I wonder why people consider "social networks" to be a "new thing". It's the "rage these days", they say. Actually, they've been around for a while but in a different disguise.
Take the case of the Erdos number:
In order to be assigned an Erdos number, an author must co-write a mathematical paper with an author with a finite Erdos number. Paul Erdos is the one person having an Erdos number of zero. If the lowest Erdos number of a coauthor is k, then the author's Erdos number is k + 1.
Erdos wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 511 direct collaborators; these are the people with Erdos number 1. The people who have collaborated with them (but not with Erdos himself) have an Erdos number of 2 (8,162 people as of 2007), those who have collaborated with people who have an Erdos number of 2 (but not with Erdos or anyone with an Erdos number of 1) have an Erdos number of 3, and so forth. A person with no such coauthorship chain connecting to Erdos has no Erdos number (or an undefined one).
There is room for ambiguity over what constitutes a link between two authors; the Erdos Number Project website says "Our criterion for inclusion of an edge between vertices u and v is some research collaboration between them resulting in a published work. Any number of additional co-authors is permitted," but they do not include non-research publications such as elementary textbooks, joint editorships, obituaries, and the like. The “Erdos number of the second kind” restricts assignment of Erdos numbers to papers with only two collaborators.
Erdos numbers have been a part of the folklore of mathematicians throughout the world for many years. Amongst all working mathematicians at the turn of the millennium who have a finite Erdos number, the numbers range up to 15, the median is 5, the average Erdos number is 4.65; and almost everyone with a finite Erdos number has a number less than 8.
So, Erdos numbers is essentially a social network that counts the degrees of separation.
Somewhat relatedly, there is also an interesting theory called Dunbar's number:
Dunbar's number is the supposed cognitive limit to the number of individuals with whom any one person can maintain stable social relationships: the kind of relationships that go with knowing who each person is and how each person relates socially to every other person.
Dunbar has argued that 150 would be the mean group size only for communities with a very high incentive to remain together. For a group of this size to remain cohesive, Dunbar speculated that as much as 42% of the group's time would have to be devoted to social grooming.
I wonder what would be the Dunbar number of the social circles that I know of.