# The Value of a Network

A friend at Spark Ryan Shmeizer sent me an interesting paper by Andrew Odlyzko and Benjamin Tilly from the middle of the last decade positing that Metcalfe’s law is a over-estimate of the value of a network. Metcalfe’s law says the value of a network is proportional to the square of the number of users in the network. In formula form that’s: n^2.  Instead, the paper makes an argument that actual total value of a network is closer to n*log(n).  I then found a PDF from Business Communications Review that takes the meat of the paper and does a very readable review of its conclusions, which is worth a read if you find this topic interesting.

The n*log(n) value claim a dramatic claim in my view because n*log(n) in its asymtotic limit diverges away from simple linear n alone, but for all practical applications in networks where all nodes are humans, the difference between n and n*log(n) is almost trivial when compared to the difference between n and n^2.  In plain English, n*log(n) is functionally much closer to pure linear value, which would imply that networks have very little value at all. Instead networks are just worth a little bit more than the sum of their parts. That’s a dramatic statement in my world where investors and entrepreneurs talk about the value of network effects all day long.

I was curious to see if the public markets agree with Metcalfe. So I looked at a pretty simple example, Facebook and Twitter.  If a network value behaves in line with Metcalfe’s Law, then in theory, the following equation should hold (or at least be pretty close to accurate):

FB’s enterprise value / TWTR’s enterprise value = FB’s Monthly Users ^ 2 / TWTR’s Monthly Users ^2

Or, if the authors of this paper are right, then the stocks should try according to a ratio in line with the n * log(n) valuation:

FB’s enterprise value / TWTR’s enterprise value = FB’s Monthly Users * log(FB’s Monthly Users) / (TWTR’s Monthly Users * log(TWTR’s Monthly Users))

So, I went ahead and just did the comparison in Excel:

As you can see, public investors think the value of a network is closer to the n*log(n) thesis than the n^2 thesis; although 6.36 is far enough away from the n*log(n) thesis, then I don’t think you can simply crown a victor and be done.

There is an endless number of caveat’s that need to be applied to this analysis (which company is growing faster? does one company skew to more valuable interactions than the other company? etc…), but it’s a fun thought experiment nonetheless.

I ultimately told Ryan that thought n*log(n) was too similar to linear to be an accurate measure of network value, so I disagree with the authors of the paper in that regard. Networks have to be worth more than the sum of their parts. But that doesn’t mean that n^2 is a perfect proxy for network value either.  Instead, I think network value comes down to two tipping points: