In a recent Medium post derived from a talk he gave at private invitation-only event for the IT industry, Dan Hon presents one view of Metcalfe’s Law, the theory espoused by Ethernet inventor Bob Metcalfe that “the value of a telecommunications system increases as the square of the number of participants”. Hon looks at the (no pun intended) value judgment embedded in talking about the “value” of a network, and considers purely market-oriented measures lacking.

I’d like to step back a bit and look at it from a different angle. Instead of “value”, let’s consider “utility”: what benefit arises to the users from their use of the network? Metcalfe’s claim can be restated simply: the global utility (sum over all users) of a network is quadratic in the number of users. You don’t even need graph theory to prove that this is trivially true, if you accept what I take to be Metcalfe’s presuppositions: first, that utility sums linearly over all users (a view which would be understandable to Jeremy Bentham), and second, that each user’s utility is linear in the number of other users on the network.

The real problem with Metcalfe’s Law, as I see it, is precisely in this second presupposition. While it is true to a first approximation, for small networks, once the network reaches a sufficient penetration of that community with which any individual user has an interest in communicating, the marginal utility of additional communications partners diminishes quite rapidly, and ultimately goes negative. We see this even with old technologies like the telephone network: nearly all of the value I get from the telephone derives from being able to communicate with family, friends, and current and potential employers, vendors, and service providers in my immediate vicinity. While connecting a billion people in India or China to the rest of the world is laudable, there cannot be more than a thousand of them that make the telephone network more valuable to me. (One thing that this analysis does not consider, and a more sophisticated analysis would, is economies of scale: do those billions of users actually make it easier or cheaper to provide me with the service that I value. To be left for another day.)

In the social network case, it’s clear how additional users can have negative marginal utility: the additional noise generated can drown out the intended communication (whether that noise is trolls, pile-ons, or just way too many well-meaning people making the same comment in a reply). Twitter is a great demonstration of this; users bearing the vaunted “blue checkmark” — a distinction given out entirely at Twitter’s discretion to a small subset of users, mostly celebrities, journalists, government officials, and corporate marketing departments — are given a variety of tools to screen out communications from the masses. One of the tools which is frequently employed by these “verified” users screens out all notifications from the remaining users, allowing them to give the appearance of using the platform to communicate with others while in actuality paying attention only to a small number of similarly privileged people. This screening was not part of the original Twitter service: it was only deployed after Twitter gained a sufficiently large and noisy user community that it was driving away users Twitter actually had a business reason to want to retain. Of course, even “old tech” had to come up with similar mechanisms: when telephone calls became cheap enough that scammers were willing to spam a thousand people at dinnertime in the hope of finding a single mark, caller ID became a necessity and more and more people began to screen their calls. (Compare also the Eternal September.)

In conclusion: Metcalfe’s Law is wrong because the marginal utility to the existing users of a communications network is not constant: while it is large and positive for small networks, as networks grow beyond the scale of normal human social circles, the utility drops off quite rapidly, and eventually goes negative. When you sum up this function over all users, unlike the linear utility posited by Metcalfe, overall value does not scale as the square of the number of users. (It might not even be asymptotically linear — I leave that analysis to someone with better mathematical chops.)