Some carefully selected excerpts:
Of all the popular ideas of the Internet boom, one of the most dangerously influential was Metcalfe's Law. Simply put, it says that the value of a communications network is proportional to the square of the number of its users.Well worth reading if you were ever a fan of Metcalfe's Law.
The foundation of his eponymous law is the observation that in a communications network with n members, each can make (n-1) connections with other participants. If all those connections are equally valuable ... the total value of the network is ... roughly n ^ 2.
We propose, instead, that the value of a network of size n grows in proportion to n log(n).
The fundamental flaw underlying ... Metcalfe's [Law] is in the assignment of equal value to all connections or all groups .... In fact, in large networks, such as the Internet ... most are not used at all. So assigning equal value to all of them is not justified.
This is our basic objection to Metcalfe's Law, and it's not a new one: it has been noted by many observers, including Metcalfe himself.
If Metcalfe's Law were true, it would create overwhelming incentives for all networks relying on the same technology to merge ... These incentives would make isolated networks hard to explain .... Yet historically there have been many cases of networks that resisted interconnection for a long time.
Further ... if Metcalfe's Law were true, then two networks ought to interconnect regardless of their relative sizes. But in the real world of business and networks, only companies of roughly equal size are ever eager to interconnect. In most cases, the larger network believes it is helping the smaller one far more than it itself is being helped ... The larger network demands some additional compensation before interconnecting. Our n log(n) assessment of value is consistent with this real-world behavior of networking companies; Metcalfe's n ^ 2 is not.
We have, as well, developed several quantitative justifications for our n log(n) rule-of-thumb valuation of a general communications network of size n. The most intuitive one is based on yet another rule of thumb, Zipf's Law .... To understand how Zipf's Law leads to our n 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 .... [The] total value to you will be the sum of the decreasing 1/k values of all the other members of the network.
By the way, the authors also have a fun bit in the article about the advantage Amazon.com and other online stores have over traditional stores:
Incidentally, this mathematics indicates why online stores are the only place to shop if your tastes in books, music, and movies are esoteric. Let's say an online music store like Rhapsody or iTunes carries 735 000 titles, while a traditional brick-and-mortar store will carry 10 000 to 20 000. The law of long tails says that two-thirds of the online store's revenue will come from just the titles that its physical rival carries. In other words, a very respectable chunk of revenue -- a third -- will come from the 720 000 or so titles that hardly anyone ever buys. And, unlike the cost to a brick-and-mortar store, the cost to an online store of holding all that inventory is minimal. So it makes good sense for them to stock all those incredibly slow-selling titles.[via Paul Kedrosky]