Unpacking Bitcoin's Power Laws

Thanks to Sami Shams (@samishams_ on X) for this guest publication.

Sami is a curious and engaged Finance and Computer Science student, investing since his early 20's, and passionate about the synthesis of disciplines to draw new insights.

We live in a world of contradictions. We crave stability, yet our financial systems are built on shifting sands. Stability as we often think of it - steady, unchanging states - is actually foreign to natural systems. Governments preach sound money while trillions are printed, eroding the very foundation of value they claim to uphold. But what if there was a way to see through the illusion? What if we could measure the silent theft of our purchasing power with mathematical precision? Enter Bitcoin - a digital asset with a fixed supply, offering a transparent lens into the true state of fiat currencies. It's not just a digital currency; it's an economic experiment unlike any other, governed by immutable laws and offering a unique perspective on the true nature of value.

Bitcoin introduces a paradigm shift with its immutable cap of 21 million coins. Unlike fiat currencies, which governments and financiers can produce at will, Bitcoin's scarcity is encoded in its very architecture. This scarcity instils confidence among its users, as they can trust that their holdings won't be devalued by unexpected increases in supply. Bitcoin's decentralised nature eliminates the need for a central authority, placing control back into the hands of individuals

Bitcoin, as it turns out, operates within a system of power laws; mathematical relationships that govern its growth and price dynamics. These laws, evident in the relationship between Bitcoin's price, its adoption rate, and the computational power securing its network, reveal a compelling story.

Power laws describe mathematical relationships where one quantity varies as a power of another, following the form y = ax^b. In nature, they appear with remarkable frequency, revealing deep patterns in seemingly unrelated phenomena.

Cities follow power laws: each doubling of city size leads to a predictable increase in measures like wages and innovation. Earthquakes follow the Gutenberg-Richter law, where the number of earthquakes decreases exponentially with magnitude. The sizes of craters on the moon, the frequency of words in languages, and even the distribution of wealth in societies all exhibit power law behaviour.

These relationships are scale-invariant, meaning the same mathematical pattern holds true whether looking at small or large scales. This self-similarity often produces fractal-like patterns, seen in objects like coastlines, tree branches, and blood vessel networks.

Power laws suggest underlying universal principles in complex systems and have become essential tools in physics, biology, economics, and network science.

Bitcoin, it turns out, is no different. Its price, for example, doesn't just rise linearly; it follows a power law relationship with time, meaning it grows as adoption increases.

Source: https://giovannisantostasi.medium.com/the-bitcoin-power-law-theory-962dfaf99ee9

There are two compelling power laws in bitcoin, among others:

Price vs. Number of Addresses: As more people use Bitcoin, its value increases, reflecting the network effect – a phenomenon where a network becomes more valuable as more people join it (above).

Price vs. Hash Rate: The more computational power dedicated to securing the Bitcoin network (measured by hash rate), the higher its price tends to be, highlighting the link between security and value (below).

Source: https://giovannisantostasi.medium.com/the-bitcoin-power-law-theory-962dfaf99ee9

These power laws, woven into the fabric of Bitcoin's code, provide a glimpse into a new economic order, one where value is determined by decentralized consensus and mathematical certainty, not by the whims of central banks.

For decades, fiat currencies like the US dollar have been the backbone of global economies. However, their value isn't as stable as it appears. Inflation, driven by factors such as monetary and fiscal largesse, slowly chips away at purchasing power. While a modest inflation rate is deemed acceptable (still very destructive in compounding terms), aggressive monetary expansion can lead to significant devaluation. Consumers may notice rising prices, but the underlying cause-the dilution of currency value-is less apparent.

* * *

If BTC/USD follows a power law, then USD/BTC (the reciprocal) also follows a power law, but with an inverted exponent:

If BTC/USD follows the form:

y = ax^b (where b is negative based on the dataset)

Then USD/BTC follows:

y = (ax^b)^(-1) = a^(-1)x^(-b)

This means:

1. If BTC/USD shows an exponential growth pattern over time, USD/BTC shows an exponential decay pattern, along a power law.

