Metcalfe's Law says that the value of a communications network is proportional to the square of the number of its users. IEEE Spectrum explain that it is wrong. In March 2005, Andrew Odlyzko and Benjamin Tilly published a preliminary paper which concludes Metcalfe's law significantly overestimates the value of adding connections. The rule of thumb becomes: "the value of a network with n members is not n squared, but rather n times the logarithm of n." Their primary justification for this is the idea that not all potential connections in a network are equally valuable. For example, most people call their families a great deal more often than they call strangers in other countries, and so do not derive the full value n from the phone service.
Metcalfe's original point (from a 35mm slide circa 1980) was to establish the existence of a cost-value crossover point—critical mass—before which networks don't pay. The trick is to get past that point, to establish critical mass.
David P Reed proposed that the value of networks that allow the formation of groups, such as AOL's chat rooms or Yahoo's mailing lists, grows proportionally with 2**n.
There are common-sense arguments that suggest Metcalfe's and Reed's laws are incorrect. For example, Reed's Law says that every new person on a network doubles its value. Adding 10 people, by this reasoning, increases its value a thousandfold (2**10). But that does not even remotely fit our general expectations of network values—a network with 50 010 people can't possibly be worth a thousand times as much as a network with 50 000 people.
the N* log N rule is related to Zipf's law and the long tail.
To understand how Zipf's Law leads to the log(n) law, consider the relative value of a network near and dear to you—the members of your e-mail list. Obeying, as they usually do, Zipf's Law, the members of such networks can be ranked in the same sort of way that Zipf ranked words—by the number of e-mail messages that are in your in-box. Each person's e-mails will contribute 1/k to the total "value" of your in-box, where k is the person's rank.