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.
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. He published more papers during his lifetime (at least 1,525) than any other mathematician in history. (Leonhard Euler published more total pages of mathematics but fewer separate papers: about 800.) 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. 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. 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. For example, only 134,007 mathematicians have an Erdős number, with a median value of 5, according to Oakland University's Erdős Number Project. 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 lower. As time passes, the smallest 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, renown 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.
Definition and application in mathematics
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; 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), 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:
... 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.
The Erdős number was most likely first defined in print by Casper Goffman, an analyst whose own Erdős number is 2. Goffman published his observations about Erdős' prolific collaboration in a 1969 article entitled "And what is your Erdős number?" See also some comments in an obituary by Michael Golomb.
The median Erdős number among Fields medalists is as low as 3. 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, however Endre Szemerédi is an Abel Prize Laureate with Erdős number 1.
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).
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As of 2016, 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.
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.
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. Synthetic biologist Jeff Hasty has an Erdős number of 3, having co-authored publications with chemist Charles Cantor and mathematician Ruth Williams, both of whom have an Erdős number of 2.
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 (1996), 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. 
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.
Social network analysis
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; 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. 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.
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 Pierre-Simon Laplace (born 1749, Erdős number 14), Richard Dedekind (born 1831, Erdős number 7), or Ferdinand Georg Frobenius (born 1849, Erdős number 3), depending on the standard of publication eligibility. It seems that older historic figures such as Leonhard Euler (born 1707) do not (yet) have finite Erdős numbers.
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 however did not intend to pay but just 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.
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 and her Bacon number is 2.
Further generalization is possible. For example, Erdős–Bacon–Sabbath numbers include the band Black Sabbath in the measure. The lowest known Erdős–Bacon–Sabbath number is 8, for physicist Stephen Hawking who has an Erdős number of 4, a Bacon number of 2, and a Sabbath number of 2.
- Small-world experiment
- Small-world network
- Six degrees of separation
- List of people by Erdős number
- List of topics named after Paul Erdős
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- 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.
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- 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.
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- American Mathematical Society, MR Collaboration Distance. A search engine for Erdős numbers and collaboration distance between other authors. As of 18 November 2011 no special access is required.
- Microsoft Academic Search features Co-Author Path which by default shows visually a researcher's path to Paul Erdős, effectively estimating his or her Erdős number.
- Numberphile video. Ron Graham on imaginary Erdős numbers.