Erdős number

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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[edit]

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 lifetime (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 tribute to his enormous output. Since then, it has gained prominence and several projects are devoted to studying connectivity among researchers, using the Erdős number as a proxy.[3] The average Erdős number of Fields Medalists is 3.21, with a standard deviation of 0.87 and a median of 3.[4] Those values will necessarily increase over long time scales, as mathematicians with low Erdős numbers die and become unavailable for collaboration. The American Mathematical Society provides a free tool to determine the Erdős Number of every mathematical author listed in the Mathematical Reviews catalogue.[5]

Definition[edit]

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;[6] 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[7]), 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.[8]

The Erdős number was most likely first defined in print by Casper Goffman, an analyst whose own Erdős number is 2.[7] Goffman published his observations about Erdős' prolific collaboration in a 1969 article entitled "And what is your Erdős number?"[9] See also some comments in an obituary by Michael Golomb.[10]

Most frequent Erdős collaborators[edit]

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).[11]

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

Erdős in Mathematics[edit]

The median Erdős number among Fields medalists is as low as 3.[12] 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 and Terence Tao.

Erdős in Physics[edit]

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.[13]

Erdős in Chemistry[edit]

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

Erdős in Medicine[edit]

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

Erdős in Mathematical Finance[edit]

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, Lloyd S. Shapley. Other Financial Mathematicians with Erdős number of 3 include Michael Harrison, Darrell Duffie, David Easley, Hélyette Geman, Peter Carr, Mark Pinsky, Pierre-Louis Lions, Robert Elliott, Bernt Oksendal, Hans Föllmer, and L. C. G. Rogers.

Erdős in Social Network Analysis[edit]

Sociologist Barry Wellman has an Erdős number of 3 via social network analyst and statistician Ove Frank[14] who collaborated with graph theorist Frank Harary.[15]

Impact[edit]

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 Erdős number is 4.65;[6] 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.[16] 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,[17][18] and since Kleitman has an Erdős number of 1,[19] 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),[20] William Labov (3),[21] Mark Liberman (3),[22] Geoffrey Pullum (3),[23] or Ivan Sag (4).[24]There are also connections with arts fields.[25]

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

Tompa[27] 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.[28][29]

Variations[edit]

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

Target Measured via
Paul Erdős, mathematician (1913–1996) Erdős number
Albert Einstein, theoretical physicist (1879–1955) Einstein number[30]
Kevin Bacon, actor (° 1958) Bacon number
Paul Erdős & Kevin Bacon Erdős–Bacon number
Paul Morphy, chess player (1837–1884) Morphy number
Honinbo Shusaku, Go player (1829–1862) Shusaku number[31]
Joseph Stiglitz, economist (° 1943) Stiglitz number[32]
Black Sabbath, rock band (active 1968 – present) Sabbath Number[33]
Paul Erdős & Kevin Bacon & Black Sabbath Erdős–Bacon–Sabbath number[34][35]

Bacon number[edit]

Main article: 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[36] and her Bacon number is 2.[37] 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.[38]

Generalizing this to Erdős-Bacon-Sabbath numbers can be done along the same principles as above. The lowest Erdős–Bacon-Sabbath number known is shared by physicist Stephen Hawking [39] and inventor Ray Kurzweil,[40] both of whom have an Erdős number of four, a Bacon number of two, and a Sabbath number of two.

Shusaku number[edit]

The Shusaku number represents the "distance" between a go player and Honinbo Shusaku, measured in Go opponents.[31] 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.[41]

See also[edit]

References[edit]

  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 Erdős". Retrieved 1 Feb 2011. 
  3. ^ http://www.oakland.edu/enp
  4. ^ The Erdős Number Project http://www.oakland.edu/enp/erdpaths
  5. ^ http://www.ams.org/mathscinet/collaborationDistance.html
  6. ^ a b Erdős Number Project
  7. ^ a b Erdos2, Version 2010, October 20, 2010.
  8. ^ Grossman et al.Erdös numbers of the second kind,” in Facts about Erdös Numbers and the Collaboration Graph. The Erdős Number Project, Oakland University, USA. Retrieved July 25, 2009.
  9. ^ Goffman, Casper (1969). "And what is your Erdős number?". American Mathematical Monthly 76 (7): 791. doi:10.2307/2317868. JSTOR 2317868. 
  10. ^ Erdős' obituary by Michael Golomb
  11. ^ Grossman, Jerry, Erdos0p, Version 2010, The Erdős Number Project, Oakland University, USA, October 20, 2010.
  12. ^ The Erdős Number Project http://www.oakland.edu/enp/erdpaths
  13. ^ "Some Famous People with Finite Erdös Numbers". oakland.edu. Retrieved 4 April 2014. 
  14. ^ Barry Wellman, Ove Frank, Vicente Espinoza, Staffan Lundquist and Craig Wilson. "Integrating Individual, Relational and Structural Analysis". 1991. Social Networks 13 (Sept.): 223-50.
  15. ^ 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.
  16. ^ Grossman, Jerry. "Some Famous People with Finite Erdös Numbers". Retrieved 1 February 2011. 
  17. ^ A dictionary-based approach for gene annotation. [J Comput Biol. 1999 Fall-Winter] - PubMed Result
  18. ^ Prof. Daniel Kleitman's Publications Since 1980 more or less
  19. ^ 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. 
  20. ^ My Erdős Number is 8, 2004.[dead link]
  21. ^ "Aaron Dinkin has a web site?". Ling.upenn.edu. Retrieved 2010-08-29. 
  22. ^ "Mark Liberman's Home Page". Ling.upenn.edu. Retrieved 2010-08-29. 
  23. ^ "Christopher Potts: Miscellany". Stanford.edu. Retrieved 2010-08-29. 
  24. ^ "Bob's Erdős Number". Lingo.stanford.edu. Retrieved 2010-08-29. 
  25. ^ 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. 
  26. ^ Erdős Number Project - Paths to Erdős
  27. ^ 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. 
  28. ^ 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
  29. ^ Phys Author Rank Algorithm.
  30. ^ People quoting their Einstein numbers: Sameen Ahmed Khan and Jonathan D. Victor
  31. ^ a b "how low is your Winning Shusaku Number". EuroGoTV. Retrieved 20 May 2011. 
  32. ^ Mentions in Freakonomics and the Wall Street Journal
  33. ^ The Black Sabbath Game
  34. ^ Erdős-Bacon-Sabbath Numbers: The People at the Center of the Universe
  35. ^ Erdős Bacon Sabbath Project
  36. ^ 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.
  37. ^ Danica McKellar was in "The Year That Trembled" (2002) with James Kisicki , who was in "Telling Lies in America" (1997) with Kevin Bacon.
  38. ^ Daniel J. Kleitman, "My Career in the Movies,", Notices of the American Mathematical Society, 45, 502 (April 1998)
  39. ^ the EBS project - Stephen Hawking
  40. ^ EBS project - Ray Kurzweil
  41. ^ Shusaku Number.

Further reading[edit]

External links[edit]