“A Refutation of Metcalfe’s Law and a Better Estimate for the Value of Networks and Network Interconnections”, 2005-03-02 (; backlinks):
[published later, minimally changed, as et al 2006] Metcalfe’s Law states that the value of a communications network is proportional to the square of the size of the network. It is widely accepted and frequently cited. However, there are several arguments that this rule is a large overestimate. (Therefore Reed’s Law is even more of an overestimate, since it says that the value of a network grows exponentially, in the mathematical sense, in network size.)
This note presents several quantitative arguments that suggest the value of a general communication network of size n grows like n log(n). [Specifically: connections clearly diminish in value; some additional connections can be of negative value, like ones enabling email spam; and real-world networks often resist merger despite the presumed large incentive Metcalfe’s law establishes. That seems to be the whole of their ‘quantitative arguments’…?]
This growth rate is faster than the linear growth, of order n, that, according to Sarnoff’s Law, governs the value of a broadcast network. On the other hand, it is much slower than the quadratic growth of Metcalfe’s Law, and helps explain the failure of the dot-com and telecom booms, as well as why network interconnection (such as peering on the Internet) remains a controversial issue.