Metcalfe's Law and Bitcoin: Network Effect and Valuation

Metcalfe's law says: the value of a network grows roughly with the square of its participants. Twice as many users means about four times the value. Applied to Bitcoin: if the number of users grows steadily as a power of time, the value grows with twice that power. That is exactly how the Bitcoin Power Law can be explained. It is not a proof, because nobody can count Bitcoin users cleanly.

Anyone who knows the Bitcoin Power Law eventually asks: why should the price follow such a simple curve at all? A common answer is the network effect, and the best-known formula for it is Metcalfe's law. This article explains what the law says, how it is applied to Bitcoin, how it connects to the Power Law and where it can lead you astray.

What Metcalfe's law says

Robert Metcalfe co-invented Ethernet, the technology behind almost every wired network. Around 1980 he wanted to convince companies to buy network cards. His argument: the cost of a network rises only a little with each new device. The benefit rises faster, because each new device can talk to all the others. The name "Metcalfe's law" was coined later by the writer George Gilder.

Think of telephones. A single telephone is useless. Two telephones make one connection. Five telephones already make ten possible connections, ten telephones 45. The number of possible connections grows much faster than the number of devices. If every connection has some benefit, the value of the network grows just as fast.

For large networks the factor for doubling the size is almost exactly 4. That is why it is written in short form as: value ~ n².

Not every connection is worth the same

The law met objections early on. In 2006 the mathematicians Andrew Odlyzko, Bob Briscoe and Benjamin Tilly argued that n² overstates the value of large networks. You do not call everyone on the network, mostly just a few people. The thousandth connection is worth less than the first. They proposed n × log(n), a growth faster than n but much slower than n².

In 2013 Metcalfe himself tested his formula against Facebook's user numbers and revenue and found that n² described the revenue well over several years. The debate is open. Only this much is clear: networks become disproportionately more useful with more participants. How much exactly depends on the network.

How Metcalfe's law is applied to Bitcoin

Bitcoin is a network for money. Anyone who holds or accepts bitcoin can pay and be paid by every other participant. The more people use Bitcoin, the more useful it is for each of them: more merchants, more buyers, deeper markets, more people who recognise it as valuable. That is exactly the kind of network effect Metcalfe describes.

A calculation needs a number for n, though. Because Bitcoin has no accounts with names, analysts use stand-ins:

• Active addresses per day: how many Bitcoin addresses appear as sender or receiver in a transaction on a given day.
• Addresses with a balance: how many addresses hold more than zero or more than a minimum amount.
• Transactions per day: how often the network is used.
• Estimated users: extrapolations from surveys or exchange accounts.

For V, people usually take the market capitalisation, the price times the amount of bitcoin mined so far. Several studies have shown that log(market capitalisation) and log(active addresses) move closely together over long periods, for example Wheatley and colleagues in Royal Society Open Science in 2019. They did not fix the exponent at 2 but estimated it from the data, a "generalised" Metcalfe's law. How steep the relationship turns out depends on which addresses you count and which years you take.

A typical result of such studies: in bubbles the price swings far above the Metcalfe value and in bear markets falls back close to it. That strongly resembles the role the floor plays in the Power Law.

The bridge to the Power Law

The Power Law describes the price using only the time since the genesis block on 3 January 2009. The formula bitcoin-powerlaw.com calculates with is: price = 10^−16.493 × days^5.688. The exponent 5.688 means: if the time since the start doubles, the line's value is multiplied by about 52.

Metcalfe's law offers one possible reason for this. Suppose the number of users does not grow exponentially, by a fixed percentage per year, but as a power of time: n ~ days^k. Then by Metcalfe the value grows with n², that is with days^(2k). A power of a power is again a power. On a chart with two logarithmic axes, both are a straight line.

The physicist Giovanni Santostasi, who made the Power Law known in 2018, argues exactly this way: the number of users grows roughly with the third power of time, value with the square of users, so together roughly with the sixth power. The measured exponent is 5.7.

On this chart both axes are logarithmic. A power of time becomes a straight line here, and prices have followed it remarkably well since 2010.

What matters in this example: the yearly growth rate falls from year to year. With a power of time, each year is a smaller share of the time elapsed so far. That fits a network that is maturing. Exponential growth at a fixed percentage, by contrast, would never slow down.

The weaknesses

The idea is convincing. Measuring it is the problem. These are the main objections.

Addresses are not people

An address is only a field to receive into, not a person. The Bitcoin white paper already recommends using a new key pair for every transaction. Modern wallets create a new address for every receipt and send change to yet another. One person can have hundreds of addresses this way. Someone who uses Bitcoin carefully and privately creates many addresses. Someone who buys once and then holds almost never shows up in active addresses.

Exchanges and ETFs bundle millions of users

The opposite happens with custodians. An exchange with millions of customers holds their bitcoin in comparatively few addresses. Purchases and sales between its customers run only in its internal database, not on the blockchain. The same applies to Bitcoin ETFs: hundreds of thousands of investors, but the coins sit with a few custodians. The more people come in through such providers, the more address counts underestimate users. The number of addresses can therefore stagnate while the number of holders grows.

Lightning and other layers

Payments in the Lightning Network run outside the blockchain. A channel is opened once and closed once, and thousands of payments can flow in between that no address count sees. Use as a means of payment in particular becomes less and less visible in on-chain data.

Noise in the data

Not every transaction is real use. When fees are low, spam waves, experiments or new protocols on the blockchain can push up the number of active addresses sharply for a while without a single new saver joining. Conversely, exchanges bundle many withdrawals into one transaction.

Chicken and egg

Metcalfe's law says: more users, more value. With Bitcoin the reverse also holds: when the price rises, new users arrive, and speculators create transactions. That price and addresses rise together does not prove that the addresses drive the price.

No network grows forever

At about 8 billion people it ends. A number of users growing like t^2.8 cannot do so without limit. At some point growth has to slow, and with it, under Metcalfe, the value. This objection hits the Power Law just as much. More on this in Criticism of the Bitcoin Power Law.

What you can take from it as a holder

Metcalfe's law is not a tool for finding the right moment to buy. It explains why a money network can grow with its adoption over the long run. You can draw three things from it:

1. Value depends on adoption. As long as more people hold, accept and recognise Bitcoin, the long-term uptrend has a foundation. If adoption stalls, the Power Law loses its explanation too.
2. Growth slows, it does not speed up. A power of time means falling yearly returns. Do not plan with the gains of the early years.
3. The network effect favours the largest network. A money network is more useful the more people use it. That is one reason why Bitcoin has so far kept its lead over imitators.

What you cannot read from it:

• No fair price for tomorrow. The Metcalfe calculation needs a constant and a count of n, both uncertain. Small changes to them shift the "fair" value by a multiple.
• No signal from short-term address counts. A month with many active addresses can be spam, a month with few an inflow into ETFs.
• No proof of the Power Law. The derivation only shows that the curve fits a plausible story. Other stories fit too. Which models exist and how they differ is shown in Bitcoin price models compared.