A power law is a relationship in which a relative change in one quantity gives rise to a proportional relative change in the other quantity, independent of the initial size of those quantities.
An example is the area of a square region in terms of the length of its side. If we double the length we multiply the area by a factor of four. Similarly, if we double the length of a side of a cube, we multiply the volume of the cube by a factor of eight. Each of these is an example of a power law relationship. What is important is that the factors don't depend on how large the square or cube is to begin with.
These examples show that power laws differ by a quantity such as the dimension of the space. The dimensions of a square and a cube are 2 and 3 respectively, so when we multiply the length of a side by 2, we multiply the area and volume by 2^2 and 2^3 respectively. The higher the dimension of an object, the greater the multiple we would use. The dimension in a power law can be any number: positive, negative, or fractional. Fractional dimensions have given rise to the concept of fractals (though we could also think of a side of a cube as having a fractional dimension of 1/3 relative to the volume, as multiplying the volume by a factor of 8 increases the side by a factor of 8^(1/3), or 2).
A power law can be turned into a linear relationship if we plot the variables on logarithmic axes. Plotting two quantities against each other in this way is how we generally determine if they have a power law relationship.
Power laws are very important because they reveal an underlying regularity in the properties of systems. Often highly complex systems have properties where the changes between phenomena at different scales is independent of which particular scales we are looking at. The picture we take at one scale is therefore similar in some way to the picture we take at another scale. This self-similar property underlies power law relationships.
The properties of many natural and human systems follow power laws. A particular example of a power law is an inverse relationship, which has a dimension of -1. The frequency of earthquakes varies inversely with their intensity. The number of cities with a certain population varies inversely as a function of that population. The number of people having a given income, is also approximately inversely related to that income.
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