BTC Trolololo model revisited (still valid today) is really a Power Law in disguise

Giovanni Santostasi
4 min readJan 12, 2024
The Power Law Model in all its glory.

Amazingly, this model from 2014 is still valid today. Not in its exact formulation but in the equation itself. Here is the mythical original post:

https://bitcointalk.org/index.php?topic=831547.0

It turns out it is actually a POWER LAW (or SCALING LAW), the Trolololo formula can be written, with some simple algebra, as Estimated Price=A * (days from GB)^n, where GB is Genesis Block, A is 10^-17 and n=5.83.

Trolololo had instead log10(Price)= D *logn(days from GB)-C, the values for D and C that Trolololo regression gave are a little off because he had less data and also calculated the fit just after the top of one of the cycles and that skewed the result.

Usually, power laws are expressed as a quantity y vs the power of another quantity x (or y~x^n) but Trolololo had log10(Price)=A*ln(days)-C but this can be rewritten as Price=10^(A*ln(days)-C) or Price=days^(A ln(10))/10^C or price=B*days^n where B=1/10^C and n=A*ln(10).

Also, the use of 2 different logs, one in base 10 and one in natural base threw me off and I could not recognize his formula as a power law but it was indeed a power law and it should have been obvious to me even in how it was constructed.

I’m not sure about Trolololo’s background but I guess he didn’t know about the significance of Power Laws. Power Laws are really important in science and technology. See this talk to understand why:

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

But he had a very good intuition that was to focus on how BTC scales up with increments of 10, 100, 1000s. So he used the log10 in the y-axis and noticed that it was not a straight line (that it would be if the growth was exponential) but a curve. He still kept the time in a linear scale so he decided to use a non-linear regression that indicated that the best fit was a log progression in time.

If he indeed plotted the log10 of price and the log10 of days from GB he would have noticed the incredible fact that the price progression then looks like a straight line! That means we are dealing with a power law.

There is no reason to use 2 different bases for the log and it is more natural to use the base of 10.

I didn’t give much consideration to Trolololo’s model because it seemed initially an arbitrary model (why log progression?).

I worked on many BTC price models but as a physicist, we are trained to plot data that do not fit initial simple linear or exponential models in a log-log chart because another form of natural law that is ubiquitous is a power law of the form y=A x^n.

But it is not usual to plot assets in a log-log chart given almost no asset is a power law.

But as Kepler did when he looked for the laws for planetary motion I decided one day to plot the price of BTC in a log-log chart.

I almost fell from the chair when I realized it was indeed a straight line!

It completely makes sense because BTC is a network and one of the characteristics of networks is that some of their parameters show power law behavior, but it is an amazing fact and something almost nobody knows and for sure not the media or the general public.

It is as a fundamental discovery as BTC itself. I’m not kidding. It means that BTC price is not random but follows a predictable behavior over the long run that is similar to the behavior of mountains, rivers, cities and nations. BTC is a force of nature itself.

I take the credit for identifying it as a power law because as far as I know, I was one of the first people to point out this and to post the discovery on reddit 5 years ago (hcbuger used my idea and popularized it, he quoted my post on reddit but for some reason everybody attributes the power law model to him).

But the credit for the first intuition that BTC scaling is important goes to the great Trolololo with his grandfather model in 2014. Just amazing. Forget about S2F, The BTC Scaling Law is the true law of BTC.

If you are interested this is my first post about BTC being a power law:

https://www.reddit.com/r/Bitcoin/comments/9cqi0k/bitcoin_power_law_over_10_year_period_all_the_way/

And here is an update after 5 years from that first post:

https://www.reddit.com/r/Bitcoin/comments/18z04kp/15_years_of_btc_power_law/

Also, you can follow me on X with updates on the model and its significance here:

https://twitter.com/Giovann35084111

--

--

Giovanni Santostasi

Physicist, neuroscientist, financial analyst. CEO and Director of Research at Quantonomy: https://www.quantonomy.fund/giovanni-santostasi-phd