Power Law Bitcoin

Bitcoin is more akin to a city and an organism rather than a financial asset. This assertion stems from the power-law behavior of Bitcoin. If you delve deep into the world of Bitcoin, you will develop a geometric intuition about Bitcoin’s power-law price behavior.

Of course, this intuition is not enough. We need to further establish a comprehensive theory of Bitcoin behavior that can explain all major on-chain parameters in a scientific, coherent, and falsifiable manner. This is what this article seeks to explore: the power-law theory of Bitcoin.

The operation of Bitcoin is depicted by the power law for several reasons:

1. Initially, Bitcoin was adopted and embraced by the first wave of users in the Satoshi Nakamoto community. The “value” (now observable as “price” 24/7) increased with the square of the number of users (network effects) (empirical measurements are closer to 1.95, but for simplicity, rounded to an integer here) — this confirms the theoretical result known as Metcalfe’s law. Metcalfe’s law posits that if each user in a network can connect with all other users, then theoretically, the number of connections in the network when there are N users is approximately N(N-1)/2, which is close to a scale of N².

2. The increase in Bitcoin price brings more resources, especially mining power. The price increase reduces the time for mining blocks, but due to “difficulty adjustment,” the required hash rate for mining blocks iteratively changes — since mining is almost unprofitable, the compensation mechanism needs to be proportional to the price increase, where P = users² and the reward itself, thus logically and dimensionally, the hash rate = price² (this also aligns with empirical evidence: the empirical value of the power is close to 2, and price = hash rate^1/2). Here, the physical significance of the hash rate is a comprehensive indicator used to measure network processing capacity, security, mining difficulty, and energy consumption.

3. The increase in hash rate brings more security to the system, thereby attracting more users. Although some people do not buy Bitcoin because of “security,” clearly no one would invest significant effort into it if it were not a secure system. The security of the system directly or indirectly brings new users.

4. Bitcoin differs from common S-curve types of growth (such as the growth curves of TV, refrigerators, cars, and phones); it follows time power-law growth with t=3. The difference here is that if there are significant inhibitory mechanisms, the power law will be more pronounced. For Bitcoin, any type of investment’s “difficulty adjustment” and risk serve as such inhibitory mechanisms, which aligns with empirical observations.

In conclusion, we have the following power-law model relationships: users = t³, price = users² = (t³)² = t⁶, hash rate = price² = (t⁶)² = t¹². The above cycle repeats infinitely and creates bubbles — bubbles are an important and necessary component of this cycle.

Furthermore, by setting this cycle in motion, the hash rate now perpetually influences the hash rate within the infinite loop. Therefore, employing power-law predictions and control over Bitcoin’s behavior is the most astonishing discovery — in reality, the three parameters observed in the phase space of individual Bitcoins almost perfectly align with our model:

Power law prediction

The power-law theory of Bitcoin opens a window for us — the ability to explain and predict the long-term behavior of Bitcoin leads to some interesting implications.

One of the most surprising, and often misunderstood by most ordinary Bitcoin investors, is the scale invariance (or using fractals to understand the reflexivity of “fractals and power laws” arbitrage explanations, “beast edge” cycles, quantification, and fractal trading principles).

Scale invariance is a typical characteristic of systems governed by power laws.

We can accurately predict that reaching 1 million BTC will take about 10 years. Although this sounds unbelievable, in the long run, core parameters such as users, price, and hash rate are predictable — Bitcoin’s scale has followed power-law growth by nine orders of magnitude, so we should not be surprised if it continues to follow power law for another 1-2 orders of magnitude.

Furthermore, scale invariance allows us to understand the role and importance of events, such as recent inflows of investment into the Bitcoin system from large institutional ETFs — scale invariance tells us that these events will not significantly affect Bitcoin’s price trajectory but rather the system will continue its scale-invariant growth — this is one of the most shocking predictions of the theory.

We cannot predict the long-term future of Bitcoin, but under the assumption that the current power-law mechanism remains intact, the theory suggests that the path of Bitcoin prices is deterministic — it will not change unless we experience catastrophic events, especially during the expansion phase of the next 1-2 orders of magnitude — this is only a small part of Bitcoin’s overall historical growth.

If Bitcoin remains scale invariant for the next 15 years, it may continue to exist for another 10 years to reach the next order of magnitude — here, we can rely on the Lindy effect to play a role (the underestimated Soros and the mythical Buffett).

From a different perspective, we consider that the bubble of Bitcoin is unrelated to scarcity — it’s more akin to Moore’s Law.

Moore’s Law posits that every 18 months, the computing speed and integration of chips double. If the computing power remains constant, the price and size of chips decrease by half. Currently, humanity has gone through 38 Moore cycles, yet we haven’t reached the limit of development.

If the automotive industry followed the development of Moore’s Law like computers, then today you could probably purchase a Rolls-Royce for $10, a liter of gasoline could run for millions of kilometers, and its power could drive the Queen Elizabeth II cruise ship.

Satoshi Nakamoto must have been aware of the cycles of Moore’s Law. He claimed that computing power doubles every 2 years, and the “difficulty adjustment” mechanism ensures that you need to spend a considerable amount of money and effort to obtain some additional bitcoins.

The theoretical limit of semiconductor capability is only constrained by the size of the Planck constant — the smallest scale unit in the universe, and the scale limit currently controlled by humans is still 17 orders of magnitude higher than the Planck constant, promising a bright future.

But Moore’s Law gives you an unfair advantage — in 4 years, you will have 4 times the hashing power, essentially with the same energy cost as machines from 4 years ago (roughly). Due to wear and tear, you need to update your equipment anyway, and the cost of the machine is only a part of the operational cost.

