# Erdős number

(Redirected from Erdős Number)
Paul Erdős in 1992

The Erdős number (Hungarian pronunciation: [ˈɛrdøːʃ]) describes the "collaborative distance" between mathematician Paul Erdős and another person, as measured by authorship of mathematical papers.

The same principle has been applied in other fields where a particular individual has collaborated with a large and broad number of peers. The American Mathematical Society provides a free online tool to determine the Erdős number of every mathematical author listed in the Mathematical Reviews catalogue.[1]

## Overview

Paul Erdős (1913–1996) was an influential mathematician who spent a large portion of his later life writing papers with a large number of colleagues, working on solutions to outstanding mathematical problems.[2] He published more papers during his lifetime (at least 1,525[3]) than any other mathematician in history.[2] (Leonhard Euler published more total pages of mathematics but fewer separate papers: about 800.)[4] Erdős spent a large portion of his later life living out of a suitcase, visiting his over 500 collaborators around the world.

The idea of the Erdős number was originally created by the mathematician's friends as a tribute to his enormous output. However, in later years it gained prominence as a tool to study how mathematicians cooperate to find answers to unsolved problems. Several projects are devoted to studying connectivity among researchers, using the Erdős number as a proxy.[5] For example, Erdős collaboration graphs can tell us how authors cluster together, how the number of co-authors per paper evolves over time, or how new theories propagate.[6]

Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers.[7] For example, only 134,007 mathematicians have an Erdős number, with a median value of 5. In contrast, the median Erdős number of Fields Medalists is 3. Only 7,097 (about 5%) of mathematicians with a collaboration path have an Erdős number of 2 or less.[8] Collaboration distances will necessarily increase over long time scales, as mathematicians with low Erdős numbers die and become unavailable for collaboration.

## Definition and application in mathematics

If Alice collaborates with Paul Erdős on one paper, and with Bob on another, but Bob never collaborates with Erdős himself, then Alice is given an Erdős number of 1 and Bob is given an Erdős number of 2, as he is two steps from Erdős.

To be assigned an Erdős number, someone must be a coauthor of a research paper with another person who has 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;[9] 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[10]), 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). Since the death of Paul Erdős, the lowest Erdős number that a new researcher can obtain is 2.

There is room for ambiguity over what constitutes a link between two authors. The American Mathematical Society collaboration distance calculator uses data from Mathematical Reviews, which includes most mathematics journals but not other subjects, and which also includes some non-research publications. 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.[11]

The Erdős number was most likely first defined in print by Casper Goffman, an analyst whose own Erdős number is 2.[10] Goffman published his observations about Erdős' prolific collaboration in a 1969 article entitled "And what is your Erdős number?"[12] See also some comments in an obituary by Michael Golomb.[13]

The median Erdős number among Fields medalists is as low as 3.[8] Fields medalists with Erdős number 2 include Atle Selberg, Kunihiko Kodaira, Klaus Roth, Alan Baker, Enrico Bombieri, David Mumford, Charles Fefferman, William Thurston, Shing-Tung Yau, Jean Bourgain, Richard Borcherds, Manjul Bhargava and Terence Tao. There are no Fields medalists with Erdős number 1,[14] however Endre Szemerédi is an Abel Prize Laureate with Erdős number 1.[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).[15]

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

## Related fields

### Physics

Among the Nobel Prize laureates in Physics, Albert Einstein and Sheldon Lee Glashow have an Erdős number of 2. Nobel Laureates with an Erdős number of 3 include Enrico Fermi, Otto Stern, Wolfgang Pauli, Max Born, Willis E. Lamb, Eugene Wigner, Richard P. Feynman, Hans A. Bethe, Murray Gell-Mann, Abdus Salam, Steven Weinberg, Norman F. Ramsey, Frank Wilczek, David Wineland. Fields Medal-winning physicist Ed Witten has an Erdős number of 3.[16]

### Chemistry

Nobel Prize laureates in Chemistry with an Erdős number of 3 include Lars Onsager, Kenichi Fukui, Herbert A. Hauptman, Walter Kohn.

### Medicine

Nobel Prize laureates in Medicine with an Erdős number of 3 include John Carew Eccles, Hamilton O. Smith, John E. Sulston.

