IEEE Spectrum: Metcalfe's Law is Wrong

Metcalfe's Law attempts to quantify this increase in                 value. It is named for no less a luminary than Robert M.                 Metcalfe, the inventor of Ethernet. During the Internet                 boom, the law was an article of faith with                 entrepreneurs, venture capitalists, and engineers,                 because it seemed to offer a quantitative explanation                 for the boom's various now-quaint mantras, like "network                 effects," "first-mover advantage," "Internet time," and,                 most poignant of all, "build it and they will come."

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.

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).

We admit that our n log(n) valuation of a                 communications network oversimplifies the complicated                 question of what creates value in a network; in                 particular, it doesn't quantify the factors that                 subtract from the value of a growing network, such as an                 increase in spam e-mail. Our valuation cannot be proved,                 in the sense of a deductive argument from first                 principles. But if we search for a cogent description of                 a network's value, then n log(n) appears to be the                 best choice. Not only is it supported by several                 quantitative arguments, but it fits in with observed                 developments in the economy

The fundamental                 flaw underlying both Metcalfe's and Reed's                 laws is in the assignment of equal value to all                 connections or all groups.

In general, connections                 are not all used with the same intensity. In fact, in                 large networks, such as the Internet, with millions and                 millions of potential connections between individuals,                 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, or at least to interconnect. These incentives would make isolated networks hard to explain.

To see this, consider two networks, each with n members. By Metcalfe's Law, each one's value is on the order of n ², so the total value of both of these separate networks is roughly 2n ². But suppose these two networks merge. Then we will effectively have a single network with 2n members, which, by Metcalfe's Law, will be worth (2n)² or 4n ²—twice as much as the combined value of the two separate networks.

Zipf's Law is one of those empirical rules that characterize a surprising range of real-world phenomena remarkably well. It says that if we order some large collection by size or popularity, the second element in the collection will be about half the measure of the first one, the third one will be about one-third the measure of the first one, and so on. In general, in other words, the kth-ranked item will measure about 1/k of the first one.

Although Zipf originally formulated his law to                 apply just to this phenomenon of word frequencies,                 scientists find that it describes a surprisingly wide                 range of statistical distributions, such as individual                 wealth and income, populations of cities, and even the                 readership of blogs.

That total value to you will be the sum of the                 decreasing 1/k values of all the                 other members of the network. So if your network has                     n                 members, this value will be proportional to 1 + 1/2 +                 1/3 +… + 1/(n–1), which                 approaches log(n). More precisely,                 it will almost equal the sum of log(n) plus a constant                 value. Of course, there are n-1 other members who                 derive similar value from the network, so the value to                 all n of                 you increases as n log(n).

Another important                         consideration is the degree to which groups that                         value each other highly are already contained                         within the networks being combined, a factor                         called clustering.

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