Paul Erdős in 1992.

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 people in other fields.

## Overview

Paul Erdős (1913–1996) was an influential 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]

The idea of the Erdős number was created by the mathematician's friends as a humorous tribute to his enormous output as one of the most prolific modern writers of mathematical papers. The Erdős number has become well known in scientific circles as a tongue-in-cheek measurement of mathematical prominence.

## 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.

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 509 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:

... 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.[5]

The Erdős number was most likely first defined in print by Casper Goffman, an analyst whose own Erdős number is 2.[4] Goffman published his observations about Erdős' prolific collaboration in a 1969 article entitled "And what is your Erdős number?"[6] See also some comments in an obituary by Michael Golomb.[7]

## Most frequent Erdős collaborators

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 who most frequently co-authored with Erdős and their number of papers co-authored with Erdős (i.e. their number of collaborations).[8]

Co-author Number of collaborations
András Sárközy 62
András Hajnal 56
Ralph Faudree 50
Richard Schelp 42
Cecil C. Rousseau 35
Vera T. Sós 35
Alfréd Rényi 32
Pál Turán 30
Endre Szemerédi 29
Ronald Graham 28

## 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, 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. There are also connections with arts fields.[13]

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),[14] William Labov (3),[15] Mark Liberman (3),[16] Geoffrey Pullum (3),[17] or Ivan Sag (4).[18]

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.[19] It seems that older historic figures such as Leonhard Euler (born 1707) do not have finite Erdős numbers.

Tompa[20] 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.[21][22]

## Variations

A number of variations on the concept have been proposed to apply to other fields.

Field Target person(s) Date of death Measured via
Mathematics Paul Erdős 1996 Erdős number
Physics Albert Einstein 1955 Einstein number[23]
Acting Kevin Bacon Living Bacon number
Math+Acting Paul Erdős & Kevin Bacon N/A Erdős–Bacon number
Chess Paul Morphy 1884 Morphy number
Go Honinbo Shusaku 1862 Shusaku number[24][24]
Economics Joseph Stiglitz Living Stiglitz number[25]

### 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. 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 by taking their sum. 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[26] and her Bacon number is 2.[27] 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.[28]

### Shusaku number

The Shusaku number represents the "distance" between a go player and Honinbo Shusaku, measured in Go opponents.[24] 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.[29]

## References

1. ^ a b Newman, M. E. J. The structure of scientific collaboration networks. In: Proc. Natl. Acad. Sci. USA, 2001. doi:10.1073/pnas.021544898
2. ^ Grossman, Jerry. "Publications of Paul Erdös". Retrieved 1 Feb 2011.
3. ^ a b Erdős Number Project
4. ^ a b Erdos2, Version 2010, October 20, 2010.
5. ^ Grossman et al.Erdös numbers of the second kind,” in Facts about Erdös Numbers and the Collaboration Graph. The Erdös Number Project, Oakland University, USA. Retrieved July 25, 2009.
6. ^ Goffman, Casper (1969). "And what is your Erdős number?". American Mathematical Monthly 76 (7): 791. doi:10.2307/2317868. JSTOR 2317868.
7. ^ Erdős' obituary by Michael Golomb
8. ^ Grossman, Jerry, Erdos0p, Version 2010, The Erdos Number Project, Oakland University, USA, October 20, 2010.
9. ^ Grossman, Jerry. "Some Famous People with Finite Erdös Numbers". Retrieved 1 February 2011.
10. ^ A dictionary-based approach for gene annotation. [J Comput Biol. 1999 Fall-Winter] - PubMed Result
11. ^ Prof. Daniel Kleitman's Publications Since 1980 more or less
12. ^ Erdős, Paul; Daniel Kleitman (April 1971). "On Collections of Subsets Containing No 4-Member Boolean Algebra". Proceedings of the American Mathematical Society 28 (1): 87–90. doi:10.2307/2037762. JSTOR 2037762.
13. ^ Bowen, Jonathan P.; Wilson, Robin J. (10–12 July 2012). "Visualising Virtual Communities: From Erdős to the Arts". In Stuart Dunn, Jonathan P. Bowen, and Kia Ng. EVA London 2012: Electronic Visualisation and the Arts. Electronic Workshops in Computing. British Computer Society. pp. 238–244.
14. ^ My Erdős Number is 8, 2004.
15. ^ "Aaron Dinkin has a web site?". Ling.upenn.edu. Retrieved 2010-08-29.
17. ^ "Christopher Potts: Miscellany". Stanford.edu. Retrieved 2010-08-29.
18. ^ "Bob's Erdos Number". Lingo.stanford.edu. Retrieved 2010-08-29.
19. ^ Erdős Number Project - Paths to Erdős
20. ^ 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.
21. ^ Kashyap Dixit, S Kameshwaran, Sameep Mehta, Vinayaka Pandit, N Viswanadham, Towards simultaneously exploiting structure and outcomes in interaction networks for node ranking, IBM Research Report R109002, February 2009; also appeared as Kameshwaran, S.; Pandit, V.; Mehta, S.; Viswanadham, N.; Dixit, K. (2010). "Outcome aware ranking in interaction networks". Proceedings of the 19th ACM international conference on Information and knowledge management (CIKM '10): 229–238. doi:10.1145/1871437.1871470. ISBN 978-1-4503-0099-5. edit
22. ^
23. ^ People quoting their Einstein numbers: Sameen Ahmed Khan and Jonathan D. Victor
24. ^ a b c "how low is your Winning Shusaku Number". EuroGoTV. Retrieved 20 May 2011.
25. ^ Mentions in Freakonomics and the Wall Street Journal
26. ^ McKellar's co-author L. Chayes published a paper with E.H. Lieb, who in turn co-authored a paper with D.J. Kleitman, a co-author of Paul Erdős.
27. ^ Danica McKellar was in "The Year That Trembled" (2002) with James Kisicki , who was in "Telling Lies in America" (1997) with Kevin Bacon.
28. ^ Daniel J. Kleitman, "My Career in the Movies,", Notices of the American Mathematical Society, 45, 502 (April 1998)
29. ^