PowerLaws

Power laws describe relationships in which a quantity varies as a power of another. In its simplest form, a variable x follows a power law if its probability density function or frequency obeys P(x) ∝ x^-α for x ≥ x_min, with α > 0. This form implies scale invariance: multiplying x by a constant rescales P(x) by a predictable factor, leaving the functional form essentially unchanged. Power-law distributions can be continuous (Pareto distribution) or discrete (Zipf's law).

The exponent α governs tail heaviness: smaller α yields a fatter tail. For a continuous density, the mean is finite only if α > 2, and the variance is finite only if α > 3. The lower cutoff x_min is often required because real data do not extend to zero and to stabilize estimates. Power laws are typically identified in broad ranges of x, sometimes spanning several orders of magnitude.

Common examples include wealth distributions (Pareto), word frequencies (Zipf's law), city sizes, earthquake energies, and the degree distributions of many networks, which often show power-law tails over several orders of magnitude. These forms have been observed in diverse fields, and different mechanisms can produce similar tail behavior.

Identification and estimation involve fitting with maximum likelihood methods that incorporate a lower bound x_min, followed by goodness-of-fit tests and comparisons with alternative heavy-tailed models. Log-log plots can be misleading with small samples or improper cutoffs, so rigorous statistical testing is recommended.

Origins and mechanisms include multiplicative growth processes, preferential attachment in networks, self-organized criticality, and scaling symmetries in complex systems. Not all heavy-tailed data are power laws; distinguishing them from lognormal or stretched exponential tails requires careful analysis.