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Closeness of someone's association with mathematician Paul Erdős

The Erdős number 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 (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—over 500—working on solutions to outstanding mathematical problems. He published more papers during his lifetime (at least 1,525) than any other mathematician in history. Erdős spent most of his career with no permanent home or job. He traveled with everything he owned in two suitcases, and would visit mathematicians he wanted to collaborate with, often unexpectedly, and expect to stay with them. 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. 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. Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers (i. e. high proximity). 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. Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co written. He had 509 direct collaborators;), 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. 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. The Erdős number was most likely first defined in print by Casper Goffman, an analyst whose own Erdős number is 2. See also some comments in an obituary by Michael Golomb. The median Erdős number among Fields medalists is as low as 3. however, Endre Szemerédi is an Abel Prize Laureate with Erdős number 1. 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

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. The table below summarizes the Erdős number statistics for Nobel prize laureates in Physics, Chemistry, Medicine, and Economics. 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 MedianFields Medal5656100.0%263.363Nobel Economics764761.84%284.114Nobel Chemistry1724224.42%3105.485Nobel Medicine2105827.62%3125.505Nobel Physics20015979.50%2125.635

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.

Computational biologist Lior Pachter has an Erdős number of 2. 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.

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). 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.

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. Philosophers John P. Burgess and Brian Skyrms have an Erdős number of 2. Cryptographers Ron Rivest, Adi Shamir, and Leonard Adleman, inventors of the RSA cryptosystem, all have Erdős number 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; 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, and since Kleitman has an Erdős number of 1, 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), William Labov (3), Mark Liberman (3), Geoffrey Pullum (3), or Ivan Sag (4). There are also connections with arts fields. 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. Martin Tompa 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." 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 refused to pay and only placed the bid to stop what he considered a mockery.

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, and her Bacon number is 2. 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, and actress Natalie Portman has one of 11 (her Erdős number is 5). 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. In go, the Shusaku number describes a player's connection to Honinbo Shusaku, the strongest player of his time. In video games, the Ryu number describes a video game character's connection to the Street Fighter character Ryu.