🔍Deep Dive: The Power Law - All Noise And No Signal?

In the previous edition, we provided a primer on Power Laws; what they are, where they’re found, and how the math can be applied towards Bitcoin.

Building upon this foundation, we stress test the model further and compare it the infamous Stock to Flow model, to help us decide if the Power Law is a model we should be paying attention to.

Lets dive in!

Table of Contents

1. Unique Power Laws in Bitcoin

   1. Power Law in Time

   2. Power Law in Bitcoin Addresses

   3. Metcalfe’s Law

2. Power Law vs. Stock to Flow

   1. The Original Stock to Flow Model

   2. Evolution of Stock to Flow

3. Key Findings

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Unique Power Laws in Bitcoin

Given’s Bitcoin unique status as both a monetary phenomenon and a technology layer, several power law relationships have recently come to light, including:

1. The number of total addresses, which follows a power law distribution in time; and

2. The Bitcoin spot price, which follows a power law distribution in Bitcoin addresses and time.

Power Law in Time

The below analysis covers the relationship between Bitcoin spot price and time. If we take the log base 10 of Bitcoin’s price and compare it to the days since Genesis, we can run a regression analysis.

The resulting equation based on the above table is:

Log(Price) = -16.58 + 5.71 Log(Days)

which can be further simplified to:

Price = 10^-16.58 Days^5.71

The R² is 0.95, meaning that this simple relationship predicted 95% of the variation in the log of price. This is a remarkably high fit and certainly not something that occurs by chance.

Below is what the model’s predictions look like on a chart.

We can then apply the graph on a normal (i.e. non-logged) scale and extrapolate the curve into the future to get the chart below.

The above model implies the following average future prices for Bitcoin:

• $100,000 in 2025

• $500,000 in 2030

• $1,000,000 in 2033

Note: These are the average prices i.e. speculative asset bubbles will cause prices to fluctuate widely around these average values to both the upside and downside.

Predictions from a regression model such as the above are more useful as when considered as ranges of values with implied probabilities, rather than specific numerical figures.

In the following graph we have indicated the range of predictions (shaded blue area) as well as the price implications (black line). Using this chart, the model implies a $1M price prediction on:

• February 17, 2028 at the earliest; or

• August 14, 2033 on average; or

• September 6, 2040 at the latest.
  

Power Law in Bitcoin Addresses

Interestingly, the number of unique addresses (holding any amount of Bitcoin) also correlates strongly with Bitcoin’s spot price. If we run a similar log-log regression that we ran above, we obtain the following results:

Based on the above, the R2 is 0.9492 and the coefficient for the log of bitcoin addresses is 1.78. As such, the implied equation is:

Price = 10^-9.25 Addresses^1.78

This means that the Bitcoin price is proportional to the number of bitcoin addresses to the power of 1.78.

The chart below shows the regression fit with the 95% confidence intervals shaded in light blue. Points are colored according to the deviation from the black prediction line.

Metcalfe’s Law

Metcalfe's law states that the value of a telecommunications network is proportional to the square of the number of connected users. The law is named after Robert Metcalfe and was first proposed in 1980.

What’s particularly interesting though is that the co-efficient of 1.78 in our Power Law model above is strikingly close to what Metcalfe’s law would suggest, which is 2.

Metcalfe's law is a representation of these network effects. In simple terms, the more users a network has, the more valuable it becomes. For example, in a telephone communication network with n nodes the number of connections that can be made is n(n-2) which approaches n² at large numbers. Two telephones can make only one connection, five can make 10 connections, and twelve can make 66 connections.

Similarly, the value of Bitcoin increases as its users increase. More people adopting Bitcoin means there are more ways to spend it, and it is more salable. Therefore, it becomes even more valuable. More value brings in new users, again making the network more valuable. And, so on.

Using the estimates we obtained above, we can say that every 50% increase in the addresses doubles the value of Bitcoin. Or, every 1% in addresses increases the value by about 2% (1.78% to be exact).

Power Law vs. Stock to Flow

The Original Stock to Flow Model

In 2019, pseudonymous Plan B introduced a Bitcoin price model inspired by the concept of stock-to-flow (S2F) that was first popularized within Bitcoin circles by Saifedean Ammous.

The model implies that price is a function of S2F based on the following equation:

Price = 0.4 * S2F³.

Below, we replicate this model.

This model was introduced when the price was below $10,000 and predicted an average price of $55K for Bitcoin’s fourth cycle (spanning 2020-2024). The actual price in the fourth cycle averaged at $32K. So, the model had a ~42% error.

The S2F model is also a power law, but only due to the fact that it incorporates the S2F ratio. S2F itself is exponential in time (it roughly doubles every cycle), essentially combining a power law together with an exponential law.

This combination allows it to benefit from the power law’s good fit and yet makes its predictions exceptionally excessive to the upside over the long run. For instance, if we were to take the S2F at face value, the model would predict a Bitcoin price of $500K for the next cycle and $3.6M for the following cycle.

Evolution of Stock to Flow

Plan B also introduced multiple versions of the S2F model, including one “cross-asset model” that predicted $288K for the previous cycle, and yet another that predicted $100K.

As the price appreciated in the previous four year Bitcoin cycle, he introduced the more aggressive price models, raising his target from $55K to $100K and then to as high as $288K.

But then, as those models proved overly bullish, he pivoted back to the more conversative price predictions from his original model.

https://x.com/100trillionUSD/status/1778724142784598180

It therefore appears that the S2F model’s parameters change substantially over time to fit the prevailing price narrative. This is one of the major pitfalls in forecasting and is easily explained by one of the most prevalent biases in human misjudgment, namely that of ego.

Over the course of roughly 12 months (Dec 2020 - Dec 2021), Plan B had grown to one of the most popular accounts on Twitter, at one point gaining more than 100,000 followers in a single month. One would understand why he likely suffered from “Excessive Self-Regard Tendency” as Charlie Munger so eloquently describes it.

As his original model proves inaccurate, the analyst (in this case Plan B) is tempted to tweak the model gradually and force it to fit again. However, if a model is not inherently correct, it will keep producing wrong predictions even if each tweak temporarily fixes it.

One way to test this is to perform a cross-validation test. This involves modeling a subset of the data, not all of it, and testing the results on new data.

If we had used data up to June 2016 to train the S2F model, we would have gotten the following predictions for the last cycle.

The very large deviation shows that the model is not robust and does not have strong long-term predictive power. The R^2 of the predictions drops to a mere 0.13, substantially lower than the originally reported 0.94.

By contrast, if we train the Power Law model on the same partial dataset up to mid-2016, it will continue to predict accurately even eight years into the future!

Below is what it would have predicted for the last cycle. The parameters remain practically unchanged even when we use pre-June 2016 data. This indicates that the power law model does have strong predictive power.

Key Findings

In this article, we covered the math behind two of the most interesting Bitcoin power laws, those that are based on both time and the number of bitcoin addresses.

We also discussed Metcalfe’s law and empirically showed that it holds true when compared to Bitcoin.

Finally compared the Power Law model to the Stock-to-Flow model and stress-tested both, with the results indicating that the Power Law model has strong predictive power, whereas the Stock-to-Flow model does not.