There’s something about the Bitcoin Power Law

Is Bitcoin powerful enough to be immune from power laws?

There are downsides to being a lawyer.

Being the frequent butt of jokes is one of them. Everyone has their favourite lawyer joke. A classic of the genre goes something like this: what do you do if you see a lawyer buried up to his neck in cement? Get more cement.

I’ll wait while you re-stitch your sides.

Thankfully there are some upsides to being a lawyer, too. Probably the best one is that people assume you’re intelligent without you having to convince them that their assumption is correct, and their assumption might not be correct anyway. This means you can be persuasive and influential even in domains which have nothing to do with your legal expertise and which you know absolutely nothing about.

For example, if a lawyer were to give a bitcoin price prediction it might carry a disproportionate and undeserved amount of weight. The working assumption might be that lawyers are typically very intelligent but also conservative and non-committal, prone to fence-sitting. And so – the thinking might go – if this intelligent, ultra-conservative species of human is prepared to stick his neck out on something as complex and credibility-threatening as a price prediction then it must really mean something. On the flipside, though, others will pause for a moment and ask why we should listen to what a lawyer has to say in the context of a separate domain of expertise. They may also be driven by a general dislike and distrust of lawyers, which could be especially pronounced when it comes to financial matters. Like I say, being a lawyer is tough.

With all that lawyerly throat-clearing out of the way let’s dive into the crazy world of mathematics and science. This post is not about price predictions specifically. It is about the so-called Bitcoin Power Law, which if it is true does have implications for understanding bitcoin’s price history and, by extension, its future.

What is a power law?

The University of Wikipedia tells me the following:

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’t be observed on a simple linear graph because the range of data is too vast to make any sense of what is happening on a linear scale.

Bitcoiners will be familiar with the concept of “zooming out” to observe historical price action: it is only when price is plotted on a log basis that the noise dissipates and the signal reveals itself. Hence bitcoiners will often come across graphs that plot the price scale in orders of magnitude of say $1, then $10, then $100, then $1,000 and so on, rather than on a linear basis of say $10,000, then $20,000, then $30,000, then $40,000.

See below how the y-axis (vertical, price) first has a linear scale (Fig. 1), and then a log scale (Fig. 2):

Fig. 1: Noise. Bitcoin price linear graph (USD). From https://en.wikipedia.org/wiki/File:Bitcoin_usd_price.svg

Fig. 2: Signal. Bitcoin price log graph (USD). From https://www.reddit.com/r/dataisbeautiful/comments/1064snb/oc_bitcoin_value_over_time_from_2010_to_2023/

Simply changing the scale of the y-axis from linear to log enables us to make the highly scientific observation that Fig. 2 is prettier than Fig. 1, and that Fig. 2 seems to be trying to tell us something about the relationship between bitcoin’s price and time.

However, you can also play tricks with the x-axis (horizontal, time) to make that a log scale, too. This gives you a “log-log” chart. And this is where the magic happens.

Fig. 3: The Magic. Bitcoin price log-log graph (USD). From https://giovannisantostasi.medium.com/a-scientific-approach-to-estimating-the-size-of-bitcoin-bubbles-1204a0032739

It turns out that when you plot certain sets of data on a log-log graph it reveals a power law relationship. That power law relationship is characterized by a linear relationship. In layman’s terms: a straight line appears. Powerful stuff.

There is something about the discovery of a linear relationship – of any description – which stops us in our tracks and captures our imagination. Everybody recognizes them and likes them: mathematician, non-mathematician, lawyer, non-lawyer. They are simple and elegant.

But when that linear relationship reflects a power law dynamic it is a thing of unexpected beauty: it feels like stumbling upon treasure which was previously buried under seemingly unrelated datasets. Here, the linear relationship manages to capture and express the vast range of a dataset, and the relationships between different datasets, in a vivid and striking format. Complex but simple. A bit like Bitcoin.

Power laws don’t just apply to bitcoin’s price and time. They also apply to its other metrics like adoption (measured by the number of addresses), and hashrate.

The wild thing about power laws is the breadth of their application. They can be observed in biology. For example, there is a power law governing the relationship between an animal’s mass and its metabolic rate (how much energy it uses), which applies as you compare different species to one another.

One might assume, reasonably, that if one species has double the mass of another, its energy requirement will also be double that of the smaller species. But no. As mass doubles (i.e. increases by 100%), its energy requirement increases not by 100% but by 75%. This applies as you go from bacteria to insects to dogs to humans to elephants. Woof woof.

