fbpx
Wikipedia

Erdős number

January 07, 2023

The Erdős number (Hungarian: [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.

Paul Erdős in 1992

Overview

Paul Erdős (1913–1996) was an influential Hungarian mathematician who in the latter part of his life spent a great deal of time writing papers with a large number of colleagues, working on solutions to outstanding mathematical problems.[1] He published more papers during his lifetime (at least 1,525[2]) than any other mathematician in history.[1] (Leonhard Euler published more total pages of mathematics but fewer separate papers: about 800.)[3] Erdős spent a large portion of his later life living out of a suitcase, visiting 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. Later 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.[4] For example, Erdős collaboration graphs can tell us how authors cluster, how the number of co-authors per paper evolves over time, or how new theories propagate.[5]

Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers.[6] 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 lower.[7] As time passes, the lowest Erdős number that can still be achieved will necessarily increase, as mathematicians with low Erdős numbers die and become unavailable for collaboration. Still, historical figures can have low Erdős numbers. For example, renowned Indian mathematician Srinivasa Ramanujan has an Erdős number of only 3 (through G. H. Hardy, Erdős number 2), even though Paul Erdős was only 7 years old when Ramanujan died.[8]

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. The American Mathematical Society provides a free online tool to determine the collaboration distance between two mathematical authors listed in the Mathematical Reviews catalogue.[8]

Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 509 direct collaborators;[4] 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 (12,600 people as of 7 August 2020[9]), 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 covers other subjects only in a limited way, and which also includes some non-research publications. The Erdős Number Project web site says:

... One drawback of the MR system is that it considers all jointly authored works as providing legitimate links, even articles such as obituaries, which are not really joint research. ...

[10] It also 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 excludes 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.[9] 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.[7] 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, Jean-Pierre Serre 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.[6]

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]

Related fields

As of 2022, all Fields Medalists have a finite Erdős number, with values that range between 2 and 6, and a median of 3. In contrast, the median Erdős number across all mathematicians (with a finite Erdős number) is 5, with an extreme value of 13.[16] The table below summarizes the Erdős number statistics for Nobel prize laureates in Physics, Chemistry, Medicine and Economics.[17] The first column counts the number of laureates. The second column counts the number of winners with a finite Erdős number. The third column is the percentage of winners with a finite Erdős number. The remaining columns report the minimum, maximum, average and median Erdős numbers among those laureates.

Statistics on Mathematical Collaboration, 1903-2016
#Laureates #Erdős %Erdős Min Max Average Median
Fields Medal 56 56 100.0% 2 6 3.36 3
Nobel Economics 76 47 61.84% 2 8 4.11 4
Nobel Chemistry 172 42 24.42% 3 10 5.48 5
Nobel Medicine 210 58 27.62% 3 12 5.50 5
Nobel Physics 200 159 79.50% 2 12 5.63 5

Physics

Among the Nobel Prize laureates in Physics, Albert Einstein and Sheldon 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, and David Wineland. Fields Medal-winning physicist Ed Witten has an Erdős number of 3.[7]

Biology

Computational biologist Lior Pachter has an Erdős number of 2.[18] Evolutionary biologist Richard Lenski has an Erdős number of 3, having co-authored a publication with Lior Pachter and with mathematician Bernd Sturmfels, each of whom has an Erdős number of 2.[19]

Finance and economics

There are at least two winners of the Nobel Prize in Economics with an Erdős number of 2: Harry M. Markowitz (1990) and Leonid Kantorovich (1975). 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 (1972), Milton Friedman (1976), Herbert A. Simon (1978), Gerard Debreu (1983), John Forbes Nash, Jr. (1994), James Mirrlees (1996), Daniel McFadden (2000), Daniel Kahneman (2002), Robert J. Aumann (2005), Leonid Hurwicz (2007), Roger Myerson (2007), Alvin E. Roth (2012), and Lloyd S. Shapley (2012) and Jean Tirole (2014).[20]

Some investment firms have been founded by mathematicians with low Erdős numbers, among them James B. Ax of Axcom Technologies, and James H. Simons of Renaissance Technologies, both with an Erdős number of 3.[21][22]

Philosophy

Since the more formal versions of philosophy share reasoning with the basics of mathematics, these fields overlap considerably, and Erdős numbers are available for many philosophers.[23] Philosophers John P. Burgess and Brian Skyrms have an Erdős number of 2.[9] Jon Barwise and Joel David Hamkins, both with Erdős number 2, have also contributed extensively to philosophy, but are primarily described as mathematicians.

