Bitcoin and Network Theory

In the previous month's installment, I covered the power law model, which was initially introduced by Italian physicist Giovanni Santostasi. To quickly recap, if you regress the logarithm of bitcoin’s price against the logarithm of time, you get a strong correlation, which indicates that bitcoin’s price adheres to a power law.

After chatting with Giovanni and poking around his research a bit more, I discovered that bitcoin addresses also conform to a power law. This suggests that not only does the price follow a power law, but the accumulation of bitcoin addresses over time does as well.

Giovanni’s latest findings on the power law contribute to a broader trend of applying network theory to model bitcoin. The Bitcoin network plays a pivotal role in maintaining decentralized consensus. Much like the internet adheres to Metcalfe’s Law, so does bitcoin’s value increase as the network expands.

There are different ways to gauge the size of a network. Traditionally, this is done by counting the number of nodes within the network, where a full node refers to a Bitcoin machine that stores a complete blockchain copy and validates transactions as they spread through the network. Giovanni adopts a broader definition of the network, considering each bitcoin address as a node, with transactions between addresses as links. This expanded viewpoint results in a vastly greater number of nodes, as the potential quantity of bitcoin addresses is theoretically infinite. Anyone can generate a bitcoin address in a permissionless manner by creating a public-private key pair.

Now, there are some considerations to keep in mind when using bitcoin addresses as nodes. Certain behaviors can inflate the number of bitcoin addresses without genuinely increasing bitcoin’s adoption. For example, if a user transfers 10 bitcoin from a single address to 10 addresses they control, each holding one bitcoin, it doesn’t reflect increased adoption, yet the number of addresses rises. Similarly, using a mixing service that redistributes bitcoin to new addresses doesn’t signify more active network usage but technically expands the network’s address-count.

Excluding such edge cases, the count of bitcoin addresses should serve as a reasonable indication of bitcoin’s usage. While the relationship might not be strictly linear, in general, greater engagement with the Bitcoin network should correlate with an increase in bitcoin addresses over time.

Causation versus Correlation

That said, can the power law tell us the cause of bitcoin’s value? No. The power law functions as a reduced-form statistical model that establishes a relationship between external metrics of bitcoin (price, time, addresses, etc.). It does not elucidate the underlying economic factors that influence these metrics. Thus, despite the increase in bitcoin addresses over time, the power law does not clarify why there has been a rise in the creation of bitcoin addresses.

To achieve that understanding, economists would require a “structural” model of bitcoin rather than a “reduced-form” statistical model like the power law. A structural model would pinpoint essential economic constructs that govern the buying and selling dynamics of bitcoin. The price of bitcoin is determined in markets through the laws of supply and demand, similar to all market behaviors. Hence, to genuinely explain bitcoin’s value and price, one must break down what drives individuals to purchase bitcoin.

To illustrate a different perspective, consider analyzing Nvidia’s stock price over the past few years. You could create graphs comparing price to time, log price to log time, log price to time, or various other transformations. While these would provide statistical representations of price, they are not indicative of causality. The true causal factor we recognize is the demand for neural networks. However, calculating the impact of neural networks in a regression alongside Nvidia’s stock price is a complex process. Nevertheless, this does not undermine the reality that neural networks represent the core technology fueling generative AI, which in turn drives the demand for the specialized computing capabilities Nvidia delivers to the market. For Bitcoin, scarcity embodies that causal factor.

Nonetheless, there is potential. It might be possible to construct a structural economic model of bitcoin demand at a more abstract level. Imagine categorizing bitcoin buyers into four groups: short-term traders, long-term holders, corporations, and nation-states. Each of these categories possesses distinct objectives, time horizons, budgets, and risk profiles. Typically, long-term holders are the first to buy, followed by corporations and then nation-states, with short-term traders interspersed throughout. Long-term holders may influence the bitcoin price level as measured by, say, a 180-day moving average, while short-term traders dictate fluctuations on a weekly or monthly basis.

I am optimistic that a more nuanced agent-based model could enhance the understanding provided by the power law. This presents an exciting frontier for research intertwining physical and social sciences, much like Bitcoin itself.