### Finance and economics

Harry M. Markowitz is the only Nobel Prize laureate in Economics with an Erdős number of 2. Other financial mathematicians with Erdős number of 2 include David Donoho, Marc Yor, Henry McKean, Daniel Stroock, and Joseph Keller.

Nobel Prize laureates in Economics with an Erdős number of 3 include Kenneth J. Arrow, Herbert A. Simon, Gerard Debreu, James Mirrlees, Daniel Kahneman, Robert J. Aumann, Alvin E. Roth, and Lloyd S. Shapley.

### Law

Judge Richard Posner, having coauthored with Alvin E. Roth, has an Erdős number of at most 4.

### Social network analysis

Sociologist Barry Wellman has an Erdős number of 3 via social network analyst and statistician Ove Frank,[17] who collaborated with graph theorist Frank Harary.[18]

## Impact

Paul Erdős teaching Terence Tao in 1985 at the University of Adelaide. Tao, who was 10 years old at the time, became a professional mathematician. He received the Fields Medal in 2006 and was elected a Fellow of the Royal Society in 2007. Tao has an Erdős number of 2

Erdős numbers have been a part of the folklore of mathematicians throughout the world for many years. Among 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 is 4.65;[9] 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.[19] 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,[20][21] and since Kleitman has an Erdős number of 1,[22] 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, and many linguists have finite Erdős numbers, many due to chains of collaboration with such notable scholars as Noam Chomsky (Erdős number 4),[23] William Labov (3),[24] Mark Liberman (3),[25] Geoffrey Pullum (3),[26] or Ivan Sag (4).[27] There are also connections with arts fields.[28]

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.

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

Tompa[30] 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.

It has been argued that "for an individual researcher, a measure such as Erdős number captures the structural properties of [the] network whereas the h-index captures the citation impact of the publications," and that "One can be easily convinced that ranking in coauthorship networks should take into account both measures to generate a realistic and acceptable ranking."[31] Several author ranking systems based on eigenvector centrality have been proposed, for instance the Phys Author Rank Algorithm.[31][32]

In 2004 William Tozier, a mathematician with an Erdös number of 4, auctioned off an co-authorship on eBay and hence providing the buyer with an Erdös number of 5. The winning bid of \$1031 was posted by Spanish mathematician, who however did not intend to pay but just placed the bid to stop what he considered a mockery.[33][34]

## Variations

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

The best known is Bacon number (as in the game Six Degrees of Kevin Bacon), connecting actors that appeared in a film together to the actor Kevin Bacon. It was created 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[35] and her Bacon number is 2.[36] The lowest known Erdős–Bacon number is 3 – for Daniel Kleitman, a mathematics professor at MIT – his Erdős number is 1 and his Bacon number is 2.[37]

Further generalization is possible. For example, Erdős–Bacon–Sabbath numbers include the band Black Sabbath in the measure.[38] The lowest known Erdős–Bacon–Sabbath number is 8, a value shared by physicist Stephen Hawking,[39] neuroscientist Daniel Levitin[40] and inventor Ray Kurzweil,[41] all of whom have an Erdős number of 4, a Bacon number of 2, and a Sabbath number of 2.

English mathematician Peter Swinnerton-Dyer has an Erdős-Morphy number of 5.

Other targets include:

Target Measured via
Albert Einstein, theoretical physicist (1879–1955) Einstein number[42]
Paul Morphy, chess player (1837–1884) Morphy number
Honinbo Shusaku, Go player (1829–1862) Shusaku number[43]
Joseph Stiglitz, economist (1943- ) Stiglitz number[44]
Black Sabbath, rock band (active 1968 – present) Sabbath Number[45]