Law

Judge Richard Posner, having coauthored with Alvin E. Roth, has an Erdős number of at most 4. Roberto Mangabeira Unger, a politician, philosopher and legal theorist who teaches at Harvard Law School, has an Erdős number of at most 4, having coauthored with Lee Smolin.

Politics

Angela Merkel, Chancellor of Germany from 2005 to 2021, has an Erdős number of at most 5.[14]

Engineering

Some fields of engineering, in particular communication theory and cryptography, make direct use of the discrete mathematics championed by Erdős. It is therefore not surprising that practitioners in these fields have low Erdős numbers. For example, Robert McEliece, a professor of electrical engineering at Caltech, had an Erdős number of 1, having collaborated with Erdős himself.[24] Cryptographers Ron Rivest, Adi Shamir, and Leonard Adleman, inventors of the RSA cryptosystem, all have Erdős number 2.[18]

Linguistics

The Romanian mathematician and computational linguist Solomon Marcus had an Erdős number of 1 for a paper in Acta Mathematica Hungarica that he co-authored with Erdős in 1957.[25]

Impact

 
Paul Erdős in 1985 at the University of Adelaide teaching Terence Tao, who was then 10 years old. Tao became a math professor at UCLA, received the Fields Medal in 2006, and was elected a Fellow of the Royal Society in 2007. His Erdős number is 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;[4] 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.[26] 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,[27][28] and since Kleitman has an Erdős number of 1,[29] 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),[30] William Labov (3),[31] Mark Liberman (3),[32] Geoffrey Pullum (3),[33] or Ivan Sag (4).[34] There are also connections with arts fields.[35]

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 Antoine Lavoisier (born 1743, Erdős number 13), Richard Dedekind (born 1831, Erdős number 7), or Ferdinand Georg Frobenius (born 1849, Erdős number 3), depending on the standard of publication eligibility.[36]

Martin Tompa[37] 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."[38]

In 2004 William Tozier, a mathematician with an Erdős number of 4, auctioned off a co-authorship on eBay, hence providing the buyer with an Erdős number of 5. The winning bid of $1031 was posted by a Spanish mathematician, who however did not intend to pay but just placed the bid to stop what he considered a mockery.[39][40]

Variations

A number of variations on the concept have been proposed to apply to other fields, notably the Bacon number (as in the game Six Degrees of Kevin Bacon), connecting actors to the actor Kevin Bacon by a chain of joint appearances in films. 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,[41] and her Bacon number is 2.[42]

Further extension is possible. For example, the "Erdős–Bacon–Sabbath number" is the sum of the Erdős–Bacon number and the collaborative distance to the band Black Sabbath in terms of singing in public. Physicist Stephen Hawking had an Erdős–Bacon–Sabbath number of 8,[43] and actress Natalie Portman has one of 11 (her Erdős number is 5).[44]

In chess, the Morphy number describes a player's connection to Paul Morphy, widely considered the greatest chess player of his time and an unofficial World Chess Champion.[45]

In video games, the Ryu number describes a video game character's connection to the Street Fighter character Ryu.[46][47]

