I hadn't come across it in some time, yet there it was, in print, somebody quoting Metcalfe's "Law" to justify some extravagant claim.
It is a great example of a small idea that morphs from the obvious to the wrong. How did it make that jump? Well, it became abused by jumping from one situation, a network of nodes, to another system, a network of users.
Robert Metcalfe has done enough work to merit respect, so I was surprised to see him defend what is little more than a heuristic. His main defence is that it made him lots of money! He does come across as quite prickly to the research done at the time showing his naive "law" to be wrong.
Honestly, the original idea was related to Ethernet nodes; the more nodes you have, the larger the number of potential connections and hence the cheaper the cost per connection. Conversely, if each connection has the same value to the users then as those connections grow the whole network has more value than the cost of building it. Metcalfe still claims this is a good model to discuss the critical mass of users needed to make the network effect positive.
However, a business presentation is not the same as a proper mathematical analysis. Let us never forget that Steem used Metcalfe's Law to justify post rewards proportional to rshares-squared until HF19. The result was a whale bonanza and a horrific Gini coefficient. Metcalfe doesn't care, but you should.
"Because my law allegedly over-estimates the values of networks, it might be used to inflate a second Internet Bubble, probably the imminent Social Networking Bubble, which will then inevitably burst. Can’t have that." Again, defending the indefensible is not winning Bob any new fans - but at least he got one prediction right.
It is mathematically obvious that a network with N nodes has potentially N(N-1)/2 connections, and as N grows we can approximate this to N2. However, this makes the assumption that such potential connections are actually active - this is obviously not true. Social networks tend to coallesce around key individuals and fan outwards, forming clusters and sub-clusters. I have even seen completely disjointed networks within the same platform.
The article Metcalfe despises so much, Metcalfe’s Law is Wrong, can be followed up with more quantitative analysis, such as On the Value of a Social Network. The function N.log(N) is a much better approximation to real network effects and, strangely, is also a consequence of something known as Zipf's Law, which in turn has deep connections to the Pareto distribution.
In truth, all these basic formulas demand some constant of proportionality to match them to real-life networks, so that we are really looking at some kN.log(N), where k depends on the actual network under investigation.
I'll leave you with one thought: would Steem have evolved differently had it not embraced Metcalfe's so-called Law?
There are consequences to believing things that are just not true.
PS. Anybody with knowledge of more recent papers on this topic, please share in a comment.
image: wikimedia
