Metcalfe's law

File:Metcalfe-Network-Effect.svg

Two telephones can make only one connection, five can make 10 connections, and twelve can make 66 connections.

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 (Template:Var²). 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).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> It later became associated with users on the Ethernet after a September 1993 Forbes article by George Gilder.<ref>Template:Cite book</ref>

Network effects

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.<ref name="Briscoe Odlyzko Tilly">Template:Cite journal</ref> Mathematically, Metcalfe's Law shows that the number of unique possible connections in an <math>n</math>-node connection can be expressed as the triangular number <math>n(n-1)/2</math>, which is asymptotically proportional to <math>n^2</math>.

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.<ref>Template:Cite journal</ref> 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.

History and derivation

Metcalfe's law was conceived in 1983 in a presentation to the 3Com sales force.<ref>Template:Cite journal</ref> It stated Template:Var would be proportional to the total number of possible connections, or approximately Template:Var-squared.

The original incarnation was careful to delineate between a linear cost (Template:Var), non-linear growth(Template:Var²) and a non-constant proportionality factor affinity (Template:Var). The break-even point point where costs are recouped is given by:<math display="block">C \times n=A\times n(n-1)/2</math>At some size, the right-hand side of the equation Template:Var, value, exceeds the cost, and Template:Var describes the relationship between size and net value added. For large Template:Var, net network value is then:<math display="block">\Pi=n(A \times (n-1)/2 - C)</math>Metcalfe properly dimensioned Template:Var as "value per user". Affinity is also a function of network size, and Metcalfe correctly asserted that Template:Var must decline as Template:Var grows large. In a 2006 interview, Metcalfe stated:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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.{{#if:|{{#if:|}}

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Growth of Template:Var

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 Template:Var reaches a saturation point. With technologies, substitutes, competitors and technical obsolescence constrain growth of Template:Var. Growth of n is typically assumed to follow a sigmoid function such as a logistic curve or Gompertz curve.

Density

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 <math> c=2m / n </math>.

The maximum possible number of edges in a simple network (i.e. one with no multi-edges or self-edges) is <math> \binom{n}{2}=n(n-1)/2</math>. Therefore the density ρ of a network is the faction of those edges that are actually present is:

which for large networks is approximated by <math> \rho=c/n </math>.<ref>Template:Cite book</ref>

Limitations

Metcalfe's law assumes that the value of each node <math>n</math> is of equal benefit.<ref name="Briscoe Odlyzko Tilly" /> 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. Likewise, in social networks, if users that join later use the network less than early adopters, then the benefit of each additional user may lessen, making the overall network less efficient if costs per users are fixed.

Modified models

Within the context of social networks, many, including Metcalfe himself, have proposed modified models in which the value of the network grows as <math>n \log n</math> rather than <math>n^2</math>.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name="Briscoe Odlyzko Tilly" /> ReedTemplate:Non sequitur inline 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.<ref>Template:Cite journal</ref>

Validation in data

For more than 30 years, there was little concrete evidence in support of the law. Finally, in July 2013, Dutch researchers analyzed European Internet-usage patterns over a long-enough timeTemplate:Specify and found <math>n^2</math> proportionality for small values of <math>n</math> and <math>n \log n</math> proportionality for large values of <math>n</math>.<ref>Template:Cite journal</ref> A few months later, Metcalfe himself provided further proof by using Facebook's data over the past 10 years to show a good fit for Metcalfe's law.<ref>Template:Cite journal</ref>

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). The functions for the two sites were <math>V_\text{Tencent}=7.39\times10^{-9}\times n^2 </math> and <math>V_\text{Facebook}=5.70\times 10^{-9}\times n^{2}</math> respectively.<ref>Template:Cite journal</ref> One of the earliest mentions of the Metcalfe Law in the context of Bitcoin was by a Reddit post by Santostasi in 2014. He compared the observed generalized Metcalfe behavior for Bitcoin to the Zipf's Law and the theoretical Metcalfe result.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The Metcalfe's Law is a critical component of Santostasi's Bitcoin Power Law Theory.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> In a working paper, Peterson linked time-value-of-money concepts to Metcalfe value using Bitcoin and Facebook as numerical examples of the proof,<ref>Template:Cite journal</ref> 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.<ref>Template:Cite journal</ref>

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. For this, he cited Metcalfe's law to support this perspective. Tao further stated, "my whole career experience has been sort of the more connections equals just better stuff happening".<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

See also

• Anti-rival good
• Beckstrom's law
• List of eponymous laws
• Matching (graph theory)
• Matthew effect
• Pareto principle
• Reed's law
• Sarnoff's law

Further reading

• Template:Cite magazine

External links

• A Group Is Its Own Worst Enemy. Clay Shirky's keynote speech on Social Software at the O'Reilly Emerging Technology conference, Santa Clara, April 24, 2003. The fourth of his "Four Things to Design For" is: "And, finally, you have to find a way to spare the group from scale. Scale alone kills conversations, because conversations require dense two-way conversations. In conversational contexts, Metcalfe's law is a drag."