# Erdős number

The Erdős number (IPA: [ɛrdøːʃ]), honouring the late Hungarian mathematician Paul Erdős, one of the most prolific writers of mathematical papers, is a way of describing the "collaborative distance", in regard to mathematical papers, between an author and Erdős.

## 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[1]; 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 infinite (or undefined) Erdős number. There is of course room for ambiguity over what constitutes a link between two authors; the Erdős 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 Erdős number was most likely first defined by Casper Goffman, an analyst whose own Erdős number is 1.[2] Goffman published his observations about Erdős's prolific collaboration in a 1969 article entitled "And what is your Erdős number?"[3]

## 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;[4] 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 via John Tukey, who has Erdős number 2, to Erdős. Similarly, the prominent geneticist Eric Lander and the mathematician Daniel Kleitman have collaborated on papers,[5][6] and since Kleitman has an Erdős number of 1,[7] 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.

Jerry Grossman, Marc Lipman, and Eddie Cheng have been looking at some questions in pure graph theory motivated by these collaboration graphs.

Tompa[8] 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 Georg Frobenius (born 1849, Erdős number 3), depending on the standard of publication eligibility[9]. It seems that older historic figures such as Leonhard Euler do not have finite Erdős numbers.

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

## Effect of Erdős' death on the Erdős number

Given that Erdős died in 1996 and no works of his remain to be published, it is no longer possible for a person to be newly assigned an Erdős number of 1. Likewise, when everybody with an Erdős number of 1 passes on, it will be impossible for a new person to obtain an Erdős number of 2, and so on. As a result of this drift, the mean Erdős number of living people must increase over time.

## Outside mathematics

### Pauli number

The Pauli number is an application of the same idea in Physics, connecting authors of physical papers with co-authors connected with Wolfgang Pauli.

### 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 a 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.

### 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.

### eBay auctions

On April 20, 2004, Bill Tozier, a researcher with Erdős number 4, offered the chance for collaboration to attain an Erdős number 5 in an auction on eBay. The final bid was \$1,031, though apparently the winning bidder had no intention to pay[10]. 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. This is noteworthy because with the exception of a few co-written articles to be published posthumously, 2 is the lowest number that can now be achieved.

### Cultural anecdotes

It is jokingly said that the famous 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 Bela Bollobas, 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]

## References

1. ^ Erdős Number Project
2. ^ [1] Michael Golomb's obituary of Paul Erdős
3. ^ Goffman, Casper (1969). "And what is your Erdős number?". American Mathematical Monthly 76.
4. ^ according to the Erdős Number Project
5. ^ http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=pubmed&cmd=search&term=lander+kleitman
6. ^ http://www-math.mit.edu/~djk/list.html
8. ^ Tompa, Martin (1989). "Figures of merit". ACM SIGACT News 20 (1): 62–71. DOI:10.1145/65780.65782.  Tompa, Martin (1990). "Figures of merit: the sequel". ACM SIGACT News 21 (4): 78–81. DOI:10.1145/101371.101376.
9. ^ Erdős Number Project - Paths to Erdős
10. ^ Decrease Your Erdős Number