Metcalfe's law

Metcalfe's law states that the value of a telecommunications network is proportional to the square of the number of users of the system (n²). First formulated by Robert Metcalfe in regard to Ethernet, Metcalfe's law explains many of the network effects of communication technologies and networks such as the Internet, social networking, and the World Wide Web. It is related to the fact that the number of unique connections in a network of a number of nodes (n) can be expressed mathematically as n(n-1)/2, which is proportional to n² asymptotically.

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.

Metcalfe's law is more of a heuristic or metaphor than an iron-clad empirical rule. In addition to the difficulty of quantifying the "value" of a network, the mathematical justification measures only the potential number of contacts, i.e., the technological side of a network. However the social utility of a network depends upon the number of nodes in contact. For instance, if Chinese and non-Chinese users don't understand each other, the utility of a network of users that speak the other language is at zero, and the law has to be calculated for the two networks separately.

The n² Growth

A graph that has a number of edges, q, can only have edges

$0\; le\; q\; le\; binom\{n\}\{2\}$

Where n is the number of vertexes in the graph. By definition,

$binom\{n\}\{2\}\; =\; frac\{n(n-1)\}\{2\}$

By using limits and the ratio of the maxium number of edges to n²

$lim\_\{nrightarrow\; infty\}\; frac\{frac\{n(n-1)\}\{2\}\}\{n^2\}$

$=\; lim\_\{nrightarrow\; infty\}\; left(frac\{n^2\}\{2\}\; -frac\{n\}\{2\}right)\; *\; frac\{1\}\{n^2\}$

$=\; lim\_\{nrightarrow\; infty\}\; frac\{1\}\{2\}\; -\; frac\{1\}\{2n\}$

The second term goes to zero as n goes to infinity leaving only a constant which implies that the number of unique connections follows $n^2$.

See also

• The generalized Network effect of microeconomics.
• Reed's law
• Sarnoff's law
• List of eponymous laws

References

External links

• Metcalfe's Law: More Misunderstood Than Wrong? A co-worker of Bob Metcalfe puts the IEEE Spectrum critique in perspective. Republished here
• Metcalfe's Law is Wrong Bob Briscoe, Andrew Odlyzko, and Benjamin Tilly, July 2006 IEEE Spectrum. Points out that Metcalfe's Law is wrong, that the value is closer to n log(n)
• Metcalfe’s Law Recurses Down the Long Tail of Social Networking by Bob Metcalfe
• ZDNet: Metcalfe's Law overshoots the mark
• Andrew Odlyzko and Benjamin Tilly paper
• Metcalfe's Law in Reverse, applying Metcalfe's law to form an argument in favour of large, unified networks.
• George Church. The Personal Genome Project. Molecular Systems Biology. 13 December 2005
• The Semantic Web and Metcalf's [sic] Law
• 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."