Facebook Data to the Rescue of Metcalfe's Law

The December 2013 issue of IEEE"s Computer magazine has as its theme: Computing Laws Revisited. Our hardcopy issue (my husband is a subscriber) arrived yesterday, on New Year's Eve, and, today, New Year's Day, I sat down to read the article, "Metcalfe's Law after 40 Years of Ethernet," written by none other than Bob Metcalfe, Ethernet's inventor, himself. The Ethernet is the physical transport level of today's Internet. On May 22, 2013, Ethernet industry leaders gathered at the Computer History Museum in California to celebrate its 40th birthday and to catch up on what has become a $100 billion industry (do I now have your attention?).

I have been interested in Metcalfe's Law since, as an Associate Editor of over a dozen journals, and a frequent reviewer of journal articles, as well as a Co-PI on a Future Internet Architecture NSF grant: Network Innovation Through Choice (which I most recently spoke on at the Network Frontiers Workshop at Northwestern University), I not only encounter related controversies but also write and review papers that touch upon this law.

I guess that Metcalfe must have been reading over my shoulder!

Now, back to a controversy: In 2006, Briscoe, Odlyzko (I do like some of his math models a lot), and Tilly published an article, "Metcalfe's Law is Wrong" in the IEEE Spectrum, which you can read online without a subscription. These authors described Metcalfe's Law as both "wrong" and "dangerous."

So, what is Metcalfe's law, you may be asking (for those not so much into hardware). As explained by Metcalfe himself in his very clear and erudite December Computer magazine article: The "law" bearing his name states that the value of a network grows as the square number of users: V= NxN. The law started as a high-concept Ethernet sales tool in the early 1980s and entered into the public discourse in the new millennium when Gilder "championed" Metcalfe's law in his book, "Telecosm: The World After Bandwidth Abundance," published in 2000.

During  a presentation to the 3Com sales force (Metcalfe cofounded this company in 1979), Metcalfe showed the figure below (thanks to his article in Computer magazine) to argue that if  a network is too small its cost exceeds its value, but if a network gets large enough to reach a critical mass, then the sky is the limit.

Valuation of Internet-based companies such as Google, LinkedIn, Facebook, Twitter, Snapchat, and others is clearly important and detractors have said that Metcalfe's law may be overvaluing such firms.

The main critique of Metcalfe's law, noted by Metcalfe, himself, was that Briscoe, Odlyzko, and Tilly countered that not all network connections are of equal value and that the growth in the value of a network is approximately Nxln(N) and not NxN as Metcalfe had stated.

Moreover, neither "law" had been validated empirically until Metcalfe did recently and reported the results in his article in the December issue of the IEEE Computer magazine.

And he used 10 years of real data from Facebook and fitted the data looking at N as a function of time. To estimate the growth of users and individual groups of friends, he first applied a sigmoid function (S-shaped adoption function). He then generalized it to what he calls a "netoid" function using the Python programming language. (It looks as though Metcalfe also thinks in code, which is cool.) He plotted Facebook annual revenue (as the value V) for the last 10 years and used a Python slider to the Metcalfe's function to "get a pretty good visual fit to this data" as he displayed in the figure below reported in Computer magazine.

Metcalfe also notes that a more elaborate model of network growth and value would involve using "nesting netoids." Fascinatingly, he observed that Facebook groups seem to be netoiding (a new verb) towards Dunbar's number, which is due to the anthropologist Robin Dunbar who theorized that humans have a cognitive limit on the number of people with whom they can maintain stable social relationships with this figure being about 150. And, since Facebook has about 1.06 billion users and 150 billion friend connections this implies  an average of 141 friends per user.

So, as we begin to celebrate the New 2014 Year, it is quite interesting that we are also marking technological advances that challenge and intrigue us in both theory and practice and that are still very much with us such as the Ethernet, which is now 40 years old!

There is also a great interview with Bob Metcalfe, thanks to Charles Severance of Computer magazine, which can be viewed below.