Logically and empirically, because the price (or general reward) = hash rate^1/2. So, four times the hash rate can only bring twice the benefit. However, due to wear and tear and costs, everything is geared towards keeping miners on the edge of profitability — there’s never a free lunch. This pricing mechanism is too perfect to be accidental — perhaps Satoshi had planned it this way from the beginning.

Four years, instead of two years or continuous reduction of rewards, is a stroke of genius, as the chip industry needs time to update and progress, giving miners time to plan updates and let devices naturally depreciate. This setting is very pragmatic, and Satoshi always knew how to hit the nail on the head.

Power Law Regression

This stunning image depicts the rapid local price surge — almost exhibiting exponential growth.

However, exponential growth is unsustainable; once it overshoots its phase, it will revert to a power-law form.

For power-law growth, the rate of growth increases with time, but at a slower pace than exponential growth. In the short term, the graph of exponential growth is almost symmetric — the speed of price decline and rise is equally fast (sometimes faster). After the bubble bursts, the graph returns to the long-term trend of power-law form — short-term exponential growth leads to bubbles, while the long-term power-law form is determined by the inherent characteristics of Bitcoin.

It’s like the mutations of species — evolution occurs in sudden bursts rather than following Darwin’s envisioned slow but steady path. For species to go extinct or new species to emerge, periods of inactivity in long-term evolution are interrupted by intermittent bursts of mutation.

Similarly, bubbles are also part of the Bitcoin story — they are not the main backdrop of overall power-law growth, but short-term exponential noise is also an essential component of the market.

Overall, Bitcoin’s power law works alongside stable inflation — if prices rise rapidly with the inflation rate, the issue is not with the power law itself — the power law is an independent backdrop, but with inflation itself. It’s like Newton telling us that gravity causes objects to fall, but you’re wondering what to do when a hurricane comes? — The answer is, a cow can fly in a hurricane, but that doesn’t violate the law of universal gravitation.

D. Sornette has a similar stance on this phenomenon (The Dragon King or Black Swan: Predictable Financial Crises), and his portrayal of Bitcoin’s bubble behavior is also brilliant:

The S2F (Stock-to-Flow) price model predicts Bitcoin’s price based on scarcity, evaluating the scarcity of an asset by calculating the ratio of stock (existing supply) to flow (new supply). More specifically, the halving mechanism of Bitcoin’s production every four years significantly impacts scarcity. However, scarcity plays no role in our new power-law theory — in this fascinating market of Bitcoin, scarcity holds no explanatory power. S2F is fraught with mathematical and conceptual errors.

New consensus is continually being discovered, and indeed, more people have independently uncovered this new power-law principle. For instance, Harvard astrophysicist Stephen Perrenod introduced the Lindy effect and developed his own FSM (Future Supply Model) valuation model, while renowned cryptocurrency analyst Nic Carter also pointed out that the Lindy effect (power law) applies to Bitcoin.

The number of Bitcoin ATMs has grown over 20 times in the past five years, equivalent to a power-law exponent of 6.

The Lindy model is phenomenological — it lacks exact underlying motives. The Lindy effect reflects the growth of the Bitcoin ecosystem and the longevity of Bitcoin, supported by its anti-fragility. It implicitly reflects the increasing security of a continuously expanding blockchain, with the rapid increase in hash power behind it.

Where Is The Singularity?

Every One Knows, Bitcoin Cannot Forever Ascend.

Virus Growth Typically Follows an Exponential Pattern in the Early and Midst of an Outbreak, Rather than a Power Law; Eventually, Its Proliferation is Constrained by External Environment, and when Inhibition Mechanisms Arise, Viral Infections Become Power Law.

This is Why Exponential Propagation of Viruses Cannot Continue Indefinitely — Through Immunization, Behavioral Changes, Vaccination, Physical Isolation, etc. ( “Just A Hair’s Breadth” Big Data Predicts the Global Spread of COVID-19, Is Airport Screening of Non-Infected Travelers Really Effective? ).

We don’t know when this singularity will occur, as we don’t know how much future value will continue to transfer into Bitcoin. In an extreme scenario, if we were to begin mining asteroids, engage in interstellar migration, or invent nanotechnology, ushering in a new era of abundance and wealth, Bitcoin could potentially continue to rise for at least several centuries. It’s worth mentioning that Taleb also offered a somewhat pessimistic prediction for Bitcoin’s future (see “Currency and Bubbles” Taleb discusses Bitcoin, the inevitable downfall).

The current power-law model of Bitcoin is unaffected by bubbles; it’s simple and effective, and there’s no urgent pressure to adjust it.

With mining difficulty adjustments from proof-of-work, Metcalfe’s Law, the propagation of social information networks, and interactions among users, we witness the true game dynamics of power-law behavior in the Bitcoin world. With simple causal components, we can predict its long-term behavior.

Therefore, we study Bitcoin as a natural process, much like physics, without considering complex reflexivity mechanisms or the autocorrelation characteristics of prices. In fact, some researchers are already studying it in this way.

Naturally, some may wonder what would happen if the US dollar were to experience hyperinflation. Would the model explode?

However, we should maintain confidence in Bitcoin. The power-law observed in this article is inherent to Bitcoin and should be independent of inflation itself. We must remember that Bitcoin is not a product of the conventional economic models we’ve relied on for so long.

In the world of Bitcoin, any form of momentary manipulation can cause prices to rise or fall, but it cannot be sustained. Overall, the trend of power-law will eventually garner everyone’s respect.

For the fundamental principles underlying the formation of the power-law in Bitcoin, external factors are unlikely to break them, at least not in the short term, even in the face of economic crises. To take it to a more extreme scenario, would a global nuclear war disrupt the power-law mechanism? If such a situation were to occur, it would truly be an unprecedented experimental observation, offering insights into the ultimate secrets of the universe regarding power-law principles, regardless of humanity’s ultimate fate.

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