Power-Law Distributions
P(x) ∝ x−α — the "heavy tail" where a few enormous events dominate. Zipf's law, Pareto principle, earthquake magnitudes, city sizes.
Linear scale — events
Log-log scale — straight line reveals power law
Exponential (for comparison)
Power laws arise from multiplicative processes, preferential attachment, and self-organized criticality. Unlike the Gaussian where extreme events are exponentially rare, in a power law the top 20% often hold 80% of the "wealth" (Pareto principle). The key diagnostic: a straight line on a log-log plot with slope −α.