There's a fascinating short article in IEEE Spectrum this month summarizing an interesting debate about Metcalfe's Law. Metcalfe's Law says that the value of a communication technology grows with the square of the number of users of that technology. Informally, the rapid growth is because the value grows with the number of users with whom you can communicate.
The debate started with a July feature article by Briscoe, Odlyzko, and Tilly that argues that Metcalfe's Law overstates the growth rate. The heart of the argument is that the growth is actually proportional to the value of the users with whom you can communicate, and that the value of being able to communicate with users probably falls off like a power law function. (Many features of social networks fall off with a power function. Some people believe that this is a fundamental law of social networks, that separates them from other types of networks in which value falls off more slowly.) If the value of new people to communicate with falls off as a power law, then the value of the communications network scales as n * log (n) — much, much slower than n squared.
There's now an interesting debate going on at VC Mike's Blog on this issue.