powerlaw

Power-law distributions describe quantities where large events are rare but not negligible. In a power-law distribution, the probability that a random variable X takes the value x (for x above a lower bound xmin) is proportional to x^-α, with exponent α > 1. The probability density is p(x) = (α-1) xmin^(α-1) x^-α for x ≥ xmin (continuous case); the discrete form uses a similar normalization over integers x ≥ xmin. The tail is heavy and scale-invariant: if x is rescaled by a constant, probabilities scale by a power of that constant. The cumulative distribution P(X ≥ x) follows ~ x^{1-α} for large x.

Common in empirical data are quantities such as city sizes, word frequencies (Zipf’s law), wealth distributions, solar flares, and network degree distributions (often described as scale-free). For finite moments, α must exceed certain values: the mean is finite if α > 2, the variance finite if α > 3.

Fitting and testing: estimate α and xmin from data, typically by maximum likelihood estimation with xmin chosen by minimizing the Kolmogorov–Smirnov distance between the data and the fitted model. For continuous data, α_hat = 1 + n [sum_{i: x_i≥xmin} ln(x_i / xmin)]^{-1}. Discrete data require adapted estimators. Goodness-of-fit can be evaluated with KS tests; model comparison with likelihood ratio tests against alternative heavy-tailed distributions (e.g., log-normal, exponential).

Limitations: finite sample size and measurement limits can bias results; many datasets that appear to follow a power law may be better described by other heavy-tailed distributions. Robust claims require rigorous testing and transparent data.

See also: Zipf’s law, Pareto distribution, Clauset–Shalizi–Newman method, scale invariance.