# Metcalfe's law

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

Metcalfe's law states that the value of a telecommunications network is proportional to the square of the number of connected users of the system (n2). First formulated in this form by George Gilder in 1993,[1] and attributed to Robert Metcalfe in regard to Ethernet, Metcalfe's law was originally presented, c. 1980, not in terms of users, but rather of "compatible communicating devices" (e.g., fax machines, telephones).[2] Only later with the globalization of the Internet did this law carry over to users and networks as its original intent was to describe Ethernet connections.[3]

## 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 Internet.[4] Metcalfe's Law is related to the fact that the number of unique possible connections in a network of ${\displaystyle n}$ nodes can be expressed mathematically 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 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.[5] Likewise, in social networks, 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.[6]  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 A “Affinity.”  The breakeven 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=A*n2, 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.”[7]

### Growth of n

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.

### 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 ${\displaystyle c=2m/n}$.

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

${\displaystyle \rho =c/(n-1)}$ which for large networks is approximated by ${\displaystyle \rho =c/n}$.[8]

## Limitations

Metcalfe’s law assumes that the value of each node ${\displaystyle n}$ is of equal benefit.[9] If this is not the case, for example because one fax machine serves 50 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 ${\displaystyle n\log n}$ rather than ${\displaystyle n^{2}}$.[10][11] Reed 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.[12]

## Validation in data

Despite many arguments about Metcalfe' law, no real data based evidence for or against was available for more than 30 years. Only in July 2013, Dutch researchers managed to analyze European Internet usage patterns over a long enough time and found ${\displaystyle n^{2}}$ proportionality for small values of ${\displaystyle n}$ and ${\displaystyle n\log n}$ proportionality for large values of ${\displaystyle n}$.[13] A few months later, Metcalfe himself provided further proof, as he used Facebook's data over the past 10 years to show a good fit for Metcalfe's law (the model is ${\displaystyle n^{2}}$).[14]

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 ${\displaystyle V_{Tencent}=7.39\times 10^{-9}\times n^{2}}$ and ${\displaystyle V_{Facebook}=5.70\times 10^{-9}\times n^{2}}$ respectively.[15]

In a working paper, Peterson linked time-value-of-money concepts to Metcalfe value using Bitcoin and Facebook as numerical examples of the proof[16] 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.[17]

## References

1. ^ Carl Shapiro and Hal R. Varian (1999). Information Rules. Harvard Business Press. ISBN 978-0-87584-863-1.
2. ^ Simeon Simeonov (July 26, 2006). "Metcalfe's Law: more misunderstood than wrong?". HighContrast: Innovation & venture capital in the post-broadband era.
3. ^ James Hendler and Jennifer Golbeck (2008). "Metcalfe's Law, Web 2.0, and the Semantic Web" (PDF).
4. ^ Bob Briscoe, Andrew Odlyzko and Benjamin Tilly (July 2006). "Metcalfe's Law is wrong" (PDF). Retrieved 2022-09-15.
5. ^ R. Tongia. "The Dark Side of Metcalfe's Law: Multiple and Growing Costs of Network Exclusion" (PDF). Retrieved 2017-12-19.
6. ^ Metcalfe, Bob (December 2013). "Metcalfe's Law after 40 Years of Ethernet". Computer. 46 (12): 26–31. doi:10.1109/MC.2013.374. ISSN 1558-0814.
7. ^ Metcalfe, Robert (18 August 2006). "Guest Blogger Bob Metcalfe: Metcalfe's Law Recurses down the Long Tail of Social Networks". VC Mike's Blog.{{cite web}}: CS1 maint: url-status (link)
8. ^ Newman, Mark E.J. (2019). "Mathematics of Networks" in Networks. Oxford: Oxford University Press. pp. 126–128. ISBN 978-0-19-880509-0.
9. ^ Andrew Odlyzko; Bob Briscoe (1 Jul 2006). "Metcalfe's Law is Wrong". IEEE Spectrum: Technology, Engineering, and Science News. Retrieved 25 November 2016.
10. ^ "Guest Blogger Bob Metcalfe: Metcalfe's Law Recurses Down the Long Tail of Social Networks". 18 August 2006. Retrieved 2010-06-20.
11. ^ B. Briscoe, A. Odlyzko, and B. Tilly, Metcalfe’s law is wrong, IEEE Spectrum 43:7 (2006), pp. 34–39.
12. ^ Rahul Tongia and Ernest Wilson (September 2007). "The Flip Side of Metcalfe's Law: Multiple and Growing Costs of Network Exclusion". Retrieved 2013-01-15.
13. ^ Madureira, António; den Hartog, Frank; Bouwman, Harry; Baken, Nico (2013), "Empirical validation of Metcalfe's law: How Internet usage patterns have changed over time", Information Economics and Policy, doi:10.1016/j.infoecopol.2013.07.002
14. ^ Metcalfe, Bob (2013). "Metcalfe's law after 40 years of Ethernet". IEEE Computer. 46 (12): 26–31. doi:10.1109/MC.2013.374.
15. ^ Zhang, Xing-Zhou; Liu, Jing-Jie; Xu, Zhi-Wei (2015). "Tencent and Facebook Data Validate Metcalfe's Law". Journal of Computer Science and Technology. 30 (2): 246–251. doi:10.1007/s11390-015-1518-1.
16. ^ Peterson, Timothy (2019). "Bitcoin Spreads Like a Virus". Working Paper. doi:10.2139/ssrn.3356098.
17. ^ Peterson, Timothy (2018). "Metcalfe's Law as a Model for Bitcoin's Value". Alternative Investment Analyst Review. 7 (2): 9–18. doi:10.2139/ssrn.3078248.