Everything you wanted to know about The Bitcoin Power Law Theory, but were afraid to ask!OP post on Bitcointalk.org:
The BTC Scaling Law
During the last few weeks, I have been increasingly involved in studying a new model by an astrophysics professor turned neuroscientist, which describes bitcoin according to Power Laws.
Giovanni Santostasi, aka
BTCdragosfera described a model where he explains why Power Laws governing price and other Bitcoin metrics (such as Active addresses and hash rate) are not a random fact but actually the symptoms of something else, namely Bitcoin being "not a common asset but more similar to natural phenomena ruled by universal Power Laws due to recursive, infinite feedback loops fundamental to the system".
Bitcoin is more similar to a city and an organism than a financial asset.
Santostasi first posted his mumblings on Reddit more than 5 years ago:
This post went almost unnoticed: Giovanni set it aside a little bit, focusing on other things, namely his new venture on Neurosciences, but kept working on the idea and finally published the following paper:
The Bitcoin Power Law TheoryIf you prefer the video format, hear a video Summary of the Theory:
The material is not very well organized, so I will try to explain the ideas better in this thread.
The Power Law is only the first part of his full BTC model, where it also models a range for bitcoin price and a bubble indicator to detect excessive price movements:
Yes, of course, you have already seen that Rainbow Graph, and actually
Trolololo chart is a Power Law chart.
The innovation of Santostasi's article is that Bitcoin is considered a financial system that behaves like a natural system according to well-defined laws. These laws are Power Laws. In this sense, the article is not a "model" about Bitcoin but a coherent "Bitcoin theory."
According to Santostasi:
In modern science, the term "theory" refers to scientific theories, a well-confirmed type of explanation of nature, made in a way consistent with the scientific method, and fulfilling the criteria required by modern science. Such theories are described in such a way that scientific tests should be able to provide empirical support for it, or empirical contradiction ("falsify") of it. Scientific theories are the most reliable, rigorous, and comprehensive form of scientific knowledge,[1] in contrast to more common uses of the word "theory" that imply that something is unproven or speculative (which in formal terms is better characterized by the word hypothesis).[2]
(This is from Wikipedia - Ed.)<...>
It is a full theory because it explains the entire behaviour of Bitcoin. It has several points and explains the interaction of all the on-chain parameters,
it even explains why the bubble exists and why we have a bottom. He makes predictions that make the theory falsifiable.
In science, a model is a representation of an idea, an object or even a process or a system that is used to describe and explain phenomena that cannot be experienced directly. Models are central to what scientists do, both in their research as well as when communicating their explanations.
A model is very similar to a hypothesis, so it is an initial mathematical formula or algorithm to try to replicate some of the behaviour of the subject studied.
A theory is the ultimate way to understand a phenomenon: it is complete in the sense that it tries to explain all the observed behaviour.
In this case, how the on-chain parameters work, how the adoption is growing (via a virus-like mechanism), and how the adoption affects the price (via Metcalfe law).
How the price brings in more miners, how the hash rate is related to price, how the Difficulty Adjustment kicks in and creates an inhibitory loop that creates stability for the system, and so on and on
People do not comprehend it yet: it tells us everything we need to know about Bitcoin, and there is a corollary that even explains the bubbles and their relevance. It can be improved and made better, but it is coherent and complete: it is really a big deal.
(I made some little readjustments to make the sentences more readable)
Introduction: What is a Power Law?
According to Wikipedia:
In statistics, a Power Law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to a power of the change, independent of the initial size of those quantities: one quantity varies as a power of another.
Power Laws can be written in the following form:
Y=10^B*(x+S)^A
or:
log(y)=A*log(x+S)+B
A lot of natural, sociological, mathematical, physical, and psychological phenomena follow Power Laws.
The Power Law has a few important features.
- Scale invariance: Power Laws remain valid when the involved variables change the order of magnitude.
- Lack of well-defined average value: Power Laws have a median but lack a mean value.
- Universality: Power Laws with the same exponent produce similar dynamics.
The first property is that a Power Law drawn in a log-log chart, or a chart where both axes are on a logarithmic scale, is a straight line with slope A, B intercept, and S as a shift parameter.
Please note that a straight line in a log-log plot is necessary, but there is insufficient evidence for Power Laws. Determining a Power Law existence only because there is a straight line fitted in a log-log chart is an unsatisfactory approach to the statistical method.
|  |  |
| bitcoin price plotted against time | A graph you might have already seen Log bitcoin price, plotted against time |
|  |  |
| Now both axis are Log, and a linear relationship appears | The previous graph slightly reworked |
Please note that those 4 graphs represent the exact same data. The blue line is the same as the red one.
The only difference is visual, and this is best captured by the red line in the log-log chart, which appears to describe the Bitcoin prices.
In reality, the explanatory power is the same in all the graphs, but the linear scale "undervalues" the errors in the first steps of the graph and "overvalues" the errors in the last steps, as the prices have increased four orders of magnitude.
The log-log chart evenly "decompresses" the chart's price and time dimension, and the best-fit function appears to be a straight line.
I made those graphs myself to check the reality of the claimed properties of the log-log chart.
If you want to understand better what Power laws are and why they are so ubiquitous in science, please have a look at the following links:
|  |  |
| Geoffrey West: The surprising math of cities and corporations | Why do Power Laws Work so Widely? | Episode 2207 | Closer To Truth |
Giovanni Santostasi already published an article answering a few questions and correcting misconceptions about PLT (Power Law Theory):
Common Misconceptions and Fallacies Regarding the Bitcoin Theory of BitcoinThis is a rendition of the "bitcoin clock" using the Power Laws: the beautiful spiral well represents the soundness of the theory.
Bitcoin Price is not the only variable that has a governing Power Law; addresses and hash rate are also governed by similar laws.
This adds beauty to the model, as we can find a "cross" relationship.
There is a beautiful YouTube video about the relationship between these quantities:
Addresses and Prices are tied via a Power Law with something resembling the Metcalfe Law.
HashPower and Price are linked via a Power Law that strongly supports price via Mining Hardware cost.
The major point of this model is that scarcity is not considered anywhere in the model. The PLT doesn't consider S2F relevant for price determination.
Of course, the author soon entered into a disagreement with PlanB:
In
this article Giovanni explain why Scarcity doesn't play a role in price determination:
Scarcity is not what Drives Bitcoin Price
<...>
Here 2 models, the power law model that has zero assumption on scarcity, just based on the observation the price is going up in a predictable manner. Scarcity it is not part of the model at all. The other, S2F where scarcity is the main driver, in fact, it assumes scarcity increases with time. The power law (with no assumption on scarcity) leads to “hurry up and get some BTC now”. As you can see S2F instead says you can get in at any time and still make the same amount of gains, no matter if you joined 10 year ago (for the same amount of 4 years HODLING time) or 10 years from now (again for the same HODLING time). Both investors will make a 10x return over this time. While you notice for the Power Law model it is important to join as soon as possible. So do you see how people use of the scarcity argument leads to illogical conclusions that are not supported by facts?
If you want to deep dive in this, there is a video here:
Why BTC stock-to-flow model based on scarcity is worse than wrong, it is complete nonsense.
Santostasi is not done yet. Firstly he's trying to organise the knowledge scattered over a few articles and X posts.
The first output will be a scientific paper, possibly peer-reviewed, and a more educational book detailing his theory.
Resources:
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