Metcalfe's law states that the financial value or influence of a telecommunications network is proportional to the square of the number of connected users of the system (n²). The law is named after Robert Metcalfe and was first proposed in 1980, albeit not in terms of users, but rather of "compatible communicating devices" (e.g., fax machines, telephones). It later became associated with users on the Ethernet after a September 1993 Forbes article by George Gilder.

Metcalfe's law characterizes many of the network effects of communication technologies and networks such as the Internet, social networking and the World Wide Web. Former Chairman of the U.S. Federal Communications Commission Reed Hundt said that this law gives the most understanding to the workings of the present-day Internet. Mathematically, Metcalfe's Law shows that the number of unique possible connections in an ${\displaystyle n}$-node connection can be expressed as the triangular number ${\displaystyle n(n-1)/2}$, which is asymptotically proportional to ${\displaystyle n^{2}}$.

The law has often been illustrated using the example of fax machines: a single fax machine on its own is useless, but the value of every fax machine increases with the total number of fax machines in the network, because the total number of people with whom each user may send and receive documents increases. This is common illustration to explain network effect. Thus, in any social network, the greater the number of users with the service, the more valuable the service becomes to the community.

Metcalfe's law was conceived in 1983 in a presentation to the 3Com sales force. It stated V would be proportional to the total number of possible connections, or approximately n-squared.

The original incarnation was careful to delineate between a linear cost (Cn), non-linear growth(n²) and a non-constant proportionality factor affinity (A). The break-even point point where costs are recouped is given by:$${\displaystyle C\times n=A\times n(n-1)/2}$$At some size, the right-hand side of the equation V, value, exceeds the cost, and A describes the relationship between size and net value added. For large n, net network value is then:$${\displaystyle \Pi =n(A\times (n-1)/2-C)}$$Metcalfe properly dimensioned A as "value per user". Affinity is also a function of network size, and Metcalfe correctly asserted that A must decline as n grows large. In a 2006 interview, Metcalfe stated:

There may be diseconomies of network scale that eventually drive values down with increasing size. So, if V = An², it could be that A (for “affinity,” value per connection) is also a function of n and heads down after some network size, overwhelming n².

Network size, and hence value, does not grow unbounded but is constrained by practical limitations such as infrastructure, access to technology, and bounded rationality such as Dunbar's number. It is almost always the case that user growth n reaches a saturation point. With technologies, substitutes, competitors and technical obsolescence constrain growth of n. Growth of n is typically assumed to follow a sigmoid function such as a logistic curve or Gompertz curve.

A is also governed by the connectivity or density of the network topology. In an undirected network, every edge connects two nodes such that there are 2m nodes per edge. The proportion of nodes in actual contact are given by ${\displaystyle c=2m/n}$.