Erdos number
Encyclopedia
The Erdős number (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 persons in other fields.
measurement of mathematical prominence.
Paul Erdős was an influential and itinerant 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. He published more papers during his life (at least 1,525) than any other mathematician in history.
To be assigned an Erdős number, an author must co-write a research paper with an author with a finite Erdős number. Paul Erdős has an Erdős number of zero. Anybody else's Erdős number is where 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).
There is room for ambiguity over what constitutes a link between two authors; the Erdős Number Project web site says:
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 1. Goffman published his observations about Erdős' prolific collaboration in a 1969 article entitled "And what is your Erdős number?"
of mathematicians throughout the world for many years. Amongst 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; 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. 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. Similarly, 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).
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. It seems that older historic figures such as Leonhard Euler
(born 1707) do not 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.
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.
) 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. 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. 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.
player and Honinbo Shusaku
, measured in Go opponents. 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.
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
, as measured by authorship of mathematical papers.
The same principle has been proposed for other eminent persons in other fields.
Overview
The idea of the Erdős number was created by friends as a humorous tribute to the enormous output of Erdős, one of the most prolific modern writers of mathematical papers, and has become well known in scientific circles as a tongue-in-cheekTongue-in-cheek
Tongue-in-cheek is a phrase used as a figure of speech to imply that a statement or other production is humorously intended and it should not be taken at face value. The facial expression typically indicates that one is joking or making a mental effort. In the past, it may also have indicated...
measurement of mathematical prominence.
Paul Erdős was an influential and itinerant 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. He published more papers during his life (at least 1,525) than any other mathematician in history.
Definition
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
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature 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.
The Erdős number was most likely first defined in print by Casper Goffman, an analyst
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
whose own Erdős number is 1. Goffman published his observations about Erdős' prolific collaboration in a 1969 article entitled "And what is your Erdős number?"
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 that most frequently co-authored with Erdős and their number of papers co-authored with Erdõs (i.e. their number of collaborations).| Co-Author | Number of Collaborations |
|---|---|
| Andras Sarkozy András Sárközy András Sárközy is a Hungarian mathematician, working in analytic and combinatorial number theory, although his first works were in the fields of geometry and classical analysis. He has the largest number of papers co-authored with Paul Erdős... |
62 |
| Andras Hajnal András Hajnal András Hajnal is an emeritus professor of mathematics at Rutgers University and a member of the Hungarian Academy of Sciences known for his work in set theory and combinatorics.-Biography:Hajnal was born on 13 May 1931, in Hungary.... |
56 |
| Ralph Faudree Ralph Faudree Ralph Jasper Faudree is a mathematician, a professor of mathematics and the provost of the University of Memphis.Faudree was born in Durant, Oklahoma. He did his undergraduate studies at Oklahoma Baptist University, graduating in 1961, and received his Ph.D. in 1964 from Purdue University under... |
50 |
| Richard Schelp Richard Schelp Dr Richard H. Schelp was an American mathematician.Schelp received his Bachelors degree in mathematics and physics from the University of Central Missouri and his Masters degree and doctorate in mathematics from Kansas State University... |
42 |
| Cecil C. Rousseau Cecil C. Rousseau Cecil Clyde Rousseau is a mathematician and author who specializes in graph theory and combinatorics. He is a professor emeritus at The University of Memphis and former chair of the USAMO.Rousseau received his Ph.D... |
35 |
| Vera T. Sos Vera T. Sós Vera T. Sós is a Hungarian mathematician, specializing in number theory and combinatorics. She was a student and close collaborator of both Paul Erdős and Alfréd Rényi. She also collaborated frequently with her husband Paul Turán, the famous analyst, number theorist, and combinatorist... |
35 |
| Alfred Renyi Alfréd Rényi Alfréd Rényi was a Hungarian mathematician who made contributions in combinatorics, graph theory, number theory but mostly in probability theory.-Life:... |
32 |
| Pal Turan Pál Turán Paul Turán was a Hungarian mathematician who worked primarily in number theory. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting 46 years and resulting in 28 joint papers.- Life and education :... |
30 |
| Endre Szemeredi Endre Szemerédi Endre Szemerédi is a Hungarian mathematician, working in the field of combinatorics and theoretical computer science. He is the State of New Jersey Professor of computer science at Rutgers University since 1986... |
29 |
| Ronald Graham Ronald Graham Ronald Lewis Graham is a mathematician credited by the American Mathematical Society as being "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years"... |
28 |
Impact
Erdős numbers have been a part of the folkloreFolklore
Folklore consists of legends, music, oral history, proverbs, jokes, popular beliefs, fairy tales and customs that are the traditions of a culture, subculture, or group. It is also the set of practices through which those expressive genres are shared. The study of folklore is sometimes called...
of mathematicians throughout the world for many years. Amongst 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; 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
Steven Brams
Steven J. Brams is a game theorist and political scientist at the New York University Department of Politics. Brams is best known for using the techniques of game theory and public choice to research voting systems and fair division. He is one of the independent discoverers of approval voting...
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
John Tukey
John Wilder Tukey ForMemRS was an American statistician.- Biography :Tukey was born in New Bedford, Massachusetts in 1915, and obtained a B.A. in 1936 and M.Sc. in 1937, in chemistry, from Brown University, before moving to Princeton University where he received a Ph.D...
