Erdős 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, the numbers range up to 15, but the average is less than 5, and almost everyone with a finite Erdős number has a number less than 8.
The Bacon number is an application of the same idea to the movie industry, connecting actors that appeared in a film together.
Jerry Grossman, Marc Lipman, and Eddie Cheng have been looking at some questions in pure graph theory motivated by these collaboration graphs.
Also, Michael Barr suggests "rational Erdős numbers, generalizing the idea that a person who has written p joint papers with Erdős should be assigned Erdős number 1/p. From the collaboration multigraph of the second kind (although he also has a way to deal with the case of the first kind) -- with one edge between two authors for each joint paper they have produced -- form an electrical network with a one-ohm resistor on each edge. The total resistance between two nodes tells how "close" these two nodes are."