Erdős number

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The Erdős number (IPA: ['ɛrdøːʃ]), honoring the late Hungarian mathematician Paul Erdős, is a way of describing the "collaborative distance" between a person and Erdős, as measured by authorship of mathematical papers.

It was created by friends as a humorous tribute to the enormous output of Erdős, one of the most prolific modern writers of mathematical papers, and has become well-known in scientific circles as a tongue-in-cheek measurement of mathematical prominence.

Paul Erdős was an influential and itinerant mathematician, who spent a large portion of his later life living out of a suitcase and writing papers with those of his colleagues willing to give him room and board^[1]. He published more papers during his life than any other mathematician in history (at least 1400)^[1].

[edit] Definition

If Alice collaborates with Paul Erdős on one paper, and with Bob on another, but Bob never collaborates with Erdős himself, then Bob is given an Erdős number of 2, as he is two steps from Erdős.

In order to be assigned an Erdős number, an author must co-write a mathematical paper with an author with a finite Erdős number. Paul Erdős is the one person having an Erdős number of zero. If the lowest Erdős number of a coauthor is k, then the author's Erdős number is k + 1.

Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 511 direct collaborators^[2]; these are the people with Erdős number 1. The people who have collaborated with them (but not with Erdős himself) have an Erdős number of 2 (8,162 people as of 2007), those who have collaborated with people who have an Erdős number of 2 (but not with Erdős or anyone with an Erdős number of 1) have an Erdős number of 3, and so forth. A person with no such coauthorship chain connecting to Erdős has an Erdős number of infinity (or an undefined one).

There is room for ambiguity over what constitutes a link between two authors; the Erdős Number Project Web site 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 “Erdős number of the second kind” restricts assignment of Erdős numbers to papers with only two collaborators.^[3]

The Erdős number was most likely first defined in print by Casper Goffman, an analyst whose own Erdős number is 1.^[4] Goffman published his observations about Erdős's prolific collaboration in a 1969 article entitled "And what is your Erdős number?"^[5]

The AMS collaboration distance calculator allows an online calculation of an individual's Erdős number.

[edit] Impact

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 who have a finite Erdős number, the numbers range up to 15, the median is 5, the average Erdős number is 4.65;^[6] and almost everyone with a finite Erdős number has a number less than 8. Due to the very high frequency of interdisciplinary collaboration in science today, very large numbers of non-mathematicians in many other fields of science also have finite Erdős numbers. For example, political scientist Steven Brams has an Erdős number 2. In biomedical research, it is common for statisticians to be among the authors of publications, and many statisticians can be linked to Erdős via John Tukey, who has Erdős number 2. Similarly, the prominent geneticist Eric Lander and the mathematician Daniel Kleitman have collaborated on papers,^[7][8] and since Kleitman has an Erdős number of 1,^[9] a large fraction of the genetics and genomics community can be linked via Lander and his numerous collaborators. According to Alex Lopez-Ortiz, all the Fields and Nevanlinna prize winners during the three cycles in 1986 to 1994 have Erdős number at most 9. Similarly, many linguists have finite Erdős numbers, many due to chains of collaboration with Noam Chomsky, whose number is 4.^[10]

Tompa^[11] proposed a directed graph version of the Erdős number problem, by orienting edges of the collaboration graph from the alphabetically earlier author to the alphabetically later author and defining the monotone Erdős number of an author to be the length of a longest path from Erdős to the author in this directed graph. He finds a path of this type of length 12.

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 mathematicians 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.

Earlier mathematicians published fewer papers than modern ones, and more rarely published jointly-written papers. The earliest person known to have a finite positive Erdős number is either Richard Dedekind (born 1831, Erdős number 7) or Ferdinand Georg Frobenius (born 1849, Erdős number 3), depending on the standard of publication eligibility.^[12] It seems that older historic figures such as Leonhard Euler (born 1707) do not have finite Erdős numbers.

[edit] Outside mathematics

[edit] Bacon number

The Bacon number (as in the game Six Degrees of Kevin Bacon) is an application of the same idea to the movie industry, connecting actors that appeared in a film together to the actor Kevin Bacon.

A small number of people are connected to both Erdős and Bacon and thus have an Erdős–Bacon number. One example is the actress Danica McKellar, best known for playing Winnie Cooper on the TV series, The Wonder Years. Her Erdős number is 4 and her Bacon number is 2. The lowest known Erdős–Bacon number is three for Daniel Kleitman, a mathematics professor at MIT; his Erdős number is 1 and his Bacon number is 2.

[edit] Kibo number

The surrealist Usenet personality Kibo has had a similar number applied to people in contact with him. A Kibo number of one is defined as someone who has at minimum received e-mail from Kibo. Mail from someone with a Kibo number will make the recipient's number the sender's Kibo number plus one.^[13]

[edit] Stringfield number

The Stringfield number is an application of the same idea to the field of Ufology, connecting those who co-investigated or co-researched UFO cases with the late Leonard H. Stringfield.^[14]

[edit] eBay auctions

On April 20, 2004, Bill Tozier, a researcher with an Erdős number of 4, offered the chance for collaboration to attain an Erdős number of 5 in an auction on eBay. The final bid was $1,031, though apparently the winning bidder had no intention to pay.^[15] The winner (who already had an Erdős number of 3) considered it a "mockery", and said "papers have to be worked and earned, not sold, auctioned or bought".

Another eBay auction offered an Erdős number of 2 for a prospective paper to be submitted for publication to Chance (a magazine of the American Statistical Association) about skill in the World Series of Poker and the World Poker Tour. It closed on 22 July 2004 with a winning bid of $127.40.

[edit] Cultural anecdotes

It is jokingly said that American baseball player Hank Aaron has an Erdős number of 1 because he autographed a baseball with Erdős when Emory University awarded them both honorary degrees on the same day.

As of June 2007, the University of Memphis mathematical sciences faculty has the most faculty members with Erdős number 1 of any mathematics department. The Erdős-1 mathematicians are Béla Bollobás, Ralph Faudree, Jeno Lehel, Cecil C. Rousseau, and Richard Schelp. Three of these faculty members are among the top ten most frequent co-authors with Erdős. [2][3]

[edit] See also

[edit] Notes and references

[edit] Further reading

• Goffman, Casper (1969). "And What Is Your Erdös Number?". American Mathematical Monthly 76 (7): 791. doi:10.2307/2317868. 
• De Castro, Rodrigo; Grossman, Jerrold W. (1999). "Famous Trails to Paul Erdős". The Mathematical Intelligencer 21 (3): 51–63. MR1709679. http://www.oakland.edu/enp/trails.ps.  Original Spanish version in Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 23 (89) 563–582, 1999, MR1744115.

[edit] External links

• Jerry Grossman, The Erdös Number Project. Contains statistics and a complete list of all mathematicians with an Erdős number less than or equal to 2.
• "On a Portion of the Well-Known Collaboration Graph", Jerrold W. Grossman and Patrick D. F. Ion.
• "Some Analyses of Erdős Collaboration Graph", Vladimir Batagelj and Andrej Mrvar.
• American Mathematical Society, MR Collaboration Distance. A search engine for Erdős numbers and collaboration distance between other authors. Special access required.
• "Theorems for Sale" (From Science News, Vol. 165, No. 24, June 12, 2004)