Most things cluster around an average — heights, test scores. But earthquakes, city sizes, wealth, word frequencies, and net links follow a POWER LAW: no typical size at all, and the same shape whether you zoom in or out. On a log-log plot it’s a straight line, and that straightness IS scale-invariance — the fingerprint of the scale-free world KLÍMAX is about. Slide the exponent.
A power-law distribution p(x) ∝ x⁻ᴺ has no characteristic scale: rescaling x→ax multiplies p by a⁻ᴺ but keeps the SHAPE, so it plots as a straight line of slope −α on log-log axes (scale invariance). Such heavy-tailed laws describe earthquake magnitudes (Gutenberg–Richter), city and firm sizes (Zipf), wealth (Pareto), word frequencies, and network degrees (scale-free graphs) — where rare huge events dominate and the mean can be ill-defined. They emerge from critical points, preferential attachment, and self-organized criticality. A fail-loud self-check throws unless f(ax)/f(x) is independent of x (= a⁻ᴺ). ◆ real statistics of scale, node-verified.
True power laws are often only approximate over a finite range (log-normal or cutoff tails can mimic them); the exact scale-invariance f(ax)/f(x)=a⁻ᴺ is the mathematical signature shown here.