See also

References

  1. ^ a b Newman, Mark E. J. (2001). "The structure of scientific collaboration networks". Proceedings of the National Academy of Sciences of the United States of America. 98 (2): 404–409. doi:10.1073/pnas.021544898. PMC 14598. PMID 11149952.
  2. ^ Grossman, Jerry. "Publications of Paul Erdős". Retrieved 1 Feb 2011.
  3. ^ "Frequently Asked Questions". The Euler Archive. Dartmouth College.
  4. ^ a b c "Facts about Erdös Numbers and the Collaboration Graph". Oakland University.{{cite web}}: CS1 maint: url-status (link)
  5. ^ "Facts about Erdös Numbers and the Collaboration Graph". Erdös Number Project. Oakland University.
  6. ^ a b De Castro, Rodrigo; Grossman, Jerrold W. (1999). (PDF). The Mathematical Intelligencer. 21 (3): 51–63. doi:10.1007/BF03025416. MR 1709679. S2CID 120046886. Archived from the original (PDF) on 2015-09-24. Original Spanish version in Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 23 (89) 563–582, 1999, MR1744115.
  7. ^ a b c "Some Famous People with Finite Erdős Numbers". oakland.edu. Retrieved 4 April 2014.
  8. ^ a b "Collaboration Distance". MathSciNet. American Mathematical Society.
  9. ^ a b c Erdos2, Version 2020, 7 August 2020.
  10. ^ "Compute your Erdös number - The Erdös Number Project". Oakland University. 1999-02-22. Retrieved 2022-10-15.
  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. ^ "Paul Erdös at Purdue". www.math.purdue.edu.
  14. ^ a b "Paths to Erdös". The Erdös Number Project. Oakland University.
  15. ^ Grossman, Jerry, Erdos0p, Version 2010, The Erdős Number Project, Oakland University, US, October 20, 2010.
  16. ^ "Facts about Erdös Numbers and the Collaboration Graph - The Erdös Number Project- Oakland University". wwwp.oakland.edu. Retrieved 2016-10-27.
  17. ^ López de Prado, Marcos (2016). "Mathematics and Economics: A reality check". The Journal of Portfolio Management. 43 (1): 5–8. doi:10.3905/jpm.2016.43.1.005. S2CID 219231926.
  18. ^ a b "List of all people with Erdos number less than or equal to 2". The Erdös Number Project. Oakland University. 14 July 2015. Retrieved 25 August 2015.
  19. ^ Richard Lenski (May 28, 2015). "Erdös with a non-kosher side of Bacon".
  20. ^ Grossman, J. (2015): "The Erdős Number Project." http://wwwp.oakland.edu/enp/erdpaths/
  21. ^ Kishan, Saijel (2016-11-11). "Six Degrees of Quant: Kevin Bacon and the Erdős Number Mystery". Bloomberg.com. Retrieved 2016-11-12.
  22. ^ Bailey, David H. (2016-11-06). "Erdős Numbers: A True "Prince and the Pauper" story". The Mathematical Investor. Retrieved 2016-11-12.
  23. ^ Toby Handfield. (PDF). Archived from the original (PDF) on 2016-02-21.
  24. ^ Erdős, Paul, Robert McEliece, and Herbert Taylor (1971). "Ramsey bounds for graph products" (PDF). Pacific Journal of Mathematics. 37 (1): 45–46. doi:10.2140/pjm.1971.37.45.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  25. ^ Erdős, Paul; Marcus, Solomon (1957). "Sur la décomposition de l'espace euclidien en ensembles homogènes" [On the decomposition of the Euclidean space into homogeneous sets]. Acta Mathematica Hungarica. 8 (3–4): 443–452. doi:10.1007/BF02020326. MR 0095456. S2CID 121671198.
