The Erdős number (Hungarian pronunciation: [ˈɛrdøːʃ]) describes the "collaborative distance" between a person and mathematician Paul Erdős, as measured by authorship of mathematical papers.
The same principle has been proposed for other eminent persons in other fields.
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The idea of the Erdős number 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 (at least 1,525[2]) than any other mathematician in history.[1]
To be assigned an Erdős number, an author must co-write a research paper with an author with a finite Erdős number. Paul Erdős has an Erdős number of zero. Anybody else's Erdős number is k + 1 where k is the lowest Erdős number of any coauthor.
Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 511 direct collaborators;[3] 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 (9267 people as of 2010[4]), 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:
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.[5]
The Erdős number was most likely first defined in print by Casper Goffman, an analyst whose own Erdős number is 1.[6] Goffman published his observations about Erdős' prolific collaboration in a 1969 article entitled "And what is your Erdős number?"[7]
While Erdős collaborated with hundreds of co-authors, there were some individuals with whom he co-authored dozens of papers. This is a list of the ten persons that most frequently co-authored with Erdős and their number of papers co-authored with Erdõs (i.e. their number of collaborations).[8]
| Co-Author | Number of Collaborations |
|---|---|
| Andras Sarkozy | 62 |
| Andras Hajnal | 56 |
| Ralph Faudree | 50 |
| Richard Schelp | 42 |
| Cecil C. Rousseau | 35 |
| Vera T. Sos | 35 |
| Alfred Renyi | 32 |
| Pal Turan | 30 |
| Endre Szemeredi | 29 |
| Ronald Graham | 28 |
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, and the mean Erdős number is 4.65;[3] 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.[9] For example, political scientist Steven Brams has an Erdős number of 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 an Erdős number of 2. Similarly, the prominent geneticist Eric Lander and the mathematician Daniel Kleitman have collaborated on papers,[10][11] and since Kleitman has an Erdős number of 1,[12] a large fraction of the genetics and genomics community can be linked via Lander and his numerous collaborators. Similarly, collaboration with Gustavus Simmons opened the door for Erdős numbers within the cryptographic research community.
According to Alex Lopez-Ortiz, all the Fields and Nevanlinna prize winners during the three cycles in 1986 to 1994 have Erdős numbers of at most 9. Similarly, many linguists have finite Erdős numbers, many due to chains of collaboration with such notable scholars as Noam Chomsky (Erdős number 4),[13] William Labov (3),[14] Mark Liberman (3),[15] Geoffrey Pullum (3),[16] or Ivan Sag (4).[17]
Earlier mathematicians published fewer papers than modern ones, and more rarely published jointly written papers. The earliest person known to have a finite 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.[18] It seems that older historic figures such as Leonhard Euler (born 1707) do not have finite Erdős numbers.
Tompa[19] 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.
K. Dixit and colleagues argue that "for an individual researcher, a measure such as Erdős number captures the structural properties of network whereas the h-index captures the citation impact of the publications. One can be easily convinced that ranking in coauthorship networks should take into account both measures to generate a realistic and acceptable ranking." Several author ranking systems based on eigenvector centrality have been proposed already, for instance the Phys Author Rank Algorithm.[20]
A number of variations on the concept have been proposed to apply to other fields.
| Field | Target Person(s) | Died | Measured via |
|---|---|---|---|
| Mathematics | Paul Erdős | 1996 | Erdős number |
| Physics | Albert Einstein | 1955 | Einstein number[21] |
| Acting | Kevin Bacon | living | Bacon number |
| Maths+Acting | Paul Erdős & Kevin Bacon | n/a | Erdős–Bacon number |
| Chess | Paul Morphy | 1884 | Morphy number |
| go | Honinbo Shusaku | 1862 | Shusaku number |
| Linguistics | Noam Chomsky | Living | Chomsky 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. Although this is the most well-known numbering system of this type, it was conceived of in 1994, 25 years after Goffman's article on the Erdős number.
A small number of people are connected to both Erdős and Bacon and thus have an Erdős–Bacon number, which combines the two numbers. One example is the actress-mathematician Danica McKellar, best known for playing Winnie Cooper on the TV series, The Wonder Years. Her Erdős number is 4[22] and her Bacon number is 2.[23] 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.[24]
The Shusaku number represents the "distance" between a go player and Honinbo Shusaku, measured in Go opponents.[25] Shusaku himself has the Shusaku number 0. If a player has played against Shusaku himself, that player would have a Shusaku number of 1. And so on.[26]