The Erdà Âs number () describes the "collaborative distance" between mathematician Paul Erdà Âs 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 Erdà Âs (1913âÂÂ1996) 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 colleaguesâÂÂmore than 500âÂÂworking 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.) Erdà Âs spent most of his career with no permanent home or job. He traveled with everything he owned in two suitcases, and would visit mathematicians with whom he wanted to collaborate, often unexpectedly, and expect to stay with them.
The idea of the Erdà Âs 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 Erdà Âs number as a proxy. For example, Erdà Âs 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 Erdà Âs numbers. The median Erdà Âs number of Fields Medalists is 3. Only 7,097 (about 5% of mathematicians with a collaboration path) have an Erdà Âs number of 2 or lower. As time passes, the lowest Erdà Âs number that can still be achieved will necessarily increase, as mathematicians with low Erdà Âs numbers die and become unavailable for collaboration. Still, historical figures can have low Erdà Âs numbers. For example, renowned Indian mathematician Srinivasa Ramanujan has an Erdà Âs number of only 3 (through G. H. Hardy, Erdà Âs number 2), even though Paul Erdà Âs was only 7 years old when Ramanujan died.
To be assigned an Erdà Âs number, someone must be a coauthor of a research paper with another person who has a finite Erdà Âs number. Paul Erdà Âs himself is assigned an Erdà Âs number of zero. A certain author's Erdà Âs number is one greater than the lowest Erdà Âs number of any of their collaborators; for example, an author who has coauthored a publication with Erdà Âs would have an Erdà Âs number of 1. The American Mathematical Society provides a free online tool to determine the collaboration distance between two mathematical authors listed in the Mathematical Reviews catalogue.
Erdà Âs wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 509 direct collaborators; these are the people with Erdà Âs number 1. The people who have collaborated with them (but not with Erdà Âs himself) have an Erdà Âs number of 2 (12,600 people as of 7 August 2020), those who have collaborated with people who have an Erdà Âs number of 2 (but not with Erdà Âs or anyone with an Erdà Âs number of 1) have an Erdà Âs number of 3, and so forth. A person with no such coauthorship chain connecting to Erdà Âs has an Erdà Âs number of infinity (or an undefined one). Since the death of Paul Erdà Âs, the lowest Erdà Âs 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 Erdà Âs Number Project web site says: It also says:
but excludes non-research publications such as elementary textbooks, joint editorships, obituaries, and the like. The "Erdà Âs number of the second kind" restricts assignment of Erdà Âs numbers to papers with only two collaborators.
The Erdà Âs number was most likely first defined in print by Casper Goffman, an analyst whose own Erdà Âs number is 2. Goffman published his observations about Erdà Âs' prolific collaboration in a 1969 article entitled "And what is your Erdà Âs number?" See also some comments in an obituary by Michael Golomb.
The median Erdà Âs number among Fields medalists is as low as 3. Fields medalists with Erdà Âs 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 Erdà Âs number 1; however, Endre Szemerédi is an Abel Prize Laureate with Erdà Âs number 1.
While Erdà Âs 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 Erdà Âs and their number of papers co-authored with Erdà Âs, i.e., their number of collaborations.
, all Fields medalists have a finite Erdà Âs number, with values that range between 2 and 6, and a median of 3. In contrast, the median Erdà Âs number across all mathematicians (with a finite Erdà Âs number) is 5, with an extreme value of 13. The table below summarizes the Erdà Âs 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 Erdà Âs number. The third column is the percentage of winners with a finite Erdà Âs number. The remaining columns report the minimum, maximum, average, and median Erdà Âs numbers among those laureates.
Among the Nobel Prize laureates in Physics, Albert Einstein and Sheldon Glashow have an Erdà Âs number of 2. Nobel Laureates with an Erdà Âs 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, and Giorgio Parisi. Fields Medal-winning physicist Ed Witten has an Erdà Âs number of 3.
Several prolific scientists working in Genetics, Biomedical Engineering, Mathematical, and Computational Biology have an Erdà Âs number of 2. Among them are Zvia Agur, Joel E. Cohen, Eugene Koonin, Bruce Kristal, Eric Lander, Lior Pachter and Temple F. Smith. Through collaborations with these authors there are many biologists with an Erdà Âs number of 3 and it has been argued that almost every author on a paper in the biological sciences can be linked to Erdà Âs.
