Power Law

"The greatest shortcoming of the human race is our inability to understand the exponential function."

—Albert Allen Bartlett—

We are surrounded by potential laws. A physical process responds to a potential law when the probability of an event occurring decreases as its magnitude increases.

There are many phenomena that follow these laws: the relationship between the number of cities and inhabitants (few cities with many inhabitants and many with few), the connectivity of nodes on the Internet, the frequency of words in the language, the wealth of people, the size of living beings, internet links, terrorism, etc. It is curious that all these systems respond to potential laws quite accurately since, a priori, we could assume that people's wealth or the connection of nodes to the Internet are quite random phenomena in distribution where we would expect to obtain a bell curve.

Therefore, in a potential law the events occur with variable frequency, where many events are small-scale and a few are large- scale. This relationship between  many and few is not arbitrary, but follows precisely a concrete potential law with an exponent (k) characteristic of each physical phenomenon; there is a definite mathematical relationship that indicates how many events of each type occur—it can be calculated. Power Law is one of the most useful concepts for making predictions and decisions in business and management.

The Power Law shows how two variables—one dependent, the other independent—covary. Mathematically, one varies as a function of the other by being raised to a certain power (exponent).The diagram above shows this type of relationship. Often these are depicted on log or log-log graphs, but I show the power curve as an asymptote on both axes of the graph to highlight the non-linearity of the

relationship between the two variables. Examples: earthquakes, storms, number of sales by individual sales reps, etc.

A concrete example will help. The great majority of earthquakes are of very low magnitude. High magnitude earthquakes are much rarer than low magnitude earthquakes. In fact, their magnitude varies in inverse exponential proportion to the total number of earthquakes. In practice, this means that there are literally thousands of earthquakes every day around the world, but magnitude 6, 7, and 8 ones are much rarer. The most powerful earthquakes of all over nine on the Richter scale are very rare. They can happen only a few times a century, or even less. This doesn’t mean that the magnitude of any particular earthquake can be predicted. It does, however, imply that given a sufficiently large sample, we will eventually see a frequency-magnitude distribution that resembles the graph above.

This type of relationship is ubiquitous in nature, and that includes our human and social natures. There was a whole book written on this topic— The Long Tail, by Chris Anderson—with emphasis on the right side of the graph. In his book, Anderson described how the internet has made many businesses or ideas viable, something that would have previously gone unnoticed. He called this the long tail because there are musicians, artists, artisans, crafts workers, professionals, etc. who can provide their productions and services to people around the world, even though they can’t compete with the more traditional providers who dominate markets by occupying the left side of the power curve. This makes for much more diversity and many more opportunities to become known and appreciated, and to develop a following because it lowers traditional entry barriers and long-term viability.

This type of relationship is also depicted in the diagram. Showing the relationship between the number of clients and the number purchases, interactions, or value of each category of client that characterizes the market and product distributions of most, if not all, companies.