Metcalfe's Law says that the value of a network is proportional to the number of nodes (or people) in the network. That is, V = k n^2, for some constant k.

But we now know that there's more to the story. That valuable network attracts users, but it also attracts abusers – spammers, propagandists, trolls. They don't add to the value of the network; they detract from it.

Here's where the handwaving starts. It is my perception that the proportion of abusers rises as the size of the network rises. That means that the total number of abusers rises faster than the number of users – perhaps as the square of the number of users.

Worse, those abusers do more damage than their numbers would indicate. It's not just that you have messages that should be ignored. It's also that you have to increase the level of mistrust for every message. I'm going to guess that the abusers do damage about in proportion to the square of their number (which is itself in proportion to the square of the number of users).

That leaves us with V = k n^2 – c n^4, to account for the damage from abuse.

It follows that one essential of larger networks is keeping c as low as possible. Otherwise abuse destroys a network.

Also note that, for any given k and c, there is a number of users beyond which the value of the network is negative.

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