Metcalfe's Law was over a dozen years old when Gilder named it. As Metcalfe himself remembers it, in a private correspondence with one of the authors, "The original point of my law (a 35mm slide circa 1980, way before George Gilder named it...) 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." [See "To the Point," a reproduction of Metcalfe's historic slide.]

Metcalfe was ideally situated to watch and analyze the growth of networks and their profitability. In the 1970s, first in his Harvard Ph.D. thesis and then at the legendary Xerox Palo Alto Research Center, Metcalfe developed the Ethernet protocol, which has come to dominate telecommunications networks. In the 1980s, he went on to found the highly successful networking company 3Com Corp., in Marlborough, Mass. In 1990 he became the publisher of the trade periodical InfoWorld and an influential high-tech columnist. More recently, he has been a venture capitalist.

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—and this is the big "if" as far as we are concerned—the total value of the network is proportional to n(n–1), that is, roughly, n ². So if, for example, a network has 10 members, there are 90 different possible connections that one member can make to another. If the network doubles in size, to 20, the number of connections doesn't merely double, to 180, it grows to 380—it roughly quadruples, in other words.

If Metcalfe's mathematics were right, how can the law be wrong? Metcalfe was correct that the value of a network grows faster than its size in linear terms; the question is, how much faster? If there are n members on a network, Metcalfe said the value grows quadratically as the number of members grows.

We propose, instead, that the value of a network of size n grows in proportion to n log(n). Note that these laws are growth laws, which means they cannot predict the value of a network from its size alone. But if we already know its valuation at one particular size, we can estimate its value at any future size, all other factors being equal.

The distinction between these laws might seem to be one that only a mathematician could appreciate, so let us illustrate it with a simple dollar example.

ILLUSTRATION: SERGE BLOCH

Imagine a network of 100 000 members that we know brings in $1 million. We have to know this starting point in advance—none of the laws can help here, as they tell us only about growth. So if the network doubles its membership to 200 000, Metcalfe's Law says its value grows by (200 000²/100 000²) times, quadrupling to $4 million, whereas the n log(n) law says its value grows by 200 000 log(200 000)/100 000 log(100 000) times to only $2.1 million. In both cases, the network's growth in value more than doubles, still outpacing the growth in members, but the one is a much more modest growth than the other. In our view, much of the difference between the artificial values of the dot-com era and the genuine value created by the Internet can be explained by the difference between the Metcalfe-fueled optimism of n ² and the more sober reality of n log(n).

This difference will be critical as network investors and managers plan better for growth. In North America alone, telecommunications carriers are expected to invest $65 billion this year in expanding their networks, according to the analytical firm Infonetics Research Inc., in San Jose, Calif. As we will show, our rule of thumb for estimating value also has implications for companies in the important business of managing interconnections between major networks.