Illustration: Serge Bloch

We knew our September issue on technology and terrorism would raise hackles. But we didn’t realize that our July feature “Metcalfe’s Law Is Wrong” by Bob Briscoe, Andrew Odlyzko, and Benjamin Tilly would churn up its own little storm of controversy, drawing remarks from none other than Bob Metcalfe himself. The inventor of Ethernet, founder of 3Com Corp., and recipient of the 1996 IEEE Medal of Honor rebutted our authors’ arguments in a late summer guest posting on VC Mike’s Blog (http://vcmike.wordpress.com). The site is run by Mike Hirshland, who, like Metcalfe, is a general partner in Polaris Venture Partners, in Waltham, Mass.

When we caught up with Metcalfe, he seemed more tickled than ticked off, encouraging us to start a discussion on his response to “the Spectrum attack on my law, by which attack I am delighted.” The difference of opinion has more than mathematical significance. It has direct bearing on the value of dot-com businesses that exploit expanding social networks. To keep it all straight, we suggest you first reread the Spectrum article and then peruse Metcalfe’s post (http://vcmike.wordpress.com/2006/08/18/metcalfe-social-networks).

Ready? Then let the discussion resume with this comment on Metcalfe’s blog entry by our author Benjamin Tilly, who is a senior programmer at Rent.com in Santa Monica, Calif.

Tilly writes: “Some of Metcalfe’s criticism is justified. For example, it is inherently difficult to quantify network value, and we did not attempt it. But he seems to have missed the point of our critique. For instance, he says that someone should try to combine the phenomena of the so-called long tail with Metcalfe’s Law. He failed to notice that someone did—us.

“The derivation of Metcalfe’s Law assumes that all potential connections have equal value. Therefore, the value of a network scales as the number of potential connections, which is proportional to n² . The phenomenon of the long tail says that if you rank a large list of things from most valuable to least, you get a power law. That power law says that while value is concentrated in the most valuable items, there is also a lot of value hidden in the long tail. So what happens if you try to combine these insights?

“When we did that, we saw that the assumption that every potential connection has the same value is absurd. It contradicts common sense, and contradicts measured value distributions, which tend to follow power laws. So we tried to adjust Metcalfe’s Law by giving the potential connections that each person could make a power law distribution—just like what happens in the long tail. If we assume that each person has roughly equal value, we find that the value of the network scales as n log(n), not n² .

“It must be stressed that so far this estimate is a theoretical prediction. But it can be subjected to a number of tests. The first, and most obvious, is to ask whether the assumptions leading to the estimate are reasonable. They are indeed reasonable—significantly more so than the ones underlying Metcalfe’s Law.

“A second test was to see whether or not there were any networks for which we could independently estimate a value distribution. In fact, there is a known power law distribution on how far physical mail travels. Through that we can estimate the volume of mail that will be sent over a given area, then divide it by the population in that area. That calculation says that the volume of mail sent should be proportional to n log(n)—which is the same estimate, arrived at by a very different means. It should be stressed that this estimate is based on empirical data—the volume of mail actually sent. To the best of my knowledge, nobody has ever produced empirical data proving that Metcalfe’s Law is correct.

“A third test is to look at what the scaling says about human behavior. Metcalfe’s Law suggests very rapid improvements in value as size gets bigger. So big, in fact, that direct competitors of even slightly different sizes should have very different valuations—different enough that either they interconnect rapidly or else the larger ones will drive the smaller ones out of business. Our version predicts that size is an advantage, but that advantage is modest once you get beyond a certain threshold. Therefore, there will be pressure to interconnect, but that pressure will be fairly small. The behavior of competing networks, ranging from early telephone networks to competing IM systems today, looks like what we predict.

“Now I won’t say that n log(n) is right. In fact, its accuracy will vary by type of network. But all of the evidence that I can find, both theoretical and empirical, says that it is far closer to being right than Metcalfe’s Law.”

What do you think? If you’d like to comment on the issue of whether Metcalfe’s Law is right or wrong, please send a letter to the editor with the heading “Metcalfe’s Law” in the subject line to: n.hantman@ieee.org.