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.

To take one example, in a typical large body of English-language text, the most popular word, "the," usually accounts for nearly 7 percent of all word occurrences. The second-place word, "of," makes up 3.5 percent of such occurrences, and the third-place word, "and," accounts for 2.8 percent. In other words, the sequence of percentages (7.0, 3.5, 2.8, and so on) corresponds closely with the 1/k sequence (1/1, 1/2, 1/3…). 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.

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 in the same sort of way that Zipf ranked words—by the number of e-mail messages that are in your in-box. Each person's e-mails will contribute 1/k to the total "value" of your in-box, where k is the person's rank.

The person ranked No. 1 in volume of correspondence with you thus has a value arbitrarily set to 1/1, or 1. (This person corresponds to the word "the" in the linguistic example.) The person ranked No. 2 will be assumed to contribute half as much, or 1/2. And the person ranked kth will, by Zipf's Law, add about 1/k to the total value you assign to this network of correspondents.

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).

Zipf's Law can also describe in quantitative terms a currently popular thesis called The Long Tail. Consider the items in a collection, such as the books for sale at Amazon, ranked by popularity. A popularity graph would slope downward, with the few dozen most popular books in the upper left-hand corner. The graph would trail off to the lower right, and the long tail would list the hundreds of thousands of books that sell only one or two copies each year. The long tail of the English language—the original application of Zipf's Law—would be the several hundred thousand words that you hardly ever encounter, such as "floriferous" or "refulgent."

Taking popularity as a rough measure of value (at least to booksellers like Amazon), then the value of each individual item is given by Zipf's Law. That is, if we have a million items, then the most popular 100 will contribute a third of the total value, the next 10 000 another third, and the remaining 989 900 the final third. The value of the collection of n items is proportional to log(n).

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.

At a time when telecommunications is the key infrastructure for the global economy, providers need to make fundamental decisions about whether they will be pure providers of connectivity or make their money by selling or reselling content, such as television and movies. It is essential that they value their enterprises correctly—neither overvaluing the business of providing content nor overvaluing, as Metcalfe's Law does, the business of providing connectivity. Their futures are filled with risks and opportunities. We believe if they value the growth in their networks as n log(n), they will be better equipped to navigate the choppy waters that lie ahead.