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The Power Law

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updated 08 Feb 2023

The Power Law is a mathematical relationship between two variables, where one variable is proportional to the power of another. It is expressed mathematically as:

y = cx^k

where y is the dependent variable, x is the independent variable, c is a constant, and k is the power law exponent. The power law exponent determines the shape of the relationship between the two variables and can range from 0 to infinity.

The Power Law is often used to describe the distribution of certain phenomena, such as wealth, population, and the frequency of events. For example, the distribution of wealth in a society often follows a power law, where a small number of individuals have a large proportion of the total wealth, and the majority of individuals have very little wealth.

The Power Law is also used in physics to describe phenomena such as the distribution of earthquakes, the size of cities, and the frequency of word usage in language. In these cases, the power law exponent provides information about the underlying mechanisms that drive the behavior of the system.

It's important to note that not all phenomena follow the Power Law, and other models, such as the normal distribution or exponential distribution, may be more appropriate in certain cases. Nevertheless, the Power Law is a useful tool for understanding the distribution of certain types of data and for making predictions about complex systems.

Resources

Geoffrey West - Why Do Power Laws Work So Widely?