## References

1. ^ http://www.ams.org/mathscinet/collaborationDistance.html
2. ^ a b Newman, M. E. J. (2001). "The structure of scientific collaboration networks". Proc. Natl. Acad. Sci. USA 98 (2): 404–409. doi:10.1073/pnas.021544898. PMC 14598. PMID 11149952.
3. ^ Grossman, Jerry. "Publications of Paul Erdős". Retrieved 1 Feb 2011.
4. ^ https://www.math.dartmouth.edu/~euler/FAQ.html
5. ^ http://www.oakland.edu/enp
6. ^ Some statistics about Erdős numbers
7. ^ a b De Castro, Rodrigo; Grossman, Jerrold W. (1999). "Famous trails to Paul Erdős" (PDF). The Mathematical Intelligencer 21 (3): 51–63. doi:10.1007/BF03025416. MR 1709679. Original Spanish version in Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 23 (89) 563–582, 1999, MR 1744115.
8. ^ a b The Erdős Number Project http://www.oakland.edu/enp/erdpaths
9. ^ a b Erdős Number Project
10. ^ a b Erdos2, Version 2010, October 20, 2010.
11. ^ Grossman et al. "Erdős numbers of the second kind," in Facts about Erdős Numbers and the Collaboration Graph. The Erdős Number Project, Oakland University, USA. Retrieved July 25, 2009.
12. ^ Goffman, Casper (1969). "And what is your Erdős number?". American Mathematical Monthly 76 (7): 791. doi:10.2307/2317868. JSTOR 2317868.
13. ^ Erdős' obituary by Michael Golomb
14. ^ Paths to Erdős — The Erdős Number Project
15. ^ Grossman, Jerry, Erdos0p, Version 2010, The Erdős Number Project, Oakland University, USA, October 20, 2010.
16. ^ "Some Famous People with Finite Erdős Numbers". oakland.edu. Retrieved 4 April 2014.
17. ^ Barry Wellman, Ove Frank, Vicente Espinoza, Staffan Lundquist and Craig Wilson. "Integrating Individual, Relational and Structural Analysis". 1991. Social Networks 13 (Sept.): 223-50.
18. ^ Ove Frank; Frank Harary, "Cluster Inference by Using Transitivity Indices in Empirical Graphs." Journal of the American Statistical Association, 77, 380. (Dec., 1982), pp. 835-840.
19. ^ Grossman, Jerry. "Some Famous People with Finite Erdős Numbers". Retrieved 1 February 2011.
20. ^
21. ^ Prof. Daniel Kleitman's Publications Since 1980 more or less
22. ^ Erdős, Paul; Kleitman, Daniel (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.
23. ^ My Erdős Number is 8, 2004.
24. ^ "Aaron Dinkin has a web site?". Ling.upenn.edu. Retrieved 2010-08-29.
26. ^ "Christopher Potts: Miscellany". Stanford.edu. Retrieved 2010-08-29.
27. ^ "Bob's Erdős Number". Lingo.stanford.edu. Retrieved 2010-08-29.
28. ^ Bowen, Jonathan P.; Wilson, Robin J. (10–12 July 2012). "Visualising Virtual Communities: From Erdős to the Arts". In Dunn, Stuart; Bowen, Jonathan P.; Ng, Kia. EVA London 2012: Electronic Visualisation and the Arts. Electronic Workshops in Computing. British Computer Society. pp. 238–244.
29. ^ Erdős Number Project - Paths to Erdős
30. ^ 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.
31. ^ a b 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
32. ^
33. ^ Clifford A. Pickover: A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality. Wiley, 2011, ISBN 9781118046074, S. 33 (excerpt, p. 33, at Google Books)
34. ^ EricaKlarreich (2004). "Theorem for Sale". Science News 165 (24): 376–377. JSTOR 4015267.
35. ^ 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.
36. ^ Danica McKellar was in "The Year That Trembled" (2002) with James Kisicki , who was in "Telling Lies in America" (1997) with Kevin Bacon.
37. ^ Daniel J. Kleitman, "My Career in the Movies,", Notices of the American Mathematical Society, 45, 502 (April 1998)
38. ^ Erdosbaconsabbath.com
39. ^ The EBS project - Stephen Hawking
40. ^ The EBS project - Daniel Levitin
41. ^ The EBS project - Ray Kurzweil
42. ^ People quoting their Einstein numbers: Sameen Ahmed Khan and Jonathan D. Victor
43. ^ "how low is your Winning Shusaku Number". EuroGoTV. Retrieved 20 May 2011.
44. ^ Mentions in Freakonomics and the Wall Street Journal
45. ^ The Black Sabbath Game at the Wayback Machine (archived January 21, 2013)