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# The Collaboration Graph and Mathematical Genealogy

definition(Erdős number): Paul Erdős has an Erdős number of zero. Everyone else has Erdős number k + 1 where k is the lowest Erdős number of any coauthor.  Anyone with no coauthorship chain connected to Erdős has an Erdős number of infinity.

Of the finite Erdős numbers at the turn of the century, the median was 5, the mean was 4.65, and the standard deviation was 1.21.   There are about 268,000 people with finite Erdős number, about 50,000 published mathematicians who have collaborated but have an infinite Erdős number, and 84,000 who have never published joint works and therefore have an infinite Erdős number  All of the Fields prize winners between 1986 and 1994 have Erdős numbers of at most 9.  At one point, an Erdős number of 5 was even for sale on eBay.

Interestingly, there are four authors with more than 700 papers: Paul Erdös with 1416 (he actually wrote more in Math Reviews), Drumi Bainov with 823, Saharon Shelah with 760, and Leonard Carlitz with 730.  Many more interesting facts can be found:

http://www.oakland.edu/enp/trivia/

MathSciNet has a neat feature that finds the minimum distance between two authors in the database:

http://www.ams.org/mathscinet/collaborationDistance.html

You can also research your mathematical family tree:

http://genealogy.math.ndsu.nodak.edu/

This site keeps track of student/advisor relationships.  Interestingly, the database data viewed as a graph is non-planar because there is a $K_{3,3}$-subdivision.

As of June 2010, the largest component had 121,424 vertices (85% of the total vertex set) and 7,190 isolated vertices.