That said, the argument and result is intentionally informal, and intended to be a better rule of thumb than Metcalfe's law. Which was itself even more informally reasoned. However its key flaw we explained with:
Metcalfe’s Law is intuitively appealing, since our personal estimate of the size of a network is based on the uptake of that network among friends and family. Our derived value also varies directly with that metric. We therefore see a linear relationship between the perceived size and value of that network.
That said, different social networks may well have different scaling laws. In fact I pointed out privately that eBay likely was a counter-example which is closer to Metcalfe's law for the simple reason that you derive value there from connections to random strangers. And indeed eBay successfully maintained a market leading position for a long time despite charging significantly higher fees than competitors like Amazon Marketplace.
Sorry for being so critical, but I would find it quite interesting if you had hard evidence to support a scaling law like n log(n) for the value of a social network.