Metcalfe’s Law in Bitcoin
Cryptocurrencies are a new asset class, and researchers have just started to understand better fundamental forces which are behind their price action. A new research paper shows that Bitcoin’s price can be modeled by Metcalfe’s Law. Bitcoin (and other cryptocurrencies) are in this characteristic very similar to Facebook as their value depends on the number of active users – network size

Authors: Peterson

Title: Bitcoin Spreads Like a Virus



We illustrate, by way of example, that Bitcoin’s long-term price is non-random and can be modeled as a function of the logistic growth of number of users n over time. Using observed data for both Facebook and Bitcoin, we derive the relationships between price, number of users, and time, and show that the resulting market capitalizations likely follow a Gompertz sigmoid growth function. This function, historically used to describe the growth of biological organisms like bacteria, tumors, and viruses, likely has some application to network economics. We conclude that the long-term growth rate in users has considerable effect on the long-term price of bitcoin.

Notable quotations from the academic research paper:

“This paper offers a simple explanation of price formation in the burgeoning and oft misunderstood cryptocurrency ecosystem. Using bitcoin as an example, we provide convincing empirical evidence that price formation is not a semi-random result of emotional investing but instead is founded on economic principles of value that have only recently begun to be recognized: network economics.

An examination of bitcoin prices offers some interesting observations that directly counter the “value is a mystery” myth. The first is that, as proponents have long argued, the value of a currency is primarily dependent upon use and acceptance of that currency. This hypothesis has been tested and is also apparent from a cursory examination of the relationship between bitcoin’s price and activity associated with the Bitcoin payment network. Specifically, price changes tend to be highly correlated with changes in number of wallets, active addresses, unique addresses, and transaction activity.

Metcalfe’s law is based on the mathematical tautology describing connectivity among n users. Hence, network value V is a function of number of users n. The underlying mathematics for Metcalfe’s law is based on pair-wise connections (e.g., telephony). If there are four people with telephones in a network, there could be a total of 3 + 2 + 1 = 6 connections. As more people join a network, they add to the value of the network nonlinearly; i.e., the value of the network is proportional to the square of the number of users. Metcalfe value V = n * (n-1) / 2.

Facebook and bitcoin price
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Facebook is ideally suited to comparison to Bitcoin. The lengths of each data series are nearly the same (about ten years). Both were fairly innovative, though not entirely original (Digicash preceded Bitcoin, MySpace preceded Facebook.) Both faced bans in China, a large potential marketplace. Both received widespread publicity about their adoption. It is rare that one has an opportunity to view the gradual adoption of a currency (or other asset) over time. This is in part because many companies are private during their development stages. But using Equation 1 with well-known Facebook we can see growth follows a pattern similar to that of Bitcoin (Figure 1). Facebook pricing prior to the IPO is unavailable. The value can only be estimated with Metcalfe’s law, and Figure 1 provides the groundwork for that analysis.

daviations from metcalfe's law

Referring back to Figure 3, there are three notable exceptions where bitcoin’s price deviated from the parabolic trend. These are periods of documented price manipulation and eventual resolution to equilibrium. These peaks represent price deviations that can not be accounted for by user-related factors. User related factors are those that a driven by user growth or network usage. Such factors include transactions, active accounts, wallets, nodes, and hash rates. Each of these factors are highly correlated with each other and the effects of changes in these factors are reflected in Metcalfe Value. What remains must be attributable to other factors, which we term non-economic factors. Non-economic factors include wash trading, falsified trade volumes (known as “painting the tape”) and perhaps other activities that may include behavioral impacts.

If n grows at a constant rate, then log(n) is linear. Since we observe for both Facebook and Bitcoin (Figure 7) that log(n) is nonlinear, n grows at a non-constant rate, indicating stages of adoption. This pattern of growth rates, on a cumulative basis, gives rise to a sigmoid function (Gompertz function) – a logistic function that for decades has been used to model viral infection, bacterial growth, tumor growth, and mobile phone proliferation. The most notable application of a Gompertz function to bitcoin pricing to date is Peterson [2018] who used it to model Metcalfe’s affinity coefficient. Using daily data for n proxied as active accounts, we have fit this equation to active addresses in Figure 7.

Grow of bitcoin

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