The Bitcoin Power Law Theory

Giovanni Santostasi
17 min readMar 20, 2024
Bitcoin is similar to natural phenomena and not a common asset. Credit to this beautiful art to: @BainterSAT

Bitcoin is more similar to a city and an organism than a financial asset.

I started to study Bitcoin 12 years ago.

Reddit was the place where I posted my explorations, instead of journals, mostly because I wanted to reach the broader Bitcoin community, instead of publishing a paper that would not be read by nonprofessional scientists.

I‘m very active on X these days, please come to join me in my Bitcoin explorations there:

https://twitter.com/Giovann35084111

My main discovery was that Bitcoin is ruled by power laws. Its regularities show that it behaves more like a physical system than an asset. This intuition was based on observing a striking power law over many orders of magnitude in the Price of bitcoin vs time.

The Bitcoin Power Law Theory.

My Power Law model has evolved now into a full theory of Bitcoin behavior that can explain in a scientific coherent and falsifiable way all the main on-chain parameters and describe the growth of Bitcoin adoptions: The Bitcoin Power Law Theory or PLT.

For a quick, intuitive, understanding of the Theory, please watch this video (also remember to like and subscribe to support our Bitcoin Educational and Research Team).

It is the hard work of my son Saverio. It is beautiful and very well done.

The Theory of Bitcoin video:

The graph below explains the PTL theory in a summarized way and shows the main data that supports the theory. Price, hash rate, and addresses (we used addresses above a cut-off to eliminate dust addresses) are all power laws of each other and of time.

They all interact and affect each other in a continuous feedback loop.

Power laws are mathematical expressions of the form y=A x^n and are ubiquitous in nature but also in social phenomena and many parameters related to how a city or a nation grows.

They are so common because it can be shown mathematically and physically that they emerge any time you have some kind of process where the output becomes the new input in an iterative process.

This is exactly what happens with Bitcoin where for example the hash rate now affects the hash rate later in an infinite loop. So it is amazing but also perfectly consistent with the nature of Bitcoin to have power laws governing its behavior.

The interaction is supported by the diagram below which is well-known in the community. I didn’t invent the diagram but I used it to illustrate how the theory works.

The theory basically is a mathematical expression, based on logic, physics, and mathematics of the feedback loop below.

  1. Initially, Bitcoin needed to be accepted and adopted by the first users in the Satoshi circle.
  2. The “value” (now the “Price” that can be checked 24/7 online) increased with the square of the users (the empirically measured value is more like 1.95 but we round all the following power numbers to their integer number for simplicity). This is a confirmation of the theoretical result called the Metcalfe law.
  3. The price increase brings in more resources in particular mining power and capabilities.
  4. The increase in price decreases the time to mine a block but because of the “Difficulty Adjustment,” the hash rate necessary to mine a block is changed iteratively. Because the mining is barely profitable the compensation mechanism needs to be proportional to the increase in price that is increased by P=users² and the reward itself so logically and dimensionally we have hash-rate=Price² (this is exactly what is observed where the empirical value of the power is close to 2 or Price=hash-rate^1/2).
  5. The increase in hash-rate brings more security to the system which attracts more users. Now some readers may say that most people do not buy Bitcoin because of “security” but indirectly they do because if it was not a secure system nobody would invest enormous value in it. So yes, the security of the system directly or indirectly brings in new users.
  6. The users grow with the power of 3 in time. This is also a new result of the theory. Most models of Bitcoin adoption invoke S-curve type of growth. S-curves are typical of the adoption of many technologies like TVs, refrigerators, cars, cell phones, and so on. Bitcoin doesn’t follow a S-curve that is initially exponential. It follows a power law of 3 in time. It turns out many phenomena have an underlying S-curve mechanism of adoption or spread (in the case of a virus for example) if they have a curbing mechanism they become power laws instead. In the case of Bitcoin both the “Difficulty Adjustment” and the risk involved in any type of investment is that curbing mechanism, this is why we observed empirically that the growth of adoption of Bitcoin is a power law of 3 in time. There is extensive literature showing examples of this type of curbing phenomenon in the spread of diseases when risk is involved for example AIDS (Bitcoin is not AIDS but these studies show that if the spread of disease involves some kind of decision-making like having sex with a partner it makes the disease spread like the power of 3 in time instead of an S-curve or other type of logistic curves).
  7. The cycle repeats indefinitely. The bubbles are an important and necessary component of this cycle and they are discussed separately in the Corollary below.
  8. This power law growth of adoption then (together with the previously explained power laws) explains why we observed the other power laws in time: Addresses=t³, Price=Addresses²=(t³)²=t⁶, Hash-Rate=Price²=(t⁶)²=t¹².

Here is a graph showing all the interacting powers laws and their proposed causative explanation.

