The increasing value of a network as its size increases certainly lies somewhere between linear and exponential growth [see diagram, "Growth Curves"]. The value of a broadcast network is believed to grow linearly; it's a relationship called Sarnoff's Law, named for the pioneering RCA television executive and entrepreneur David Sarnoff. At the other extreme, exponential—that is, 2ⁿ—growth, has been called Reed's Law, in honor of computer networking and software pioneer David P. Reed. Reed proposed that the value of networks that allow the formation of groups, such as AOL's chat rooms or Yahoo's mailing lists, grows proportionally with 2ⁿ.

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 n log(n) valuation for a network provides a rough-and-ready description of the dynamics that led to the disappointingly slow growth in the value of dot‑com companies. On the other hand, because this growth is faster than the linear growth of Sarnoff's Law, it helps explain the occasional dot-com successes we have seen.

The fundamental flaw underlying both Metcalfe's and Reed's laws is in the assignment of equal value to all connections or all groups. The underlying problem with this assumption was pointed out a century and a half ago by Henry David Thoreau in relation to the very first large telecommunications network, then being built in the United States. In his famous book Walden (1854), he wrote: "We are in great haste to construct a magnetic telegraph from Maine to Texas; but Maine and Texas, it may be, have nothing important to communicate."

As it turns out, Maine did have quite a bit to communicate with Texas—but not nearly as much as with, say, Boston and New York City. 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.

There are common-sense arguments that suggest Metcalfe's and Reed's laws are incorrect. For example, Reed's Law says that every new person on a network doubles its value. Adding 10 people, by this reasoning, increases its value a thousandfold (2¹⁰). But that does not even remotely fit our general expectations of network values—a network with 50 010 people can't possibly be worth a thousand times as much as a network with 50 000 people.

At some point, adding one person would theoretically increase the network value by an amount equal to the whole world economy, and adding a few more people would make us all immeasurably rich. Clearly, this hasn't happened and is not likely to happen. So Reed's Law cannot be correct, even though its core insight—that there is value in group formation—is true. And, to be fair, just as Metcalfe was aware of the limitations of his law, so was Reed of his law's.

Metcalfe's Law does not lead to conclusions as obviously counterintuitive as Reed's Law. But it does fly in the face of a great deal of the history of telecommunications: 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.

Surely it would require a singularly obtuse management, to say nothing of stunningly inefficient financial markets, to fail to seize this obvious opportunity to double total network value by simply combining the two. Yet historically there have been many cases of networks that resisted interconnection for a long time. For example, a century ago in the United States, the Bell System and the independent phone companies often competed in the same neighborhood, with subscribers to one being unable to call subscribers to the other. Eventually, through a combination of financial maneuvers and political pressure, such systems connected with one another, but it took two decades.

Similarly, in the late 1980s and early 1990s, the commercial online companies such as CompuServe, Prodigy, AOL, and MCIMail provided e-mail to subscribers, but only within their own systems, and it wasn't until the mid-1990s that full interconnection was achieved. More recently we have had (and continue to have) controversies about interconnection of instant messaging systems and about the free exchange of traffic between Internet service providers. The behavior of network operators in these examples is hard to explain if the value of a network grows as fast as Metcalfe's n ².

There is a further argument to make about interconnecting networks. 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. Typically in such cases, 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. [See sidebar, "Making the Connection," for the mathematics behind this argument.]

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, named for the 20th-century linguist George Kingsley Zipf.