While doing one of my normal perusals of Wikipedia, I came across some assertions on the value of social networks relative to their sizes. Metcalfe’s law, which was originally applied to early Ethernet networks, states that the value of a network is proportional to the square of the number of users. This makes sense, because for a network of size n, the total number of possible connections between people is n(n-1)/2, which is proportional to n^2.

Reed’s law, however, says that size plays an even bigger factor in the value of a social network. David P. Reed, in an essay called That Sneaky Exponential (linked from the wiki article), claims that the value of a social network is not quadratic, but exponential. This is because, he says, the value of a network does not rely on the number of possible pairings, but on the number of possible subsets. It’s easy to see that the number of subsets is roughly 2^n, because for each element, one can either choose it or not choose it, and we use our multiplication rules from set theory. Communication on the internet is not merely one-to-one; there are communities of large numbers of people, so we can that it is not pairs, but groups, that are important.

This connects in with our discussion of network cascades, and how people will be more likely to join the instant messaging or other network that their friends already use. However, this looks at the total value of a network instead of its value for only one person. While if we divide by the number of people, Metcalfe’s shows the value being directly proportional to the number of users, Reed’s way of looking at things is still exponential, just with a slightly smaller exponent. Therefore, the value per person still increases very quickly with the number of people using the network. Finding the exponential the per-person value winds up is left as an exercise for the reader.