See "A Refutation of Metcalfe's Law", by Odlyzko and Tilly:
http://www.dtc.umn.edu/~odlyzko/doc/metcalfe.pdf
I'd extend the rule further and suggest that the value of a network scales as some Odlyzko and Tilly suggest, but with an additional negative function subtracting value (v) from the network:
v = n(log(n) - f(n)
That is: as the network grows, the added value of each additional member is reduced (log(n)). Further,
each additional member of the network exacts a cost to the network as a whole as well. Doing some simple modeling, I suspect that this isn't a strictly linear factor, but itself grows with n, quite possibly as the original Metcalfe's law suggestion. That is: any given member is increasingly
less likely to be a positive contribution to the network, but might well present an
equal opportunity to be a net negative to the group as a whole:
v = n(log(n)) - kn^2
Where 0 < k < 1 and n > 1.
Moreover, let's look at some group sizes which might allow us to estimate for k in various contexts.
For software team size, it's very typical that a core engineering group has a size of 5-10 members, more or less. This suggests that k is about 0.2 for software development: every added team member exacts a cost, within a single group, of about 20%. This sees value grow for 1 < n < 8, then fall with larger n, hitting negative values at about n=14.
For an elementary school classroom where the ideal class size seems to be around 22-25 students, k would be around 0.08.
For Dunbar's Number, the number of relationships people can manage (100 - 300, typically set at 150), k is between 0.028 (100), 0.2 (150), and 0.113 (300).
For city sizes, it's likely that different cities offer different matches of positive and negative factors. k of 0.0005 gives value max at n ~ 10,000, k of 0.0006 is ~ 100,000, and 0.000007 is around 1 million.
To scale to 1 billion users with net positive value means you have to keep k to less than 0.00000001. That is: any one member can have only a 1 in 10 million chance of being annoying to other members.