The Bitcoin Power Law Theory
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17 min read
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Mar 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.
- Initially, Bitcoin needed to be accepted and adopted by the first users in the Satoshi circle.
- 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.
- The price increase brings in more resources in particular mining power and capabilities.
- 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).
- 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.
- 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).
- The cycle repeats indefinitely. The bubbles are an important and necessary component of this cycle and they are discussed separately in the Corollary below.
- 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
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:
kuntah ⚡is the author of this amazing graph.
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
