Power Law Distribution

A power-law distribution is a relationship of the form f(x) ~ x^(-α), where the frequency of events decreases with their size raised to some exponent α. Unlike the Gaussian bell curve, which clusters events around a mean and suppresses extremes exponentially, power laws have no characteristic scale — there is no 'typical' event size, and the distribution's tail extends indefinitely. This produces a counterintuitive property: rare, enormous events contribute disproportionately to the distribution's variance and expected value. Power laws govern earthquakes, city sizes, income distributions, internet traffic, species extinctions, and — as Per Bak demonstrated — any system at self-organized criticality. The distribution's mathematical structure makes specific prediction impossible while making the class of events statistically inevitable.