power law

The ratio of a specific kind of clusters / pattern is the same at any scale:

$$\frac{f(kx)}{f(x)}=g(k)$$

(the distribution of cluster-size does not dedpend on scale $x$)

The only function, for which this equation holds is the power law:

$$f(x)=A{x}^{-\gamma }$$ $$g(k)=\frac{f(kx)}{f(x)}=\frac{\cancel{A}{k}^{-\gamma }\cancel{x^{-y}}}{\cancel{A}\cancel{x^{-\gamma }}}=k^{-\gamma }$$

In log-log plots, power laws are straight lines.

A defining feature: Rare / Extreme events aren’t vanishingly unlikely