2. The USD's purchasing power of Bitcoin demonstrates a consistent decay rate that follows the same mathematical principles

* * *

Over time, the amount of Bitcoin needed to purchase a single dollar has decreased significantly, and in line with power laws. This inversion highlights not just Bitcoin's appreciation but also the dollar's depreciation. Bitcoin acts as a mirror, reflecting the diminishing value of fiat currencies with stark clarity.

This mathematical relationship suggests the dollar's weakening against Bitcoin isn't random but follows a predictable exponential decay pattern.

What makes this particularly interesting is how it relates to the Federal Reserve's 2% inflation target. While 2% might sound like a modest linear decline, it actually compounds dramatically over time. This compounding effect of fiat inflation creates non-linear decay in purchasing power.

When this fiat debasement meets Bitcoin's network effect growth, we get a multiplication of forces that helps explain the power law relationship. The dollar's compound inflation rate effectively multiplies against Bitcoin's network effect growth, creating a steeper decline in dollar purchasing power than many would intuitively expect. This suggests the power law pattern we observe might be an emergent property of compounding inflation meeting network effect growth, rather than a coincidental relationship. (I will write another article on this soon. Hint: USD as global unit of account masks this). This creates a feedback loop - inflation drives capital to Bitcoin, which strengthens Bitcoin's network, which attracts more capital, which drives further appreciation, which in turn attracts more attention and capital seeking to escape inflation. Each cycle compounds on the previous one, mirroring the mathematical power law relationship we observe in the price data. Importantly, this isn't just about price - it's about the fundamental restructuring of monetary preferences as people recognise and react to the compounding effects of inflation.

The feedback loop helps explain why the power law relationship has remained remarkably consistent over Bitcoin's history. It's not merely a price phenomenon but reflects structural forces present: programmed monetary inflation meeting programmed monetary scarcity, with natural human action and network effects amplifying the relationship over time, thanks to the novel incentives introduced by the bitcoin network. This suggests the pattern isn't likely to break but rather continue strengthening as more market participants recognise and act on these factors.

Bitcoin then, with its unwavering scarcity and transparent ledger, exposes this financial sleight of hand. It provides a fixed reference point, a measuring stick against which the debasement of fiat currencies becomes undeniable. While its appreciation also accounts for inherent growth of the network, there is a proportion of the asset that is demonstrating this.

Traditional measures of inflation, like the Consumer Price Index (CPI), are often criticised for being manipulated or for not accurately reflecting the true cost of living. Gold, often touted as a safe haven asset, is subject to its own supply and demand dynamics, making it an imperfect measure of currency debasement.

The power laws governing Bitcoin's relationship with fiat currencies reveal a profound truth: our current financial system's apparent stability is an illusion that contradicts the natural order. Just as power laws govern everything from city growth to earthquake magnitude, they expose the inherent instability of artificial constructs like the 2% inflation target. Nature doesn't follow linear patterns - it follows exponential ones, and Bitcoin's power law relationships simply mirror this fundamental reality.

Through Bitcoin's mathematical transparency, we can now quantify and predict the exponential decay of fiat purchasing power with increasing precision. This isn't merely about price appreciation or investment returns - it's about recognising a fundamental shift in how value is stored and transferred in the digital age. The feedback loops created by compounding inflation meeting programmed scarcity aren't just patterns on a chart; they're the mathematical signature of a monetary revolution in progress.

As we stand at this intersection of natural law and monetary evolution, Bitcoin offers more than an alternative to fiat currency - it provides a window into how economic systems actually behave when freed from artificial constraints. The power laws we've explored suggest that Bitcoin's role in reshaping our financial landscape isn't just possible - it's mathematically probable. This isn't simply about choosing between different forms of money; it's about aligning our financial system with the natural laws that govern complex systems throughout the universe.