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A power law is a mathematical model that portrays a dynamic relationship between two objects. The power law portrays a wide variety of natural and man-made phenomena including frequencies of words in most languages, sizes of craters on the moon, the sizes of power outages, earthquakes, wars, body mass versus metabolic rate, lifespan versus heart rate, river branches and a huge number of other relationships. The power law is an important way to model form, function, and context in patterns in nature. Because one can find power law phenomena in most patterns in nature, it is theorized to be a unifying principle that underlies patterns in nature at all levels.

A power law models the nature of the relationship of one characteristic (y) with respect to another (x) by using the mathematical expression:

“y” is the result of the computation and is typically represented on the vertical axis of a graph. The symbol “a” is a constant value that defines the size, scale, or magnification of a process. The symbol “x” is generally represented on the horizontal axis of the graph. The symbol “k” is the constant characteristic number that defines the nature of the relationship between "x" and "y". That relationship is said to change by following a non-linear power law described by the constant (k).

A key feature of power laws is that, upon resizing (rescaling) of the independent variable (x) (such as body mass), the power law’s exponent that defines the dependent variable remains the same. We say that the power law relationship is the same at all scales. An example is the trend of the average metabolic rate of different species as the average species mass increases. The trend is a non-linear power law with a constant exponent of -0.75 that defines the trend.

A power law plot showing metabolic rate versus body mass to the power of -0.75 is shown below. Throughout the animal kingdom, this power law relationship (k) between body mass and metabolic rate has been shown to be the same for mice and for whales as well as the spectrum of organisms between these size extremes. The exponent remains at -0.75.

Note that the relationship is not linear. One characteristic changes non-linearly in response to a change in another characteristic. In this example, metabolic rate decreases (with increase in body mass) at a much faster rate than does a linear model.

The power law model also portrays the frequency of occurrence of patterns. An example is the relationship of the number of cities to city population. There are a lot fewer large cities than there are small cities. Likewise for the Internet, airline routes, or power line grids. There are fewer large hubs than there are smaller hubs.

Data distributed as a power law model is sometimes called the “80/20 rule” which says that 80% of effects come from 20% of the available causes. A familiar example of the “80/20” rule is that 80% of the wealth is held by 20% of the people.

In complex systems such as patterns in nature, power laws are often thought to be signatures of hierarchy and robustness. One mathematical process that serves to detect a power law relationship in data that has been collected is to replot the data on a log-log scaled graph. If a power law relationship exists, the resulting log-log graph will be a straight line. The slope of the line is the value of the power law exponent.

One common use for power laws is to study the scaling phenomena in patterns in nature. Despite the complexity and the difference between organisms, a large part of their pattern dynamics is described by a constant power law exponent. The exponent for such dynamic phenomena as metabolism versus body mass or the scaling of social phenomena remains constant even though there are complex dynamics going on within or between organisms. Geoffrey West and his colleagues note that the power law exponent that defines the relationships between various natural processes is a unifying constant characteristic amongst many diverse patterns in nature. A large number of these exponents are simple multiples of ¼. This unifying idea is called “Quarter Power Scaling”.

Another common use for power law analysis is fractal geometry. Fractal networks, like the Internet and many other social and natural phenomena, are said to be “scale invariant” or “scale free” because their power law exponent stays constant as things grow or shrink.

One can analyze a map of a coastline by overlaying it with a grid of squares of a certain size (say 1 cm). Count the number of grid cells that contain a piece of the coastline. Then lay down a second grid of squares of another size ( 0.5 cm) and count again. Do this many times then plot the grid size versus the number of occupied cells. The resulting plot will be a power law curve. The slope of the log/log plot will be the power law's exponent. In this case it is called the "fractal dimension".

As a power law, fractal dimension is a handy number for analyzing patterns. A very high fractal dimension implies a degree of complexity leading to randomness. A lower fractal dimension implies more order. The fractal dimension of a shark’s track in the water can suggest certain behaviors. A high fractal dimension of his track suggests he is in a searching mode and may not see anything. A lower fractal dimension implies a more ordered track where a sighted prey is being hunted.