Metcalfe's law

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).

[1] It later became associated with users on the Ethernet after a September 1993 Forbes article by George Gilder.

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.

[3] Mathematically, Metcalfe's Law shows that the number of unique possible connections in an

Metcalfe's law was conceived in 1983 in a presentation to the 3Com sales force.

[5] 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).

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:[6] 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.

The maximum possible number of edges in a simple network (i.e. one with no multi-edges or self-edges) is

[3] If this is not the case, for example because one fax machine serves 60 workers in a company, the second fax machine serves half of that, the third one third, and so on, then the relative value of an additional connection decreases.

[8][3] Reed[non sequitur] and Andrew Odlyzko have sought out possible relationships to Metcalfe's Law in terms of describing the relationship of a network and one can read about how those are related.

Tongia and Wilson also examine the related question of the costs to those excluded.

Finally, in July 2013, Dutch researchers analyzed European Internet-usage patterns over a long-enough time[specify] and found

[11] In 2015, Zhang, Liu, and Xu parameterized the Metcalfe function in data from Tencent and Facebook.

Their work showed that Metcalfe's law held for both, despite differences in audience between the two sites (Facebook serving a worldwide audience and Tencent serving only Chinese users).

[12] One of the earliest mentions of the Metcalfe Law in the context of Bitcoin was by a Reddit post by Santostasi in 2014.

[14] In a working paper, Peterson linked time-value-of-money concepts to Metcalfe value using Bitcoin and Facebook as numerical examples of the proof,[15] and in 2018 applied Metcalfe's law to Bitcoin, showing that over 70% of variance in Bitcoin value was explained by applying Metcalfe's law to increases in Bitcoin network size.

[16] In a 2024 interview, mathematician Terence Tao emphasized the importance of universality and networking within the mathematics community, for which he cited the Metcalfe's Law.

Tao believes that a larger audience leads to more connections, which ultimately results in positive developments within the community.

Tao further stated, "my whole career experience has been sort of the more connections equals just better stuff happening".