  26. ^ Grossman, Jerry. "Some Famous People with Finite Erdős Numbers". Retrieved 1 February 2011.
  27. ^ Pachter, L; Batzoglou, S; Spitkovsky, VI; Banks, E; Lander, ES; Kleitman, DJ; Berger, B (1999). "A dictionary-based approach for gene annotation". J Comput Biol. 6 (3–4): 419–30. doi:10.1089/106652799318364. PMID 10582576.
  28. ^ Kleitman, Daniel. "Publications Since 1980 more or less". Massachusetts Institute of Technology.
  29. ^ Erdős, Paul; Kleitman, Daniel (April 1971). "On Collections of Subsets Containing No 4-Member Boolean Algebra" (PDF). Proceedings of the American Mathematical Society. 28 (1): 87–90. doi:10.2307/2037762. JSTOR 2037762.
  30. ^ von Fintel, Kai (2004). . Semantics, Inc. Archived from the original on 23 August 2006.
  31. ^ "Aaron Dinkin has a web site?". Ling.upenn.edu. Retrieved 2010-08-29.
  32. ^ "Mark Liberman's Home Page". Ling.upenn.edu. Retrieved 2010-08-29.
  33. ^ "Christopher Potts: Miscellany". Stanford.edu. Retrieved 2010-08-29.
  34. ^ "Bob's Erdős Number". Lingo.stanford.edu. Retrieved 2010-08-29.
  35. ^ 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 (eds.). EVA London 2012: Electronic Visualisation and the Arts. Electronic Workshops in Computing. British Computer Society. pp. 238–244.
  36. ^ "Paths to Erdös - The Erdös Number Project- Oakland University". oakland.edu.
  37. ^ Tompa, Martin (1989). "Figures of merit". ACM SIGACT News. 20 (1): 62–71. doi:10.1145/65780.65782. S2CID 34277380. Tompa, Martin (1990). "Figures of merit: the sequel". ACM SIGACT News. 21 (4): 78–81. doi:10.1145/101371.101376. S2CID 14144008.
  38. ^ 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" (PDF). 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. S2CID 16370569.
  39. ^ 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)
  40. ^ Klarreich, Erica (2004). "Theorem for Sale". Science News. 165 (24): 376–377. doi:10.2307/4015267. JSTOR 4015267.
  41. ^ McKellar's co-author Lincoln Chayes published a paper with Elliott H. Lieb, who in turn co-authored a paper with Daniel Kleitman, a co-author of Paul Erdős.
  42. ^ Danica McKellar was in The Year That Trembled (2002) with James Kisicki, who was in Telling Lies in America (1997) with Kevin Bacon.
  43. ^ Fisher, Len (2016-02-17). "What's your Erdős–Bacon–Sabbath number?". Times Higher Education. Retrieved 2018-07-29.
  44. ^ Sear, Richard (2012-09-15). "Erdős–Bacon–Sabbath numbers". Department of Physics, University of Surrey. Retrieved 2018-07-29.
  45. ^ Kingston, Taylor. "Your Morphy Number Is Up" (PDF). Chesscafe. (PDF) from the original on 13 June 2006. Retrieved 9 December 2020.
  46. ^ McWhertor, Michael (November 22, 2021). "Street Fighter's Ryu and Chun-Li join Ubisoft's take on Smash Bros., Brawlhalla". Polygon. Retrieved December 3, 2022.
  47. ^ Walker, Ian (June 22, 2021). "Street Fighter's Ryu Is The Kevin Bacon Of Video Games". Kotaku. Retrieved December 3, 2022.

