Unlocking the Mystery of the Power Law: What It Means for Bitcoin

Ever heard of the “Power Law” but found it tricky to grasp? Let’s break it down. In this post, we’ll explore what the Power Law is, why it shows up in everything from nature to economics, and how it applies to Bitcoin.

What is a Power Law?

A Power Law is a mathematical formula that looks like this:

It describes how two variables, X and Y, relate to each other. While it might seem like just another equation, the Power Law is actually a key to understanding patterns in everything from natural phenomena to financial markets.

Power Laws appear across various fields, from physics to linguistics. They’re not just abstract concepts; they help us make sense of real-world patterns.

The Basics:

Here’s the basic Power Law equation:

– Y is the predicted variable.

– X is the predictor.

– a is a multiplier.

– n is the exponent or “power.”

Depending on the values of these parameters, the equation can create different curves:

A simple straight line.

A quadratic relationship.

A quadratic relationship with a multiplier.

A cubic relationship.

A square root relationship.

Each of these equations produces different curves. The key factor here is the power (n)—it shapes the curve. As n increases, the curve becomes steeper, leading to rapid growth.

When you take the logarithm of these equations, the curves straighten out, making it easier to see the underlying relationships.

Exponential vs. Power Law:

It’s crucial to distinguish between a Power Law and an exponential function. In a Power Law, the exponent (n) is fixed while X varies. In an exponential function, the base is constant, and X is the exponent. Exponential functions grow rapidly and usually describe short-term phenomena, while Power Laws explain long-term, stable relationships.

Power Laws in Nature

Power Laws are common in nature. Scientists have found them across many fields, showing up in everything from the sizes of craters on the moon to the frequency of words in languages. The existence of Power Laws suggests there are universal mechanisms at work, connecting seemingly unrelated phenomena.

Power Laws in Economic and Man-Made Systems

Power Laws also show up in human-made systems. Take Pareto’s Law of income distribution—it’s a Power Law. Or Metcalfe’s Law, which explains the value of networks like the internet. Another example is Zipf’s Law, which describes city population distributions. Rank cities by population, and the second-largest city will have about half the population of the largest. This relationship forms a straight line when plotted on a log-log scale—classic Power Law behavior.

Power Laws in Bitcoin

Now, let’s talk Bitcoin. Power Laws also apply here. For instance, the number of Bitcoin wallet addresses follows a Power Law over time. The price of Bitcoin shows a Power Law relationship when plotted against time or the number of addresses.

One key example is the relationship between Bitcoin’s price and time since its inception. By taking the logarithm of both the price and the days since Bitcoin’s launch (Genesis), we can model this relationship accurately. Here’s the equation:

This translates to:

This model explains 95% of the variation in Bitcoin’s price, suggesting a strong and predictable relationship. According to this model, Bitcoin’s average price could reach $100K by March 2026, though actual peaks might be higher due to market cycles.

Wrapping Up

This primer on Power Laws sets the stage for next week’s deep dive into Bitcoin. We’ll compare the Power Law to other models like stock-to-flow and introduce a new metric for evaluating Bitcoin’s price. Stay tuned for more insights.