A power law is a relationship between two quantities whereby a relative change i

Question

A power law is a relationship between two quantities whereby a relative change in one quantity causes a proportional relative change in the other. These relationships are ubiquitous in science and engineering applications, such as the inverse-square law of Newtonian gravity. Power laws are expressed mathematically as equalities of the form y=axk. We can determine whether data likely follows a power law by plotting a function through the data and examining the fit. For this problem, we will genereate synthetic data that roughly follows a power law, then demonstrate that the data obeys a power law. Generate data according to a power law at 20 evenly spaced points on the interval [1.1, 10]. Then, for each of your points, add random noise of up to 20% of that point's value. This simulates having noisy data. Next, plot the noisy data using a semilog plot. Finally, plot the same power law as before, without noise, at many more points (on the order of 1000) on the same semilog axes as the data. This simulates fitting the data, which we will learn how to do later in the semester. Because of the way we generated the data, the power law should be a good fit for the noisy data. In a real application, this would tell us that the process generating the data likely obeyed a power law.

Explanation / Answer

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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