Thought this social networking analogy gave some nice intuition about the power-law network structures.
Metcalfe's Law is Wrong
By Bob Briscoe, Andrew Odlyzko, and Benjamin Tilly
Communications networks increase in value as they add members—but by how much? The devil is in the details
Of all the popular ideas of the Internet boom, one of the most dangerously influential was Metcalfe's Law. Simply put, it says that the value of a communications network is proportional to the square of the number of its users....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.... We propose, instead, that the value of a network of size n grows in proportion to n log(n)....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.....
To Probe Further
David P. Reed argues for his law in "The Sneaky Exponential" on his Web site at http://www.reed.com/Papers/GFN/reedslaw.html.
Several additional quantitative arguments are made for the n log(n) value for Metcalfe's Law on the authors' Web sites at http://www.cs.ucl.ac.uk/staff/B.Briscoe and http://www.dtc.umn.edu/~odlyzko.
Chris Anderson's article "The Long Tail" was featured in the October 2004 issue of Wired. Anderson now has an entire Web site devoted to the topic at http://www.thelongtail.com.
An article in the December 2003 issue of IEEE Spectrum, "5 Commandments," which can be found at http://www.spectrum.ieee.org/dec03/5com, discusses Moore's and Metcalfe's laws, as well as three others: Rock's Law ("the cost of semiconductor tools doubles every four years"); Machrone's Law ("the PC you want to buy will always be $5000"); and Wirth's Law ("software is slowing faster than hardware is accelerating").