# Power law

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A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event (e.g. its size), the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary as a power of the size of the population, and hence follows a power law. The distribution of a wide variety of natural and man-made phenomena follow a power law, including frequencies of words in most languages, frequencies of family names, sizes of craters on the moon and of solar flares, the sizes of power outages, earthquakes, and wars, the popularity of books and music, and many other quantities.

## Contents

### Technical definition

A power law is any polynomial relationship that exhibits the property of scale invariance. The most common power laws relate two variables and have the form

where a and k are constants, and o(xk) is an asymptotically small function of xk. Here, k is typically called the scaling exponent, where the word "scaling" denotes the fact that a power-law function satisfies $f(c x) \propto f(x)$ where c is a constant. Thus, a rescaling of the function's argument changes the constant of proportionality but preserves the shape of the function itself. This point becomes clearer if we take the logarithm of both sides: