Power law

In mathematics, are power laws (English power laws ) laws, which have the form of a monomial.

They belong to the scaling laws and describe the scale invariance of many natural phenomena. They occur for example in the context of word frequencies on ( Zipfsches Act) or human perception ( Stevens's power function ). Pareto distributions are also power laws.

Mathematical details

Power laws describe polynomial dependencies between two variables and the form

It is the pre-factor, and the exponent of the power law, and by the indicated addition of terms are assumed to be negligible, and omitted.

The value of is usually less relevant - it is more interested in the exponent of the power law, as this determines whether increases decreases or increases and at what speed. In particular, the pre-factor can be integrated into the exponent. is converted to to.

Examples

Whether a given distribution can be approximated by a power function, is shown in a double - logarithmic plot. Is the graph of the function is a straight line, an approximation is made ​​possible by a power function. The slope of the line is then its exponent. A detailed derivation and example can be found in the article Pareto distribution.

Exponential growth of cities

A power law of the size distribution results in exponential growth, when both the number and the extent of the objects to be measured is growing exponentially. The size distribution of objects at any time obeys a power law:

For example, let the number of cities at the time of an exponentially increasing size:

The expansion of a city founded at the time at the time was also growing exponentially:

Therefore applies to the expansion of cities, the probability statement

By taking logarithms and transforming these scenarios:

The likelihood at the time that a random city is established before a selected time is,

Used this formula for the calculation of the distribution function (set ), the result is the distribution function

The corresponding probability density for the expansion ( derivative of the distribution function; " size distribution " ) is thus of the form of this:

That is, with.