Network value grows closer to n log n than n squared, because most users value only a few connections.
Odlyzko and Tilly's 2005 correction to Metcalfe. The n log n form is why a network of a billion is not a million times more valuable than one of a thousand, and why the dot-com valuations failed.
In practice
- A messaging app with ten thousand users in one city is worth more to each of them than one with ten million spread thinly across forty.
- A professional network's hundredth connection adds far less than its tenth, and the pricing that assumed otherwise did not survive.