# Metcalfe’s Law is Wrong

Making the Connection

A network grows not only by the addition of single members. It can jump in size by interconnecting or merging with another network. If two networks are of similar size, each would see roughly the same increase in value if they combined. You would expect them to interconnect, and indeed, networks of comparable size do so routinely. But when one network is much bigger than the other, the larger one usually resists interconnecting.

If Metcalfe’s Law were true, no matter what the relative sizes of two networks, both would gain the same amount by uniting, making the observed behavior seem irrational [see table below, “The Value of Interconnecting”]. If our n log(n) law holds (or other laws with growth rates falling between ours and Metcalfe’s), then, as the table shows, the smaller network would gain more than the larger one. For example, if the larger network is eight times as large as the smaller one, its gain would be less than half that of the smaller one. This clearly reduces the incentive for the owners of the larger network to interconnect without compensation.

This model of network interconnection is simplistic, of course, and it does not deal with other important aspects that enter into actual negotiations, such as a network’s geographical span, its balance of outgoing and incoming traffic, and any number of additional factors. All we are trying to show is that there may be sound economic reasons, besides raw market power, for larger networks to demand payment for interconnection with smaller ones. In fact, that’s a common phenomenon in real life.

Our n log( n ) law describes best the increase in value of a single network as it grows through acquisition of individual members. But our law should not be applied directly to evaluate the effects of connecting separate networks. 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.

When clustering is weak, the people you tend to communicate with the most--family members, work colleagues, fellow hobbyists, and so on--are not on the same network you are. In these cases, the value of connecting separate networks can be higher than our n log(n) law predicts.

Nonetheless, given that most networks grow organically, with people drawing in the people closest to them, the majority of networks are strongly clustered. Therefore, in most networks, n log(n) appears to be the best simple description of network value in terms of the network’s size.

B.B., A.O. & B.T.