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Power Laws

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

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Power Laws

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

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

  3. 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

  4. 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!!!

  5. How Does it do? Discreet Actual Value: 2.5 Continuous

  6. Error as a function of Xmin and n For Discreet Data For Continous Data

  7. 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

  8. How does it perform?

  9. 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?

  10. 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

  11. How does real world data stack up?

  12. 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

  13. 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

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