See the 3 parameters in a single Bitcoin Phase Space in the video linked below.

Consequences and Predictions of the PLT.

This theory explains the long-term behavior of Bitcoin and it has many consequences.

The most astonishing and relevant, and often misunderstood by most casual Bitcoin investors, is Scale Invariance.

Scale invariance is a property of objects or laws that remain unchanged when scales of length, energy, or other variables are multiplied by a common factor. It’s a feature used in physics, mathematics, and statistics.

Scale invariance is a typical feature of systems that are governed by power laws.

In essence, it says that the system will continue to scale up as it grows in the same manner and this is why we can use scale invariance to make predictions and the growth of the system given it has been expressing over 9 orders of magnitude it will continue to happen almost for sure for other 1 or 2 orders of magnitude (it will take about 10 years to reach 1 million BTC). As incredible as this sounds then everything important about the system, price, hash-rate, and adoption is predictable in the long term.

Scale invariance also allows us to understand the role and importance of events like the recent inflow of investments in the Bitcoin system by large institutions' ETFs.

Scale invariance tells us these events will not affect dramatically the price trajectory of Bitcoin but rather they are the critical events necessary for the system to continue in its scale-invariant growth.

It also means, and many people will find it very hard to comprehend, that the Power Law trend (plus bubbles) is all that you get. NO MORE, NO LESS.
This is the most astounding prediction of the theory.
All the theories are falsifiable and this is one of the ways to falsify the Theory at least in the current form.

A future theory could modified to add a changing of the slope or phase transitions but in the current form the theory says the path of Bitcoin price is already established and unless we have catastrophic events it will not be changed in particular for 1 or 2 orders of magnitude that are just a small fraction of the overall historical growth of Bitcoin. If it was scale invariant for 15 years it would continue to be likely for another 10 years (next order of magnitude).

By the way, in terms of scale, the next 10 years are not compatible with the previous 15 years because it is just another order of magnitude. It takes some time to understand how logarithmic scaling works for most people who are not familiar with these ideas.

The principle of scale invariance used in making predictions in time is simple.

Keep everything proportional in the log-log space (or scale space). It is as simple as making a triangle bigger and keeping all its sides proportional.

In the graph below I apply the principle (pedagogically) to the prediction Harold Christopher Burger made 5 years ago using the Power Law (blue dots). 5 years later the prediction turned out to be correct (red dots). You can see one could have used scale invariance to make the prediction (he did so indirectly assuming the path continued). Scale invariance is used all the time in science to make predictions.

There is much more to the theory (for example why we see so well-aligned bottoms following the power law) but they will be elaborated on in subsequent articles.

I’m in the process of writing both a science article (so I go through the process of peer review of these ideas and make sure they have scientific validity) and a popular book on the topic (in the style of my country-man Galileo “Dialogue Concerning the Two Chief World Systems”).

Corollary to the Theory.

How bubbles work.

Nothing to do with scarcity, all to do with Moore’s Law.
Here is a summarized version of the arguments below in a video format:

Satoshi was aware of the Moore’s law. It is a heuristic law that claims computing power doubles every 2 years. The “Difficulty Adjustment” mechanism guarantees that you need to spend a lot of money and energy to get some extra coins.

But Moore’s law gives you an unfair advantage. In 4 years you would have 4 times the hashing power, basically at the same energy cost as the machine of 4 years ago (roughly). You will need to update anyway because of wear and tear and the cost of the machines is just part of the operational cost. It turns out (I explained how in my theory) both logically and empirically that we have Price (or rewards in general)= hash rate¹/2. So basically 4 times the hash rate gives you only 2x the benefits. But then the halvings cut in half that benefit to leave you with zero increased benefits. It is all about keeping the miners on the edge of profitability and never allowing a free lunch. It is too perfect to be due to chance and I think Satoshi planned exactly for this.

The 4 years, instead of 2 or a continuous decrease in rewards is there because it is also a good idea in terms of logistics given it takes time for the chip industry to update and progress and also gives time to the miners to plan for the update and have the equipment depreciate naturally. It is pure genius and extremely pragmatic and to the point with anything that has to do with Bitcoin. The bubbles are a consequence of the “security attracts more adoption part of the loop”. I didn’t even invent this loop, others did and it is used all the time to explain the cycle of adoption.

It makes sense because directly increasing security brings in people and gives you confidence about the ability of Bitcoin to store value. Without this, there is no value. The best analogy I have is when people move to a growing city (Bitcoin is a shining city in the digital world like Saylor says) when a burst of activity happens. You want to move in because of bridges, houses, roads, and so on. You don’t necessarily think about these things directly but simply you are attracted to the activity. That is where all the new and good things are happening. This creates a temporary FOMO, which is good FOMO because it is based on fundamentals, not some stupid speculation, maybe FOMO is not the best word so you can help me to find a better one. But you know what I mean.

kuntah ⚡is the author of this amazing graph.

