Metcalfe's Law

Metcalfe's law is the idea that the value of a communications network grows in proportion to the square of the number of users connected to it. It is named after Robert Metcalfe, co-inventor of Ethernet, who used the argument in the 1980s to explain why networks of computers and devices become far more useful as they grow; the name "Metcalfe's law" was popularised by the writer George Gilder in the 1990s. It is less a law of nature than a rule of thumb about the network effect.

The law is widely quoted in discussions of telephone systems, social media, payment systems and cryptocurrencies, where investors use it to argue that adoption drives value. It sits alongside other technology rules of thumb such as Moore's law and the law of accelerating returns, and like them it has been criticised for overstating what growth alone can deliver.

One telephone on its own is useless, because there is nobody to call. Two phones make one possible conversation. Five phones make ten possible pairs, and ten phones make forty-five. Each new person who joins can talk to everyone already there, so the number of possible conversations grows much faster than the number of people.

Metcalfe's law takes that observation and says the usefulness, or value, of the network grows at roughly the same speed as the number of possible connections. That is why companies often fight hard to be the largest network in their market: once enough people use one service, joining a smaller rival makes little sense.

Metcalfe originally made the argument to sell networking equipment: he pointed out that the cost of a network rose roughly in line with the number of devices, while its value rose with the number of connections, so beyond a certain size a network would pay for itself. The idea spread during the growth of the commercial internet in the 1990s and was used to justify heavy spending on users ahead of profits. Some commentators later blamed that reasoning for the inflated valuations of the dot-com bubble, when companies with large audiences but no clear path to earnings collapsed.

The law also helps explain why some digital markets become winner-takes-all markets. If value rises faster than size, the largest network gains an advantage that is hard for rivals to overcome, which is one reason competition authorities pay close attention to dominant platform companies.

The number of unique pairs that can connect in a network of $n$ users is:

For large networks this is close to $n^2 \div 2$, so Metcalfe's law is usually written as:

• $V$ is the value of the network.
• $n$ is the number of connected users or devices.
• $k$ is a constant that converts connections into money and must be estimated from data.

For example, a network with 1,000 users has 1,000 × 999 ÷ 2 = 499,500 possible connections. If it grows to 2,000 users, the number becomes 2,000 × 1,999 ÷ 2 = 1,999,000, about four times as many. Under Metcalfe's law, doubling the user base roughly quadruples the network's value.

• Technology valuations. Analysts have compared the revenues or market values of social networks with the square of their user numbers, and some studies have found rough fits for large platforms over certain periods.
• Payment networks. The value of card networks and mobile money systems rises as more merchants and users join, a classic two-sided version of the effect, seen in the rapid spread of mobile money services once a critical mass of agents and users was reached.
• Cryptocurrency valuation. Researchers and analysts have modelled the market value of Bitcoin and other coins against measures of users, such as daily active addresses, sometimes finding that price has tracked a Metcalfe-style curve over long periods. Ratios such as network value to transactions draw on similar thinking.

Such models compare market capitalization with network activity to judge whether a coin looks expensive or cheap relative to its use, but they rely on the choice of user measure and on the constant $k$, both of which are open to debate.

Critics argue that Metcalfe's law overstates value because not all connections are equally valuable. Most people interact with a small circle, so a new member adds less than the formula suggests. The researchers Andrew Odlyzko and Benjamin Tilly argued in the 2000s that value grows more like $n \log n$, a much slower rate. Large networks also suffer from congestion, spam, fraud and moderation costs that the formula ignores, and value can drop sharply if users leave for a competitor.

In crypto, counting users is especially difficult. One person can control thousands of addresses, while an exchange may hold coins for millions of people in a few wallets, and some activity is generated by bots or wash trading. Fits between price and user numbers can therefore be coincidental, and they say nothing about cause: rising prices can attract users as easily as users can raise prices. For these reasons, Metcalfe's law is best treated as a useful intuition about the network effect rather than as a precise valuation tool.