External links

  • Jerry Grossman, The Erdős Number Project. Contains statistics and a complete list of all mathematicians with an Erdős number less than or equal to 2.
  • "On a Portion of the Well-Known Collaboration Graph", Jerrold W. Grossman and Patrick D. F. Ion.
  • "Some Analyses of Erdős Collaboration Graph", Vladimir Batagelj and Andrej Mrvar.
  • American Mathematical Society, MR: Search MSC database. A search engine for Erdős numbers and collaboration distance between other authors. As of 18 November 2011 no special access is required.
  • Numberphile video. Ron Graham on imaginary Erdős numbers.

January 07, 2023
erdős, number, hungarian, ˈɛrdøːʃ, describes, collaborative, distance, between, mathematician, paul, erdős, another, person, measured, authorship, mathematical, papers, same, principle, been, applied, other, fields, where, particular, individual, collaborated,. The Erdos number Hungarian ˈɛrdoːʃ describes the collaborative distance between mathematician Paul Erdos 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 Paul Erdos in 1992 Contents 1 Overview 2 Definition and application in mathematics 3 Most frequent Erdos collaborators 4 Related fields 4 1 Physics 4 2 Biology 4 3 Finance and economics 4 4 Philosophy 4 5 Law 4 6 Politics 4 7 Engineering 4 8 Linguistics 5 Impact 6 Variations 7 See also 8 References 9 External linksOverview EditPaul Erdos 1913 1996 was an influential Hungarian mathematician who in the latter part of his life spent a great deal of time writing papers with a large number of colleagues working on solutions to outstanding mathematical problems 1 He published more papers during his lifetime at least 1 525 2 than any other mathematician in history 1 Leonhard Euler published more total pages of mathematics but fewer separate papers about 800 3 Erdos spent a large portion of his later life living out of a suitcase visiting over 500 collaborators around the world The idea of the Erdos number was originally created by the mathematician s friends as a tribute to his enormous output Later 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 Erdos number as a proxy 4 For example Erdos collaboration graphs can tell us how authors cluster how the number of co authors per paper evolves over time or how new theories propagate 5 Several studies have shown that leading mathematicians tend to have particularly low Erdos numbers 6 The median Erdos number of Fields Medalists is 3 Only 7 097 about 5 of mathematicians with a collaboration path have an Erdos number of 2 or lower 7 As time passes the lowest Erdos number that can still be achieved will necessarily increase as mathematicians with low Erdos numbers die and become unavailable for collaboration Still historical figures can have low Erdos numbers For example renowned Indian mathematician Srinivasa Ramanujan has an Erdos number of only 3 through G H Hardy Erdos number 2 even though Paul Erdos was only 7 years old when Ramanujan died 8 Definition and application in mathematics Edit If Alice collaborates with Paul Erdos on one paper and with Bob on another but Bob never collaborates with Erdos himself then Alice is given an Erdos number of 1 and Bob is given an Erdos number of 2 as he is two steps from Erdos To be assigned an Erdos number someone must be a coauthor of a research paper with another person who has a finite Erdos number Paul Erdos has an Erdos number of zero Anybody else s Erdos number is k 1 where k is the lowest Erdos number of any coauthor The American Mathematical Society provides a free online tool to determine the collaboration distance between two mathematical authors listed in the Mathematical Reviews catalogue 8 Erdos wrote around 1 500 mathematical articles in his lifetime mostly co written He had 509 direct collaborators 4 these are the people with Erdos number 1 The people who have collaborated with them but not with Erdos himself have an Erdos number of 2 12 600 people as of 7 August 2020 9 those who have collaborated with people who have an Erdos number of 2 but not with Erdos or anyone with an Erdos number of 1 have an Erdos number of 3 and so forth A person with no such coauthorship chain connecting to Erdos has an Erdos number of infinity or an undefined one Since the death of Paul Erdos the lowest Erdos 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 covers other subjects only in a limited way and which also includes some non research publications The Erdos Number Project web site says One drawback of the MR system is that it considers all jointly authored works as providing legitimate links even articles such as obituaries which are not really joint research 10 It also 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 excludes non research publications such as elementary textbooks joint editorships obituaries and the like The Erdos number of the second kind restricts assignment of Erdos numbers to papers with only two collaborators 11 The Erdos number was most likely first defined in print by Casper Goffman an analyst whose own Erdos number is 2 9 Goffman published his observations about Erdos prolific collaboration in a 1969 article entitled And what is your