Power-Law & Pareto Distributions

Power-law: P(x) ∝ x^{−α}. They appear everywhere in nature — city sizes (Zipf), wealth (Pareto), earthquake magnitudes (Gutenberg-Richter), internet links, word frequencies. Heavy tails mean extreme events are far more common than Gaussian intuition suggests.

Power-law (Pareto)
Log-normal
Exponential
Power-law mean
Variance (finite?)
95th pct / mean
99th pct / mean
Real-world α: Wealth ≈ 1.5-2.5 | Word freq (Zipf) ≈ 1 | City sizes ≈ 1-2 | Web links ≈ 2.1 | Earthquake energy ≈ 1.7 | Solar flares ≈ 1.5