Metcalfe’s Law is wrong
Metcalfe’s Law states that the value of a network of size n is proportional to n2. This follows from a simple observation: that the number of possible connections for each user in a network is n - 1; and since there are n users, then the total number of connections is n × (n - 1), which is roughly n2. This all seems reasonable and makes sense.
But it’s wrong. It begs one simple question: Metcalfe’s Law assumes that each connection is equally worthwhile. This doesn’t really make sense: is my connection to a bushman in the Kalahari as useful to me as my connection to my brothers, or to my bank? Not very likely.
It turns out that there’s another law — Zipf’s Law — which addresses all sorts of distributions. The article goes into more detail, but basically the second-most-important item in a list is one half as valuable as the first; the third is one third as valuable; the fourth is one quarter as valuable; and so on an so forth. It turns out that adding up 1 + ½ + ⅓ + ¼ … 1/n approximates log n reasonably closely. One might say that the value of a network of size n to any single user is proportional to log n (that is, the sum of the value of his most important link, his second most important link and so on until we get to his link to a squid-fishing boat in the Atlantic).
Thus Briscoe, Odlyzko & Tilly suggest their own network-value law: the value of a network is in proportion to n log n. They present some economic predictions based on this law, which seem to be borne out by the facts.
Anyway, read the article. It’s good and detailed and makes sense.