There are at least two winners of the Nobel Prize in Economics with an Erdà Âs number of 2: Harry M. Markowitz (1990) and Leonid Kantorovich (1975). Other financial mathematicians with Erdà Âs number of 2 include David Donoho, Marc Yor, Henry McKean, Daniel Stroock, and Joseph Keller.
Nobel Prize laureates in Economics with an Erdà Âs 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 Erdà Âs numbers, among them James B. Ax of Axcom Technologies, and James H. Simons of Renaissance Technologies, both with an Erdà Âs number of 3.
Since the more formal versions of philosophy share reasoning with the basics of mathematics, these fields overlap considerably, and Erdà Âs numbers are available for many philosophers. Philosophers John P. Burgess and Brian Skyrms have an Erdà Âs number of 2. Jon Barwise and Joel David Hamkins, both with Erdà Âs number 2, have also contributed extensively to philosophy, but are primarily described as mathematicians.
Judge Richard Posner, having coauthored with Alvin E. Roth, has an Erdà Âs number of at most 4. Roberto Mangabeira Unger, a politician, philosopher, and legal theorist who teaches at Harvard Law School, has an Erdà Âs number of at most 4, having coauthored with Lee Smolin.
Angela Merkel, Chancellor of Germany from 2005 to 2021, has an Erdà Âs number of at most 5.
Some fields of engineering, in particular communication theory and cryptography, make direct use of the discrete mathematics championed by Erdà Âs. It is therefore not surprising that practitioners in these fields have low Erdà Âs numbers. For example, Robert McEliece, a professor of electrical engineering at Caltech, had an Erdà Âs number of 1, having collaborated with Erdà Âs himself. Cryptographers Ron Rivest, Adi Shamir, and Leonard Adleman, inventors of the RSA cryptosystem, all have Erdà Âs number 2.
The Romanian mathematician and computational linguist Solomon Marcus had an Erdà Âs number of 1 for a paper in Acta Mathematica Hungarica that he co-authored with Erdà Âs in 1957.
Erdà Âs 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 Erdà Âs number, the numbers range up to 15, the median is 5, and the mean is 4.65; almost everyone with a finite Erdà Âs 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 Erdà Âs numbers. For example, political scientist Steven Brams has an Erdà Âs 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 Erdà Âs via Persi Diaconis or Paul Deheuvels, who have Erdà Âs numbers of 1, or John Tukey, who has an Erdà Âs number of 2. Similarly, the prominent geneticist Eric Lander and the mathematician Daniel Kleitman have collaborated on papers, and since Kleitman has an Erdà Âs 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 Erdà Âs numbers within the cryptographic research community, and many linguists have finite Erdà Âs numbers, many due to chains of collaboration with such notable scholars as Noam Chomsky (Erdà Âs 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 Erdà Âs 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 Erdà Âs number is either Antoine Lavoisier (born 1743, Erdà Âs number 13), Richard Dedekind (born 1831, Erdà Âs number 7), or Ferdinand Georg Frobenius (born 1849, Erdà Âs number 3), depending on the standard of publication eligibility.
Martin Tompa proposed a directed graph version of the Erdà Âs number problem, by orienting edges of the collaboration graph from the alphabetically earlier author to the alphabetically later author and defining the monotone Erdà Âs number of an author to be the length of a longest path from Erdà Âs to the author in this directed graph. He finds a path of this type of length 12.
Also, Michael Barr suggests "rational Erdà Âs numbers", generalizing the idea that a person who has written p joint papers with Erdà Âs should be assigned Erdà Âs 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)âÂÂwith one edge between two mathematicians for each joint paper they have producedâÂÂform 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 Erdà Âs 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 Erdà Âs number of 4 auctioned off a co-authorship on eBay, hence providing the buyer with an Erdà Âs 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 Erdà Âs number.
A small number of people are connected to both Erdà Âs and Bacon and thus have an Erdà ÂsâÂÂBacon 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 Erdà Âs number is 4, and her Bacon number is 2.
Further extension is possible. For example, the "Erdà ÂsâÂÂBaconâÂÂSabbath number" is the sum of the Erdà ÂsâÂÂBacon number and the collaborative distance to the band Black Sabbath in terms of singing in public. Physicist Stephen Hawking had an Erdà ÂsâÂÂBaconâÂÂSabbath number of 8, and actress Natalie Portman has one of 11 (her Erdà Âs 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 Hon'inbà  Shà «saku, 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.