Erdos number 12 See also some comments in an obituary by Michael Golomb 13 The median Erdos number among Fields medalists is as low as 3 7 Fields medalists with Erdos 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 Jean Pierre Serre and Terence Tao There are no Fields medalists with Erdos number 1 14 however Endre Szemeredi is an Abel Prize Laureate with Erdos number 1 6 Most frequent Erdos collaborators EditWhile Erdos 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 Erdos and their number of papers co authored with Erdos i e their number of collaborations 15 Co author Number of collaborationsAndras Sarkozy 62Andras Hajnal 56Ralph Faudree 50Richard Schelp 42Cecil C Rousseau 35Vera T Sos 35Alfred Renyi 32Pal Turan 30Endre Szemeredi 29Ronald Graham 28Related fields EditAs of 2022 update all Fields Medalists have a finite Erdos number with values that range between 2 and 6 and a median of 3 In contrast the median Erdos number across all mathematicians with a finite Erdos number is 5 with an extreme value of 13 16 The table below summarizes the Erdos number statistics for Nobel prize laureates in Physics Chemistry Medicine and Economics 17 The first column counts the number of laureates The second column counts the number of winners with a finite Erdos number The third column is the percentage of winners with a finite Erdos number The remaining columns report the minimum maximum average and median Erdos numbers among those laureates Statistics on Mathematical Collaboration 1903 2016 Laureates Erdos Erdos Min Max Average MedianFields Medal 56 56 100 0 2 6 3 36 3Nobel Economics 76 47 61 84 2 8 4 11 4Nobel Chemistry 172 42 24 42 3 10 5 48 5Nobel Medicine 210 58 27 62 3 12 5 50 5Nobel Physics 200 159 79 50 2 12 5 63 5Physics Edit Among the Nobel Prize laureates in Physics Albert Einstein and Sheldon Glashow have an Erdos number of 2 Nobel Laureates with an Erdos 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 and David Wineland Fields Medal winning physicist Ed Witten has an Erdos number of 3 7 Biology Edit Computational biologist Lior Pachter has an Erdos number of 2 18 Evolutionary biologist Richard Lenski has an Erdos number of 3 having co authored a publication with Lior Pachter and with mathematician Bernd Sturmfels each of whom has an Erdos number of 2 19 Finance and economics Edit There are at least two winners of the Nobel Prize in Economics with an Erdos number of 2 Harry M Markowitz 1990 and Leonid Kantorovich 1975 Other financial mathematicians with Erdos number of 2 include David Donoho Marc Yor Henry McKean Daniel Stroock and Joseph Keller Nobel Prize laureates in Economics with an Erdos number of 3 include Kenneth J Arrow 1972 Milton Friedman 1976 Herbert A Simon 1978 Gerard Debreu 1983 John Forbes Nash Jr 1994 James Mirrlees 1996 Daniel McFadden 2000 Daniel Kahneman 2002 Robert J Aumann 2005 Leonid Hurwicz 2007 Roger Myerson 2007 Alvin E Roth 2012 and Lloyd S Shapley 2012 and Jean Tirole 2014 20 Some investment firms have been founded by mathematicians with low Erdos numbers among them James B Ax of Axcom Technologies and James H Simons of Renaissance Technologies both with an Erdos number of 3 21 22 Philosophy Edit Since the more formal versions of philosophy share reasoning with the basics of mathematics these fields overlap considerably and Erdos numbers are available for many philosophers 23 Philosophers John P Burgess and Brian Skyrms have an Erdos number of 2 9 Jon Barwise and Joel David Hamkins both with Erdos number 2 have also contributed extensively to philosophy but are primarily described as mathematicians Law Edit Judge Richard Posner having coauthored with Alvin E Roth has an Erdos number of at most 4 Roberto Mangabeira Unger a politician philosopher and legal theorist who teaches at Harvard Law School has an Erdos number of at most 4 having coauthored with Lee Smolin Politics Edit Angela Merkel Chancellor of Germany from 2005 to 2021 has an Erdos number of at most 5 14 Engineering Edit Some fields of engineering in particular communication theory and cryptography make direct use of the discrete mathematics championed by Erdos It is therefore not surprising that practitioners in these fields have low Erdos numbers For example Robert McEliece a professor of electrical engineering at Caltech had an Erdos number of 1 having collaborated with Erdos himself 24 Cryptographers Ron Rivest Adi Shamir and Leonard Adleman inventors of the RSA cryptosystem all have Erdos number 2 18 Linguistics Edit The Romanian mathematician and computational linguist Solomon Marcus had an Erdos number of 1 for a paper in Acta Mathematica Hungarica that he co authored with Erdos in 1957 25 Impact Edit Paul Erdos in 1985 at the University of Adelaide teaching Terence Tao who was then 10 years old Tao became a math professor at UCLA received the Fields Medal in 2006 and was elected a Fellow of the Royal Society in 2007 His Erdos number is 2 Erdos 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 Erdos number the numbers range up to 15 the median is 5 and the mean is 4 65 4 almost everyone with a finite Erdos 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 Erdos numbers 26 