law model as similar as possible above xmin”

• 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