Fig. 4: Metabolic rate. From https://francisco-rodrigues.medium.com/power-law-universality-in-nature-c933f271bda8

Power laws can also be observed in physics. For example, Kepler’s Third Law states that the square of the period of the orbit of a planet around the Sun is proportional to the cube of the semi-major axis of the orbit. Confused? Me too, so don’t worry because here’s another picture I have pasted with my expert legal skills:

Fig. 5: Kepler’s Third Law. From https://astronomy.stackexchange.com/questions/36866/how-did-kepler-come-to-the-potencies-in-his-third-law

Fascinating though the application of power laws to the biological and celestial realms is, there is an instinctive feeling that it is perhaps unsurprising that the natural world should be governed by universal mathematical structures. We can marvel at our human ingenuity in recognizing and articulating these mathematical relationships across multiple domains, but in some ways it does also feel intuitive that nature should be subject to overarching mathematical principles such as these.

Where things get weird, though, is the realization that power laws are observable in socio-economic systems too, such as cities. As cities scale the relationship of their various datasets is governed by power laws.

Fig. 6: City population and rank. From https://blogs.cornell.edu/info2040/2016/11/13/zipfs-law-for-cities-a-simple-explanation-for-urban-populations/

This feels harder to grapple with. Instinctively we feel that the growth of a city is subject to far too many variables and deliberate human interventions to be governed mathematically. A human city feels like a chaotic mish-mash of nature and…well, something other than nature. But where do you draw the line between nature and not nature? Humans are part of nature. In any case, facts don’t care about your feelings, or your definitions: the power laws apply.

Even if there is a Bitcoin Power Law, so what?

That’s enough of a lawyer pretending to be a mathematician or scientist. Why is any of this important for Bitcoin? Why is it causing a fuss?

The Bitcoin Power Law has been popularized by two bitcoiners in particular: Giovanni Santostasi (who was the first to see it) and Fred Kreuger. My initial reaction when I stumbled across it was that it was yet another model that might or might not be valid, to one extent or another (actually Giovanni stresses that it is a theory rather than a model).

I am not particularly interested in price models, or price theories. I have never spent my bitcoin on anything other than transaction fees for moving it to cold storage. I am a long-term hodler and I buy bitcoin at the top, the bottom, and every goddam place in between. I don’t sell it and I don’t trade it.

I veered on the side of skepticism because Bitcoin has a solid track record of making people look stupid when they even hint at the possibility of discovering some kind of truth about it, especially its price. Giovanni is a veteran bitcoiner but Fred is fairly new to the scene, so I worked on the assumption that perhaps Fred was simply the latest manifestation of the meme, “I’m new to Bitcoin and I’m here to fix it/explain it”. And although Fred does display general humility in the face of Bitcoin, and even though I quite like him, his trad-fi Wall Street background probably sharpened my sense of skepticism towards his announcing a eureka moment ever so slightly.

I didn’t think too much of it, and I didn’t really form a very strong view one way or another. I didn’t look into it. But the more I have looked into it the more I think Giovanni and Fred might just be on to something here. It’s worth noting that Fred has a background not just in trad-fi but also very high-level mathematics. My understanding is that neither Giovanni nor Fred is aware of any other financial asset that is governed by power laws, and so you can see why they are getting so excited.

I recommend watching Giovanni’s recent podcast with Bram Kanstein. It is three hours long. Giovanni puts forward a very strong case and does explain the concepts well, in very good English, but he can be frustrating to listen to on a personal level. He rarely pauses for his interviewer to ask questions and allow the audience to take stock of what has been said, and he often doesn’t even allow the interviewer to finish a question. He does get a bit defensive. Bram is a very polite podcaster and was giving Giovanni a more than free rein to argue his case but despite that, Giovanni was still hard to listen to at times. You could see Bram was getting a bit frustrated and eventually he seems to have given up and accepted that Giovanni was just going to press ahead no matter what.

Of course, having less-than-perfect social skills does not disprove Giovanni’s argument for power laws within Bitcoin, but I hope he can improve the way he comes across so he can reach as wide an audience as possible. Perhaps he is losing patience with people rejecting his arguments without good reasons (which is not something that Bram was doing), which is understandable, but I hope he can adapt his approach slightly otherwise people might switch off.

So shall we all start trading our bitcoin now, because Science™?

No, none of this means I will start trading my bitcoin, and I don’t think you should either.

I will continue stacking and hodling with a timeframe of forever. Just because there might be a long-term trend visible over fifteen years it doesn’t follow that this trend applies at the zoomed in time unit of weeks and months. You could sell some bitcoin one day “at the top” and then find you are never able to catch up if the price runs away from you over the subsequent weeks and months, rather than falling as you had hoped, or if you find yourself out of work with no disposable income to “buy the dip” if and when it finally arrives (and then face the double-whammy of selling more bitcoin to fund your living expenses). Over a longer period the power law might kick back into place but that won’t be of any comfort to you.

Fig. 7: Breaking bank: don’t trade your bitcoin

I’m quite happy to zoom out to try and understand what might or might not be happening. But I’m not going to zoom in and start playing stupid fiat games.

