Metcalfe's law

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Two telephones can make only one connection, five can make ten 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 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 follows 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.

In fact, Metcalfe's law measures 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.



[edit] 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_{n\rightarrow \infty} \frac{\frac{n(n-1)}{2}}{n^2}
= \lim_{n\rightarrow \infty} \left(\frac{n^2}{2} -\frac{n}{2}\right) * \frac{1}{n^2}
= \lim_{n\rightarrow \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 n2.

[edit] See also

[edit] External links

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