For example political scientist Steven Brams has an Erdos 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 Erdos via John Tukey who has an Erdos number of 2 Similarly the prominent geneticist Eric Lander and the mathematician Daniel Kleitman have collaborated on papers 27 28 and since Kleitman has an Erdos number of 1 29 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 Erdos numbers within the cryptographic research community and many linguists have finite Erdos numbers many due to chains of collaboration with such notable scholars as Noam Chomsky Erdos number 4 30 William Labov 3 31 Mark Liberman 3 32 Geoffrey Pullum 3 33 or Ivan Sag 4 34 There are also connections with arts fields 35 According to Alex Lopez Ortiz all the Fields and Nevanlinna prize winners during the three cycles in 1986 to 1994 have Erdos 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 Erdos number is either Antoine Lavoisier born 1743 Erdos number 13 Richard Dedekind born 1831 Erdos number 7 or Ferdinand Georg Frobenius born 1849 Erdos number 3 depending on the standard of publication eligibility 36 Martin Tompa 37 proposed a directed graph version of the Erdos number problem by orienting edges of the collaboration graph from the alphabetically earlier author to the alphabetically later author and defining the monotone Erdos number of an author to be the length of a longest path from Erdos to the author in this directed graph He finds a path of this type of length 12 Also Michael Barr suggests rational Erdos numbers generalizing the idea that a person who has written p joint papers with Erdos should be assigned Erdos 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 Erdos 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 38 In 2004 William Tozier a mathematician with an Erdos number of 4 auctioned off a co authorship on eBay hence providing the buyer with an Erdos number of 5 The winning bid of 1031 was posted by a Spanish mathematician who however did not intend to pay but just placed the bid to stop what he considered a mockery 39 40 Variations EditA number of variations on the concept have been proposed to apply to other fields notably the Bacon number as in the game Six Degrees of Kevin Bacon connecting actors to the actor Kevin Bacon by a chain of joint appearances in films It was created in 1994 25 years after Goffman s article on the Erdos number A small number of people are connected to both Erdos and Bacon and thus have an Erdos 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 Erdos number is 4 41 and her Bacon number is 2 42 Further extension is possible For example the Erdos Bacon Sabbath number is the sum of the Erdos Bacon number and the collaborative distance to the band Black Sabbath in terms of singing in public Physicist Stephen Hawking had an Erdos Bacon Sabbath number of 8 43 and actress Natalie Portman has one of 11 her Erdos number is 5 44 In chess the Morphy number describes a player s connection to Paul Morphy widely considered the greatest chess player of his time and an unofficial World Chess Champion 45 In video games the Ryu number describes a video game character s connection to the Street Fighter character Ryu 46 47 See also EditAuthor level metrics Collaboration graph Graph modeling collaboration in a social network List of people by Erdos number List of things named after Paul Erdos Scientometrics Study of measuring and analysing science technology and innovation Six degrees of separation Concept of social inter connectedness of all people Small world experiment Experiments examining the average path length for social networks Small world network Graph where most nodes are reachable in a small number of steps Sociology of scientific knowledge Study of science as a social activityReferences Edit a b Newman Mark E J 2001 The structure of scientific collaboration networks Proceedings of the National Academy of Sciences of the United States of America 98 2 404 409 doi 10 1073 pnas 021544898 PMC 14598 PMID 11149952 Grossman Jerry Publications of Paul Erdos Retrieved 1 Feb 2011 Frequently Asked Questions The Euler Archive Dartmouth College a b c Facts about Erdos Numbers and the Collaboration Graph Oakland University a href Template Cite web html title Template Cite web cite web a CS1 maint url status link Facts about Erdos Numbers and the Collaboration Graph Erdos Number Project Oakland University a b De Castro Rodrigo Grossman Jerrold W 1999 Famous trails to Paul Erdos PDF The Mathematical Intelligencer 21 3 51 63 doi 10 1007 BF03025416 MR 1709679 S2CID 120046886 Archived from the original PDF on 2015 09 24 Original Spanish version in Rev Acad Colombiana Cienc Exact Fis Natur 23 89 563 582 1999 MR1744115 a b c Some Famous People with Finite Erdos Numbers oakland edu Retrieved 4 April 2014 a b Collaboration Distance MathSciNet American Mathematical Society a b c Erdos2 Version 2020 7 August 2020 Compute your Erdos number The Erdos Number Project Oakland University 1999 02 22 Retrieved 2022 10 15 Grossman et al Erdos numbers of the second kind in Facts about Erdos Numbers and the Collaboration Graph The Erdos Number Project Oakland University USA Retrieved July 25 