Also, in historical terms, Bitcoin is a mere fetus. The power laws governing the metabolic rate of different species have had billions of years to stabilize and to find some kind of equilibrium, as have the power laws governing planetary movements, so the departures to the straight line in those contexts are very minimal.

Assuming there is a power law governing Bitcoin, and even assuming in principle that a power law continues to govern Bitcoin in the future, it doesn’t follow that the actual mathematical rate will repeat exactly what has happened over the last fifteen years. In other words the overall straight line might stay the same angle, it might get stepper, or it might get shallower. In the meantime, and especially while Bitcoin is in its infancy, it should not be surprising to see spikes above the straight line and dips below it. All the more reason not to trade.

Humility

On the face of it claiming to have unearthed some kind of universal truth about Bitcoin smacks of a lack of humility.

But what happened to the mantra of “Don’t trust, verify”? One could argue that, rather than infringing the notion of humility in the face of Bitcoin, it is a demonstration of humility to accept the application of the Bitcoin Power Law in the face of evidence supporting the theory. It lacks humility to instinctively dismiss it.

Just as financial institutions can no longer dismiss Bitcoin as a matter of course and now have to at least consider it (and then by all means reject it if they can put forward rational reasons x, y or z for doing so), I feel that bitcoiners can’t simply dismiss the power law theory without offering something in the way of a reasonable counter argument. I think there could be two possible ways to counter it.

Firstly, a purely mathematical argument that the relationships represented by the Bitcoin datasets up to this point in time do not constitute a power law. I haven’t seen anyone put forward such an argument.

Secondly, an argument that, yes, there have been power laws in place up to this point but they won’t remain in place in perpetuity and will break down when event x, y or z happens. I’m open to such arguments.

If for whatever reason your instinct is still to dismiss all this I would draw a parallel with orange-pilling, as follows.

I would never specifically urge someone simply to “buy bitcoin”. It's just not in my nature, and in any case if someone doesn’t first study it they won't know what they have purchased. This will only increase the risk that they will burn themselves. I don't need that on my conscience. Instead I would tell them that the subject of Bitcoin is fascinating and that it is an excellent use of one’s time to study it, regardless of whether they subsequently decide to buy it.

In the same vein I would say this: whether or not you accept that Bitcoin is subject to these fundamental mathematical principles I would strongly recommend diving in to the incredible world of power laws generally. Like Bitcoin, once seen they cannot be unseen.

Bullish or bearish?

My gut feeling is that one of the reasons (perhaps the main reason) why many bitcoiners reject the power law theory is that its price predictions are a little more bearish than they would like. I believe the power law theory suggests something like a 40% CAGR for the next ten years, then perhaps 30% for the ten years after that. Still pretty good, but considerably less than we have had up to this point.

Would bitcoiners be so keen to dismiss the power law if it predicted 100% CAGR for the next fifty years? My guess is probably not. So perhaps the rejection of the theory is driven in many cases by bitcoiners wanting it not to be true rather than any evidence or rational argument that it is not true, or that it will break down on the occurrence of a certain event or permutation of events.

But the discovery of a power law could well be a good thing because it suggests perpetual sustainability. It is exponential growth that is unsustainable. As Giovanni explains, a system that is subject to exponential growth collapses on itself because it simply cannot continue.

What would you prefer? A few mad cycles of exponential growth, followed by a galactic crash and burn, or a steady path into perpetuity where the interplay of Bitcoin’s datasets shares mathematical relationships with all manner of other biological and physical systems that have withstood the test of the universe’s immense timescales? Which of these two outcomes is the true bullish scenario? If Bitcoin is indeed hope, which of these outcomes is more hopeful?

Ah-ah-ah-ah, stayin’ alive, stayin’ alive

One of the more abstract interpretations of Bitcoin is that it is a living organism. Well, how fitting would it be if, like an actual living organism, Bitcoin were indeed subject to power laws? How incredibly beautiful would that be?

The sound money qualities of Bitcoin are driven chiefly by two factors: its finite supply of 21 million units, and its proof of work. Bitcoin mining is rooted in the uncompromising reality and thermodynamic constraints of physical energy.

Is it really so unthinkable that a complex adaptive system like Bitcoin, which is ultimately governed by the universal laws of physics, should be subject to power laws in the same way other complex adaptive systems constrained by energy are?

It is a demonstration of humility to accept that this thing – Bitcoin, the eighth wonder of the world – really does have, or just might have, a mind of its own regardless of what we do, and that the best thing we can ever hope to achieve is even a very small understanding of how it behaves.

If just one component of that overall understanding includes the application of power laws, then so be it. I’ll accept that not because I want to, but because I like to think I’m not in the habit of denying reality.