The price goes up fast and almost exponentially. It is the only time when the price does that instead of growing as a power law. It overshoots because of all the excitement but then it needs to go back and in fact, you can see from the chart below it is almost perfectly symmetric where the price goes down as fast as it went up (sometimes faster). The bubble bursts and it goes back to equilibrium. It is a form of punctuated evolution, necessary for Bitcoin growth.

“Punctuated equilibrium is the idea that evolution occurs in spurts instead of following the slow, but steady path that Darwin suggested. Long periods of stasis with little activity in terms of extinctions or emergence of new species are interrupted by intermittent bursts of activity.”

So the bubbles are also part of the Bitcoin story. They are not the main story that is the overall power law growth but they are an important part and necessary part of it too.

This explains then, I think, perfectly both the growth out of the bubbles (about 2 years) and during the bubbles (about 2 years) of the full cycle. Let me know what you think and if this makes sense.

This article by renowned physicist D. Sornette has a very similar position about the origin and nature of the bubbles.

Note

Scarcity plays absolutely no role in this theory. There is no mechanism or explanatory power for scarcity.

Addendum

These are models that confirm my findings. The Bitcoin Power Law Theory (BPLT) has precedence over these works by several years but it is reassuring other researchers found similar results:

https://stephenperrenod.substack.com/p/bitcoins-lindy-model

Stephen is another astrophysicist with a Ph.D. from Harvard.

Beautiful Movie Pi by Aronofsky. I hate the end, though.

Q&A

I don’t understand what is a power law

A simple concept, it is a relation of the type y=A x^n.
As simple as this type of equation is, it represents many phenomena in nature and also in human-made phenomena.

But how is it possible power law shows up in Bitcoin when it is made by human interactions?

First of all, it is not true that Bitcoin is made only by human interactions. After all is a code with precise algorithms that work with precise mathematical formulas. The “Difficulty Adjustment” is one of the many feedback loops existing in the system that act similar to a thermostat so it can be studied as a physical system. The energy demands of miners are also pure physics. But also physics that are more based on social interactions like the adoption of new Bitcoiners can be modeled by equations that are similar to the ones in find in physics and in biology, like the spread of a virus for example.

Single individuals may have free will and act independently but when you consider a huge number of agents then patterns emerge that can be studied with the tools we developed to understand natural phenomena. We call this Universality which means we can find patterns in nature that are similar independently from the particular nature of the phenomenon studied.

Scientists have applied these methods to social network growth, how cities grow, how corporations survive completion, and many others. Often these social or economic phenomena follow power laws. Even terrorist attacks follow power laws.

You can watch this video for example from physicist G. West:

https://www.youtube.com/watch?v=XyCY6mjWOPc

What about scarcity, demand, and supply

It has zero part in the BPLT and we will elaborate in future articles.

Why are not using other currencies besides the dollar?

The dollar is still stable in comparison with most currencies in the world. It is inflationary but this is a small correction for Bitcoin. When we study things in Physics we simplify at first and we exclude possible complications like friction or air drag. We can always add them later but first, we want to understand the essence of a phenomenon without distractions.

Would the power law work with inflationary currencies?

I have no idea, why should I even try? What info we get from that? I could do it but there are 300 things I want to explore about Bitcoin that it seems a waste of time, as wasteful as these currencies.

In general, though, power laws in BTC work with inflations that are stable. If you start to have something too inflationary (like the rate of inflation increases fast) then the problem is not about the power law it is about the inflationary currency.

It is like I’m telling you gravity works and makes objects go down. Then you ask what about in a hurricane? Yeah, a cow can fly during a hurricane, so you and your house. It is not a violation of the law of gravity.

Do you see the logical fallacy here?

When you create a scientific theory, in the case of Bitcoin, you always start from simple situations.

For example, using a stable numeraire like the dollar that yes is inflationary but with a small constant rate (at least in comparison with the pesos). Then you can study more special cases when you understand the simpler and general ones. But at that point, you are not studying gravity you are studying hurricane-force winds creating lift. It is a different phenomenon altogether. People are so science illiterate and lack critical thinking that is really hard to pass across basic points and ideas.

What is the price in 2060?

10²⁹⁹⁹¹²³⁵ are you happy now? The BPLT should not be used to make predictions beyond 2040 at most. Ray Kurzweil's technological singularity will come next and all the predictions are off. History has a literal singularity so nobody knows what is going to happen.

It cannot go up forever

  1. We don’t know because we don’t know how much future value is transferred into Bitcoin. We could start to mine asteroids or invent nanotech that usher a new era of abundance and wealth so Bitcoin could go up forever (well see question above).
  2. The model can be adjusted easily by adding a tapering component. The power law basically would be an approximation of that model. These models by the way do not lead to exponential behavior but in fact, they are more moderate than the power law itself. For now, it is not necessary to add this component that will make the model more complicated with no real benefit to our understanding.

