No, I haven’t forgotten how to string a sentence together to make something intelligible. The title of this post contains the ten most common words in the English language. ‘The’ is the most common word. What is fascinating is the relationship between words. ‘The’ is found twice as many times as ‘of’, which is used three times as much as ‘and’, and four times as much as ‘to’; and so on. That’s a bit of a tongue twister. The chart below makes it easier to comprehend.
If you plot the frequency of word usage, you get a chart like this.
Figure 1 - Steemit
It’s called Zipf’s law, and it doesn’t just apply to the English language, but every language, ever. Zipf’s law is an example of a wider phenomenon called power laws.
A power law is a mathematical relationship between two quantities, where one quantity varies as a power of another. In simpler terms, it describes how one thing changes as another thing changes, but not in a simple, linear way like 1-to-1 or 2-to-1. Instead, it follows a specific pattern where one quantity changes at a rate that's proportional to a power of the other quantity.
Imagine you have a bunch of data points representing the sizes of different cities and the populations of those cities. In a power law relationship, you might find that as the population of a city increases, its size increases by a power of that population. So, if a city's population doubles, its size might not just double, but maybe quadruple or increase even more. These aren’t linear relationships. If a city doubles it doesn’t require twice as many roads. A horse needs less food compared to its weight than a human, and a person needs comparatively less than a dog. Increasing size can create efficiency. Power laws also explain why there are natural limits. Godzilla would collapse under his own weight.
Power laws are common in many natural and social systems. For example, they're often found in the distribution of wealth, the sizes of earthquakes, the frequency of words in a language as we saw, and even the sizes of craters on the moon! They're used to describe phenomena where a few instances are exceptionally large or very small, while most fall somewhere in between, creating a long-tailed distribution.
Understanding power laws can help us make sense of complex systems and predict how they might change over time. They're a powerful tool in fields like economics, physics, biology, and sociology for modelling and analysing diverse phenomena.
Here’s another one which blew my mind. The average human’s heart will beat 1.5 billion times during a lifetime. So does a mouse. So does a horse. So does an elephant. So does every animal with a heart. The only difference is the frequency. The smaller the animal the faster the metabolic rate. But we all get the same number of goes. We just experience at different speeds.
Figure 2- Francisco Rodrigues - Medium
The pattern forms a straight line. But if you look closely, you will see that both the horizontal and vertical axes use log. Instead of intervals at 1,2,3,4 the vertical axis is 0.1, 1, 10, 100, 1000. The horizontal axis plots body mass at 0,01, 0.1, 1,10, 100, 1000, and 10,000.
What’s this got to do with Bitcoin?
Italian astro-physicist Giovanni Santostasi has discovered that Bitcoin has power laws which apply to multiple aspects of its growth. For example, price. If you look at a regular chart of bitcoin’s price it’s almost impossible to detect any pattern if you are looking across the whole fifteen years. That’s why many people prefer to log the price. What Santostasi has done is a log log chart. He didn’t just log the price of bitcoin, but also the horizontal time axis.
The result is this.
When you first take a look at his chart, it looks similar to the derided Stock to Flow chart. But Santostasi is at pains to point out that this chart is generated by maths. It isn’t taking data and attempting to shoehorn it into a preconceived idea. The formula is displayed at the top, and anyone can replicate it (see below). While the previous power law examples form straight lines or curves, bitcoin price goes in waves because of the halving cycle.
Santostasi’s conclusion is that the price of bitcoin will continue to go parabolic but not exponential. It’s an important distinction. It means that bitcoin will reach nearly $200k in this cycle and eventually $1m but not before 2033. If his theory is correct, then all that is required is patience. Notice that the green floor is strong. The line only breaks below it once at the start of bitcoin’s life but be aware that the floor is set at 40% below, so there is a lot of room for volatility! The upper limit though is more variable.
If Santostasi is right this means that all of the S curve analogies that you have read about are wrong. Bitcoin is not following a similar adoption curve to the phone, fax machines, or the Internet.
Santostasi also strongly disagrees with the price of bitcoin being related to scarcity. He argues that Bitcoin isn’t scarce. I would agree with that. It is easy to confuse scarcity with finite. Bitcoin is finite. There are 1.9 quadrillion sats that have been released so far. That’s 1.9 followed by fifteen zeros. Not exactly scarce. It is Bitcoin’s finite quality, plus its improvements on other assets being the only digital asset, which is the source of its value. It is the only finite asset ever.
Why do power laws apply to Bitcoin? Maybe for two reasons. Bitcoin is grounded in physical reality. Energy comes from the physical world. Raw energy is transformed into electrical energy, and from there into Bitcoin. Power laws describe natural and social phenomena. Bitcoin is natural. Bitcoin is also a network of nodes and people. Bitcoin is social.
Santostasi has discovered other power laws as the chart below shows. For example, Hash Rate versus Time.
Figure 3 - Giovanni Santostasi
I believe that Santostasi is a scientist, not a salesman. He’s seen a pattern which he considers statistically meaningful. As a scientist he embraces challenges to his model. He also accepts that his model could fail, especially if there is a catastrophic event such as a global war. It’s a model, not reality. But for the time being bitcoin price has scaled nine times from 0.01 cent to its current multiples of 10,000, and the model has a more than 95% accuracy. Is it unreasonable to scale up two further times?
For fun I asked ChatGPT4o if there was a power law between Bitcoin and the thousands of cryptocurrencies. Using the top 50 by market cap turns out there is.
The fact that the market cap distribution follows a power law suggests that Bitcoin's dominance is likely to persist. In a power law distribution, the leading entity (in this case, Bitcoin) retains a significant share of the total market cap. Bitcoin, being the first and most well-known, benefits from network effects, widespread adoption, and a strong brand identity.
References
The bitcoin price power law formula p=A*(r-t1) alpha1
Alpha1 = 5.82
A = 1.011e-17
T1 = days from Jan 3, 2009 (Genesis Block)
R2 = 0.953 (a measure of accuracy)
Live Chart https://charts.bitbo.io/long-term-power-law/
Further reading: https://giovannisantostasi.medium.com/the-bitcoin-power-law-theory-962dfaf99ee9
Best short video:
Best long read on power laws: Scale: The Universal Laws of Life and Death in Organisms, Cities and Companies by Geoffrey West
X https://twitter.com/Giovann35084111
Disclaimer: This is not financial advice. We have never and will never urge anyone to buy Bitcoin. Charts and statements derived from ChatGPT can be plain wrong. This article is only for education, and entertainment. It is always a good time to do your own research.