Power Law Detection Methods
Practical methods for determining whether distributions truly follow power laws, distinguishing them from alternatives like log-normal or exponential distributions.
Practical methods for determining whether distributions truly follow power laws, distinguishing them from alternatives like log-normal or exponential distributions.
When to Use Power Law Detection Methods
When rigorous statistical validation is needed beyond intuitive inequality assessment to confirm power law distributions.
How to Apply
Visual Log-Log Analysis
Create rank-size plots on logarithmic axes - power laws form approximately straight lines with slope equal to negative of exponent α.
Questions to Ask
- Does log-log plot show linearity?
- Over how many orders of magnitude?
- Is there curvature suggesting alternative distributions?
Outputs
- Log-log plot
- Visual linearity assessment
- Range of power law behavior
Statistical Testing
Apply maximum likelihood estimation (MLE) for power law exponent, Kolmogorov-Smirnov goodness-of-fit test, and likelihood ratio tests comparing to alternatives.
Questions to Ask
- What is the MLE estimate of α?
- What is K-S distance for fitted distribution?
- Does power law fit better than log-normal/exponential?
Outputs
- Exponent estimate
- Goodness-of-fit p-value
- Likelihood ratio test results
Avoid Common Mistakes
Guard against confusing power laws with exponential or log-normal distributions, fitting to insufficient data, ignoring truncation, and assuming stationarity.
Questions to Ask
- Is sample size sufficient (hundreds to thousands)?
- Are bounds properly accounted for?
- Has distribution shifted over time?
Outputs
- Mistake audit
- Confidence assessment
Implement Monitoring Dashboard
Track concentration metrics over time: top 10%/20% share, Gini coefficient, Pareto percentage, distribution stability, tail thickness.
Questions to Ask
- What are current concentration ratios?
- Is concentration increasing or decreasing?
- What is churn in top 10%?
Outputs
- Monitoring dashboard
- Alert thresholds
- Trend analysis