2009 Goffman Casper 1969 And what is your Erdos number American Mathematical Monthly 76 7 791 doi 10 2307 2317868 JSTOR 2317868 Paul Erdos at Purdue www math purdue edu a b Paths to Erdos The Erdos Number Project Oakland University Grossman Jerry Erdos0p Version 2010 The Erdos Number Project Oakland University US October 20 2010 Facts about Erdos Numbers and the Collaboration Graph The Erdos Number Project Oakland University wwwp oakland edu Retrieved 2016 10 27 Lopez de Prado Marcos 2016 Mathematics and Economics A reality check The Journal of Portfolio Management 43 1 5 8 doi 10 3905 jpm 2016 43 1 005 S2CID 219231926 a b List of all people with Erdos number less than or equal to 2 The Erdos Number Project Oakland University 14 July 2015 Retrieved 25 August 2015 Richard Lenski May 28 2015 Erdos with a non kosher side of Bacon Grossman J 2015 The Erdos Number Project http wwwp oakland edu enp erdpaths Kishan Saijel 2016 11 11 Six Degrees of Quant Kevin Bacon and the Erdos Number Mystery Bloomberg com Retrieved 2016 11 12 Bailey David H 2016 11 06 Erdos Numbers A True Prince and the Pauper story The Mathematical Investor Retrieved 2016 11 12 Toby Handfield Philosophy research networks PDF Archived from the original PDF on 2016 02 21 Erdos Paul Robert McEliece and Herbert Taylor 1971 Ramsey bounds for graph products PDF Pacific Journal of Mathematics 37 1 45 46 doi 10 2140 pjm 1971 37 45 a href Template Cite journal html title Template Cite journal cite journal a CS1 maint multiple names authors list link Erdos Paul Marcus Solomon 1957 Sur la decomposition de l espace euclidien en ensembles homogenes On the decomposition of the Euclidean space into homogeneous sets Acta Mathematica Hungarica 8 3 4 443 452 doi 10 1007 BF02020326 MR 0095456 S2CID 121671198 Grossman Jerry Some Famous People with Finite Erdos Numbers Retrieved 1 February 2011 Pachter L Batzoglou S Spitkovsky VI Banks E Lander ES Kleitman DJ Berger B 1999 A dictionary based approach for gene annotation J Comput Biol 6 3 4 419 30 doi 10 1089 106652799318364 PMID 10582576 Kleitman Daniel Publications Since 1980 more or less Massachusetts Institute of Technology Erdos Paul Kleitman Daniel April 1971 On Collections of Subsets Containing No 4 Member Boolean Algebra PDF Proceedings of the American Mathematical Society 28 1 87 90 doi 10 2307 2037762 JSTOR 2037762 von Fintel Kai 2004 My Erdos Number is 8 Semantics Inc Archived from the original on 23 August 2006 Aaron Dinkin has a web site Ling upenn edu Retrieved 2010 08 29 Mark Liberman s Home Page Ling upenn edu Retrieved 2010 08 29 Christopher Potts Miscellany Stanford edu Retrieved 2010 08 29 Bob s Erdos Number Lingo stanford edu Retrieved 2010 08 29 Bowen Jonathan P Wilson Robin J 10 12 July 2012 Visualising Virtual Communities From Erdos to the Arts In Dunn Stuart Bowen Jonathan P Ng Kia eds EVA London 2012 Electronic Visualisation and the Arts Electronic Workshops in Computing British Computer Society pp 238 244 Paths to Erdos The Erdos Number Project Oakland University oakland edu Tompa Martin 1989 Figures of merit ACM SIGACT News 20 1 62 71 doi 10 1145 65780 65782 S2CID 34277380 Tompa Martin 1990 Figures of merit the sequel ACM SIGACT News 21 4 78 81 doi 10 1145 101371 101376 S2CID 14144008 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 PDF 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 S2CID 16370569 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 Klarreich Erica 2004 Theorem for Sale Science News 165 24 376 377 doi 10 2307 4015267 JSTOR 4015267 McKellar s co author Lincoln Chayes published a paper with Elliott H Lieb who in turn co authored a paper with Daniel Kleitman a co author of Paul Erdos Danica McKellar was in The Year That Trembled 2002 with James Kisicki who was in Telling Lies in America 1997 with Kevin Bacon Fisher Len 2016 02 17 What s your Erdos Bacon Sabbath number Times Higher Education Retrieved 2018 07 29 Sear Richard 2012 09 15 Erdos Bacon Sabbath numbers Department of Physics University of Surrey Retrieved 2018 07 29 Kingston Taylor Your Morphy Number Is Up PDF Chesscafe Archived PDF from the original on 13 June 2006 Retrieved 9 December 2020 McWhertor Michael November 22 2021 Street Fighter s Ryu and Chun Li join Ubisoft s take on Smash Bros Brawlhalla Polygon Retrieved December 3 2022 Walker Ian June 22 2021 Street Fighter s Ryu Is The Kevin Bacon Of Video Games Kotaku Retrieved December 3 2022 External links EditJerry Grossman The Erdos Number Project Contains statistics and a complete list of all mathematicians with an Erdos number less than or equal to 2 On a Portion of the Well Known Collaboration Graph Jerrold W Grossman and Patrick D F Ion Some Analyses of Erdos Collaboration Graph Vladimir Batagelj and Andrej Mrvar American Mathematical Society MR Search MSC database A search engine for Erdos numbers and collaboration distance between other authors As of 18 November 2011 no special access is required Numberphile video Ron Graham on imaginary Erdos numbers Retrieved from https en wikipedia org w index php title Erdos number amp oldid 1130500029, wikipedia, wiki, book, books, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.