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:**

**1) Thought Leadership: **

If people identify your network is *THE* place to do X (for international hotel reviews, I’d say TripAdvisor has thought leadership) then you have thought leadership. TripAdvisor might only have a couple million unique reviewers, so their network is still relatively small compared to the size of all humans on the planet, but the network value is immense due to thought leadership, perhaps even larger than than Metcalfe’s law.

**2) Critical Mass**

You reach critical mass when you don’t have to wonder to yourself, “I wonder if my friend Bob is on this network or not.” You just assume Bob is in fact on the network and are only surprised if it turns out he’s not on the network. The value of reaching this tipping point is so much more important than the equation n*log(n) or n^2. It’s an assumed success

(1) and (2) might sound a little too similar to be separate, but I think each can be mutually exclusive. You can have critical mass without thought leadership (MySpace’s peak?) and you can have Thought Leadership without Critical Mass (Secret? or is it Yik Yak now?).

While we might quibble about equations, I think the authors of the paper would agree with me on these points (or at minimum, the latter point re: critical mass).