A Deeper Dive into the Power Law Model for Bitcoin

Bitcoin addresses follow a power law as well. But it's still all about scarcity.

Last month, I reviewed the power law model, first proposed by the Italian physicist Giovanni Santostasi. Recall that regressing the logarithm of bitcoin's price against the logarithm of time generates a tight fit, and this is consistent with bitcoin's price following a power law. After speaking to Giovanni and reading more of his work, it turns out that bitcoin addresses also follow a power law. Therefore, not only price but also the number of addresses over time fits a power law.

In fact, Giovanni's recent research on the power law is part of a broader trend to use network theory to model bitcoin. As we know, the Bitcoin network is critical for enforcing decentralized consensus. Just as the internet follows Metcalfe's law, so too does the value of bitcoin increase in the size of the network.

There are several different ways to measure the size of a network. The traditional method is to use the number of nodes on the network, where a full node is the Bitcoin machine that keeps a copy of the full blockchain on disk, and validates all transactions as they propagate through the network. Giovanni uses a broader notion of the network with a bitcoin address as the node, and transactions between addresses as links. This more generalized approach will have many more nodes, since the number of bitcoin addresses is theoretically infinite. Anyone can create a bitcoin address in a permissionless way by generating a public-private key pair.

Using bitcoin addresses as nodes does have some consequences that need to be carried in mind. There are some simple examples of behaviors that increase bitcoin addresses without actually increasing bitcoin adoption. For example, suppose a single user with 10 bitcoins in a single address sends that bitcoin to 10 addresses that he controls with one bitcoin each. This would not increase bitcoin adoption, but would increase the number of addresses. Similarly, using a mixing service that recycles bitcoin by sending it to new addresses would also not represent an increase in the usage of the network, but would technically increase the size of the network if it is measured in addresses.

Aside from edge cases like these, the number of addresses should be a rough proxy for the usage of bitcoin. The relationship may not be one-to-one, but it should be in the right direction, where more usage of the Bitcoin network leads to more bitcoin addresses over time.

Causation versus Correlation

That said, does the power law establish what causes bitcoin's value? No. The power law is a statistical model that establishes a fit between external measures of bitcoin (price, time, addresses, etc). It does not provide the underlying economic forces that drive those measures. So even though bitcoin's addresses have increased over time, the power law does not explain why people have created more bitcoin addresses over time.

That would require what economists would call a “structural” model of bitcoin, as opposed to a “reduced-form” statistical model like the power law. A structural model would identify some core economic constructs that determine the buying and selling of bitcoin. The value of bitcoin, based on its price, is mediated in markets through supply and demand, like all markets. Therefore, to truly explain bitcoin's value, and therefore its price, it is imperative to explain what leads individuals to buy bitcoin.

To see a slightly different example, imagine you seek to explain Nvidia's stock price over the last few years. You could graph price against time, log price against log time, log price against time, or any other transformation. Those would all be statistical representations of price, but they are not causal. The real causal effect that we all know about is the demand for neural networks. However, quantifying the neural network in a regression that includes Nvidia stock price is a messy business. But that does not discount the truth that neural networks are the underlying technology driving generative AI, which drives the demand for the accelerated computing that Nvidia uniquely provides to the market. For Bitcoin, scarcity is that causal effect.

But all is not lost. It may be possible to build a structural economic model of bitcoin demand at a slightly higher level of abstraction. Imagine buyers of bitcoin in four categories: short-term traders, long-term holders, corporations, and nation-states. Each of these groups has their own objectives, time preferences, budgets, and risk appetites. The long-term holders buy first, then the corporations, then the nation-states, while the short-term traders intersperse throughout. The long-term holders may drive the level of bitcoin price measured, say, by a 180-day moving average, while the short-term traders determine short-term fluctuations week to week or month to month.

I'm optimistic that a more comprehensive agent-based model could amplify the power law. This is an exciting area for future research that combines the physical and social sciences, much like Bitcoin itself.