Except Whatsapp has a very strong network effect that's difficult to replicate (the other person must also have the app). That's why we designed Upptalk to interoperate with standard telephony services like SMS and SIP.
I disagree, network effects are incredibly strong. Since Telegram is only capable of talking to other Telegram apps, its value increases not only in proportion to the number of contacts who have it installed, but also how often you need to communicate with those people.
I don't have time right now to read all that, but since I was referring generally to network effects, and not specifically Metcalf's Law, the paper probably validates my comment above.
The paper is about the scaling of the value of a network from network effects as the network scales.
It validates your point in that the value of a network depends both on how many people are in a network, and the value that each gets from it.
But it supports what I said. A variety of ways of estimating the second factor in real networks finds that it tends to scale proportionately to log(n), hence giving a total value of O(n log(n)) for the network.
This is backed up by every way we could find to look at the problem, from the volume of mail delivered by the post office, to scaling laws that have been found to be ubiquitous in social networks.
