I respect Metcalfe a lot, but halfway through undergraduate discrete math it was pretty obvious to most people in the class even before seeing a formal proof that a fully connected graph has O(n^2) edges. I just figured that people wowed by "Metcalfe's Law" were business types who didn't any formal theory into computing.
but it's a loose approximation so it's not good to overanalyze it.
The number of pairwise connections grows as the number of pairwise connections, and connections ("how many people can you talk to") are valuable, so value grows. But individual connections to networks grow the pairwise connections by N, so that's even better.
broadcast (one to many connections, like giving a speech to a crowd) is an efficiency hack, which is good, and efficiency hacks grow as the number of connections grow, so that's good too...
... is more how I think about what Metcalfe was talking about. Which aspects are x, which are x squared, which are log x is interesting, but that's not all bound up in his simple statment, despite his "as the square" wording.
and Bob Metcalfe is personally a great guy in all the ways people are saying, but it's not soooo unique, that's the way a lot of tech types were as the mantle passed from the Greatest Generation to the Boomers (and what was that one in the middle, "lost" or "invisible" or something) I'm not suggesting we've lost that (we may have) just saying that's how it was, for instance as an undergrad you could walk into any professor's office and get serious attention.
It counts connections and uses them as an estimate of value.
However not all connections are equally valuable. And therefore the "law" is incorrect. An estimate in far better agreement with the data is O(n log(n)), and you can find multiple lines of reasoning arriving at that in https://www-users.cse.umn.edu/~odlyzko/doc/metcalfe.pdf.
I see only two real 'quantitative' arguments in https://www-users.cse.umn.edu/~odlyzko/doc/metcalfe.pdf#page... . Your first argument, 'connections aren't of equal value', doesn't defeat Metcalfe. Your second argument, that Metcalfe's law would mean efficient markets would merge all networks into one, is both the most amazing overestimate of the competence & economic rationality of telecom giants I've ever seen and also not actually true as a matter of economic theory (https://gwern.net/doc/economics/automation/metcalfes-law/201...). So neither of your handwaving arguments was very good to begin with.
The gravity law argument based on geographic distribution of traffic, Zipf's Law and Bradford's law all have empirical evidence behind them. That's three. Additionally another version of the same paper Bob Briscoe contributed data from British Telecom usage that supported the same scaling rule.
The second paper that you gave is interesting. Odlyzko was the one who contributed that particular argument. It is right that there are rational reasons to not interconnect. But Metcalfe would imply more of a first mover advantage than we actually see. In social networks we had Friendster, MySpace and Facebook, each of which overtook the other. How could a new entrant supplant the king? Not once, but twice?
Since then new social networks have continued to sprout and succeed. Facebook managed to stay on top, in part through purchasing other networks. One of which (Instagram) is on track to surpass Facebook in revenue.
Now let's look at the 4 papers that you collected.
The first and third have the same flaw. They are looking at revenue over time as the network grew. But the growth of the network is not the only change that happened over time.
1. The Facebook product improved to become more compelling, even for the same users. In part by adding new channels through purchasing other networks.
2. Facebook kept adding new ways to monetize people, improving revenue.
3. People's behavior has shifted to more online over time. Thus it was easier to get value from the same users in 2014 than in 2008.
Because so much has changed, comparing users in 2008 to users in 2014 is not apples to apples.
Next, let's turn to the last paper. I'm in agreement with patio11 that Bitcoin's valuation has been driven by the largest Ponzi scheme in history. Therefore I view most of its valuation as fake. And so am not inclined to accept arguments from that valuation as valid.
And I saved the best for last. In section 2.4 the EU paper argues that Briscoe's law (I think Odlyzko should be credited, but Bob Briscoe is in the EU) is more accurate than Metcalfe's law after you hit scale.
Their argument in effect is a variant of one that was discussed privately before we wrote our paper. Our immediate perception of the size of a network is based on how much of our personal social groups are on it. The value we get from that network is based on the same. Therefore our perception of the size of the network is correlated with the value we get from it. If the network mostly contains parts of groups, you do get something like Metcalfe's Law out of this. But once the network contains a lot of completed social groups, members of those groups slow down how much value they gain as the network continues to grow.
In other words when the connections in the network are a random sampling of the connections that matter to us, growing the network adds valuable connections. Once the network contains the connections that we know matters to us, most of us only benefit marginally from continued growth.
