🔍Deep Dive: What is the Power Law?

Heard of the “Power Law” but never really understood it? We got you covered!

In this latest edition of Bitcoin News Research we define the Power Law and its relationship across all fields from the natural world, to economic systems, and most importantly - Bitcoin.

Section 1: What is a Power Law?

Power laws are mathematical expressions of the form Y = aXⁿ that describe the relationship between two variables, namely Y and X.

At first glance, they appear to be just another mathematical relationship. In reality, however, they exhibit intriguing properties and are remarkably ubiquitous across nature and a wide variety of man-made phenomena.

Research on the origins of power-law relations and efforts to observe and validate them in the real world is an active topic in many fields of science including physics, neuroscience and even linguistics!

The Basics

Let begin with the basic power law relationship which states:

Y = aXⁿ

There are two variables:

• Y is the predicted variable;

• X is the predictor.

And two parameters:

• a is a multiplier;

• n is the power.

Below are examples of five power law equations, each of which follow the same structure with slightly different parameters:

These curves that represent these equations are shown below. A is just a line, and you can see that it pales in comparison to any higher-powered relationship. B has the power of 2 and quickly separates itself from the straight line. C is the same thing, multiplied by 2, and thus progressing twice as fast as B, but nothing extraordinary.

Things get more interesting when we introduce the power of 3 in curve D. You can see that it eclipses everything else. The power is the most important component. The multiplier has some effect, but the power dominates it eventually.

Furthermore, the power does not need to be more than 1. It can be less than one or even negative, too. In curve E, we have the power of 0.5, which is equivalent to a square root. With a multiplier of 1, it would have fallen below the straight line and not be visible, so we have added a large multiplier of 100 to make it visible.

Based on the above, its clearly evident that “Power Law’ curves can take a variety of different shapes based on their corresponding equations.

What’s most interesting, is that the log of X with an exponent n is n times the log of X. So we can rewrite the above equations by taking the natural log of both sides. For simplicity, let Y’ = Ln(Y) and X’ = Ln(X). Written this way all curves become straight lines.

Is That Exponential?

Importantly, in a power law relationship, X is the based and the power (n) remains constant. By comparison, in an exponential relationship, the based remains constant and X is the power which varies. Exponentials grow so fast that they eclipse everything and exhaust all the energy/space/resources very rapidly. For that reason, they are typically temporary.

Section 2: Power Laws in Nature

Power laws are common in our natural world, and scholars have discovered them in many branches of science.

Scientific interest in power law relations stems from their emergence from even the simplest of universal mechanisms. Finding a power law in data can point towards certain shared mechanisms in the behavior of totally different phenomena and can indicate a deep underlying connection.

A wide variety of physical, biological, and human-made phenomena tend to follow a power law distribution across a broad range of magnitudes. These phenomena include:

• the sizes of craters on the moon and solar flares (see image below);

• the frequencies of words in most languages and family names;

• human judgments of stimulus intensity

3. Power Laws in Economic and Man-Made Systems

We also come across multiple power laws in the man made world. The infamous “Pareto's Law” of income distribution is in fact a power law. Ever heard of Cathy Wood mention “Metcalf’s Law”, yup that’s a power law as well!

The population of cities also follows a power law distribution with a power of -1. This is known as Zipf’s law which states that the frequency of any city is inversely proportional to its rank in the frequency table (i.e. a power of -1).

Lets assume we take a country (e.g., the United States) and order the cities by population (e.g., New York first, Los Angeles second, etc), we see that the log of rank and log of population surprisingly fall on a descending straight line.

This implies that the city of rank n has a size proportional to 1/n; for example, the population size of a city ranked 2nd is half that of the city ranked 1st.

The below figure maps this relationship perfectly, showing the power law distribution of 135 US metropolitan areas in 1991.

4. Power Laws in Bitcoin

Now onto the most important discussion - how does this all relate to Bitcoin?

Intriguingly, several power law relationships have also been found to exist in Bitcoin (shoutout to @Giovann35084111). For example, the commonly cited number of total wallet addresses follows a power law in time. Similarly, the price of Bitcoin also follows a power law when measured in addresses, and in time. Even hash rate follows a power law.

Bitcoin itself is altogether an economic, financial, and social phenomenon. The existence of power laws in its behavior indicates similarities with other natural and economic systems in the underlying mechanisms.

For now, let’s focus on the relationship between price and time. If we take the log base 10 of Bitcoin price and days since Genesis, we can run a regression analysis. The table below shows the results. The resulting equation is:

Log(Price) = -15.04 + 5.27 Log(Time)

which is equivalent to:

Price = 10^-15.04 Time^5.27

The R² is 0.95, implying this relationship predicted 95% of the variation in the log of price. This is indicates a remarkably high fit and is certainly not something that can be ascribed to pure chance.

Below is what the model’s predictions look like:

And here is the same graph on the normal (non-logged) scale. We can extrapolate the prediction curve into the future as well. For example, this model predicts that the average price will hit $100K in March 2026. Note this it just the average value. The actual cycle peaks will be considerably higher due to the bubbles.

5. Summary

In this article, we offered a primer on power laws, which will serve as a foundation for next weeks articles on the same topic where we do a further deep dive.

We also discussed the unique properties and peculiar uniquity of power laws in both nature and man-made systems and then focused on a specific example of a power law relationship in Bitcoin.

Next week, we aim to stress test the power law model and compare it to the controversial stock-to-flow model, introducing a new metric to assess the over/under-valued price ranges of Bitcoin. Stay tuned to find out more!