You say that price=hash rate^(1/2), but the equation doesn’t have the right dimensions.

For simplicity we meant that the relationship is proportional in nature, the right equation of course would b price=A hash rate^(1/2) where A is a constant with the right units to make the equation work dimensionally.

The growth of a virus typically follows an exponential pattern during the early and middle stages of an outbreak, not a power law obviously viral exponential spread doesn’t continue indefinitely (by immunity, behavioral changes, vaccinations..) and takes other forms like logistic growth, again, not a power law.

This is not the case as explained in the theory.
I will add to the references soon. There is a vast body of work showing that viral infections become power laws when there is a curbing mechanism present.

Price is autocorrelated so power law is spurious

This is one of the arguments so loved by statisticians and economics “experts”. Of course is autocorrelated we are claiming it is deterministic. So you are supporting our hypothesis? Anyway, so much more can be said about this absurd argument and you can read it in the linked article below where we debunk the debunkers.

Also, notice below a peer-reviewed article on Bitcoin that makes similar arguments in a more polite and professional way that if you start by stating you claim causation because of a plausible mechanism then you are ok with ignoring these more formal tests of causation given that if there was causation and the data was partially deterministic then the data would be obviously correlated.

All the power laws we observe are claimed to be created by causative processes like the Metcalfe law, the “Difficulty Adjustment”, power law like the spreading of social information and interactions between users of the Bitcoin network.
So we apply the same argument below about the omission of these tests and their inappropriateness in this context of studying Bitcoin as a natural process (based on principles and mechanisms similar to the ones in biology, network theory, and physics).

Also, Sinz explains:

What about S2F or other price models?

S2F is full of mathematical and conceptual errors see one of our previous discussions on this topic. Basically, it is nonsense. And no autocorrelation, or cointegration don’t kill this vampire. The basics of the concepts and the math used in the construction of the model do.

What happens if the dollar goes hyperinflationary? Will the model break upside?

One of the most common and annoying questions ever. What the questioner is trying to allude to here? That he will be soon a millionaire?

  1. upside to what? Even more worthless dollars? Do you dream of this scenario? You get to become a millionaire of a worthless currency. Happy?
  2. Do you realize that could lead to civil wars, maybe even nuclear war? What do you do with your Bitcoin then?
  3. Bitcoin charts would be the last of your problems if that happens.

All “models will be destroyed”

Saylor was not even referring to Bitcoin models at all but some generic model about the economy. I had to go back and hear that interview. Completely irrelevant. And then let’s stop to let me people how think, me included, just think for yourself. You can. I love Saylor, to death, but I’m sure he never plotted a Bitcoin chart by himself or if he did he didn’t spend years studying them. When he was talking shit about Bitcoin I was trying to understand it. Ok, let’s compromise, all the models will be destroyed the Power Law Theory is a Theory, not a model. OK?

Can it be that we are at the beginning of a S-curve?

No for several reasons (we will discuss soon).

If the knowledge actually becomes commonplace, the value will skyrocket because people will price in the future price?

No, this goes against one of the main predictions and fundamental tenets of the PLT. Momentarily manipulation of any kind can move the price up or down. But it cannot last and in general, the trend will be respected.
This is a very difficult concept to understand. You can be aware of the relationship between patents and the size of a city, that is a power law, but you cannot change it, or changing it too much is a fundamental property of the system. It is not there by chance. It is what the system really is.
The Power Laws we observe in Bitcoin are what Bitcoin is.
We cannot change them unless we change radically Bitcoin.
It is the strongest and most consequential part of the theory and it can be falsified with time or more observations will support it.

Note 2 :

For all the wannabe debunkers, please read this to see if we debunked you in the bud (or all your objections about autocorrelation and it is not allowed to take the log of time are answered).

https://medium.com/quantodian-publications/bitcoins-power-law-really-debunked-2e5add103ba9

References

  1. Why Stock Markets Crash. Critical Events in Complex Financial Systems” Didier Sornette, 2002.
  2. “Universal Cointegration and Its Applications” Tu et al., including supplemental information
  3. Phillips, P. C. (1986). Understanding spurious regressions in econometrics. Journal of econometrics, 33(3), 311–340.
  4. Scale: The Universal Laws of Life, Growth, and Death in Organisms, Cities, and Companies G. West
  5. Bitcoin power law, over 10 year period, all the way to Genesis Block. : r/Bitcoin (reddit.com)
  6. https://www.reddit.com/r/Bitcoin/comments/21pujs/bitcoin_compared_with_metcalfes_and_zipfs_law/

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