[prajit]

"There’s the famous observation that the value of a network grows as a function of the square of the number of nodes"

The value of a network is measured as O(V^2), where V is the number of nodes (users) in the network.

"and also many of these services/products double their userbase every N months."

V = 2^N (the number of users doubles every N months), so the overall function for the value of a network is O(V^2) = O((2^N)^2) = O(2^(2N)) = O(4^n). O(4^n) > O(2^n), but it is not the same as O(2^(N^2)), which was what Sam was trying to convey.