, who has an Erdős number of 2. Similarly, the prominent geneticist Eric Lander
Eric Lander
Eric Steven Lander is a Professor of Biology at the Massachusetts Institute of Technology , a member of the Whitehead Institute, and director of the Broad Institute of MIT and Harvard who has devoted his career toward realizing the promise of the human genome for medicine. He is co-chair of U.S...
and the mathematician Daniel Kleitman
Daniel Kleitman
Daniel J. Kleitman is a professor of applied mathematics at MIT. His research interests include combinatorics, graph theory, genomics, and operations research.- Biography :...
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. According to Alex Lopez-Ortiz, all the
Fields
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
and Nevanlinna
Nevanlinna Prize
The Rolf Nevanlinna Prize is awarded once every 4 years at the International Congress of Mathematicians, for outstanding contributions in Mathematical Aspects of Information Sciences including:...
prize winners during the three cycles in 1986 to 1994 have Erdős numbers of at most 9. Similarly, many linguists
Linguistics
Linguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....
have finite Erdős numbers, many due to chains of collaboration with such notable scholars as Noam Chomsky
Noam Chomsky
Avram Noam Chomsky is an American linguist, philosopher, cognitive scientist, and activist. He is an Institute Professor and Professor in the Department of Linguistics & Philosophy at MIT, where he has worked for over 50 years. Chomsky has been described as the "father of modern linguistics" and...
(Erdős number 4), William Labov
William Labov
William Labov born December 4, 1927) is an American linguist, widely regarded as the founder of the discipline of variationist sociolinguistics. He has been described as "an enormously original and influential figure who has created much of the methodology" of sociolinguistics...
(3), Mark Liberman
Mark Liberman
Mark Liberman is an American linguist. He has a dual appointment at the University of Pennsylvania, as Trustee Professor of Phonetics in the Department of Linguistics, and as a professor in the Department of Computer and Information Sciences. He is the founder and director of the Linguistic Data...
(3), Geoffrey Pullum
Geoffrey Pullum
Geoffrey Keith "Geoff" Pullum is a British-American linguist specialising in the study of English. , he is Professor of General Linguistics at the University of Edinburgh....
(3), or Ivan Sag
Ivan Sag
Ivan Sag is an American linguist and cognitive scientist. He is the Sadie Dernham Patek Professor in Humanities, Professor of Linguistics, and Director of the Symbolic Systems Program at Stanford University...
(4).
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
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
(born 1831, Erdős number 7) or Ferdinand Georg Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
(born 1849, Erdős number 3), depending on the standard of publication eligibility. It seems that older historic figures such as Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
(born 1707) do not have finite Erdős numbers.
Tompa proposed a directed graph
Directed graph
A directed graph or digraph is a pair G= of:* a set V, whose elements are called vertices or nodes,...
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
Michael Barr (mathematician)
Michael Barr is Peter Redpath Emeritus Professor of Pure Mathematics at McGill University. Although his earlier work was in homological algebra, his principal research area for a number of years has been category theory...
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
H-index
The h-index is an index that attempts to measure both the productivity and impact of the published work of a scientist or scholar. The index is based on the set of the scientist's most cited papers and the number of citations that they have received in other publications...
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.
Variations
A number of variations on the concept have been proposed to apply to other fields.| Field | Target Person(s) | Died | Measured via |
|---|---|---|---|
| Mathematics Mathematics Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity... |
Paul Erdős Paul Erdos Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory... |
1996 | Erdős number |
| Physics Physics Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic... |
Albert Einstein Albert Einstein Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history... |
1955 | Einstein number |
| Acting Acting Acting is the work of an actor or actress, which is a person in theatre, television, film, or any other storytelling medium who tells the story by portraying a character and, usually, speaking or singing the written text or play.... |
Kevin Bacon Kevin Bacon Kevin Norwood Bacon is an American film and theater actor whose notable roles include Animal House, Diner, Footloose, Flatliners, Wild Things, A Few Good Men, JFK, Apollo 13, Mystic River, The Woodsman, Trapped, Friday the 13th, Hollow Man, Tremors, Death Sentence, Frost/Nixon, Crazy, Stupid, Love.... |
living | Bacon number |
| Maths+Acting | Paul Erdős Paul Erdos Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory... & Kevin Bacon Kevin Bacon Kevin Norwood Bacon is an American film and theater actor whose notable roles include Animal House, Diner, Footloose, Flatliners, Wild Things, A Few Good Men, JFK, Apollo 13, Mystic River, The Woodsman, Trapped, Friday the 13th, Hollow Man, Tremors, Death Sentence, Frost/Nixon, Crazy, Stupid, Love.... |
n/a | Erdős–Bacon number |
| Chess Chess Chess is a two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. It is one of the world's most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments.Each player... |
Paul Morphy Paul Morphy Paul Charles Morphy was an American chess player. He is considered to have been the greatest chess master of his era and an unofficial World Chess Champion. He was a chess prodigy... |
1884 | Morphy number Morphy number The Morphy number is a measure of how closely a chess player is connected to Paul Morphy by way of playing chess games. People who played a chess game with Morphy have a Morphy number of 1. Players who did not play Morphy but played someone with a Morphy number of 1 have a Morphy number of 2... |
| Go | Honinbo Shusaku Honinbo Shusaku Honinbo Shusaku was a professional Go player and is considered by many to be the greatest player of the golden age of Go in the mid-19th century.- Biography :He was nicknamed "Invincible" after he earned a perfect score for 19 straight wins in the annual castle... |
1862 | Shusaku number |
| Linguistics Linguistics Linguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context.... |
Noam Chomsky Noam Chomsky Avram Noam Chomsky is an American linguist, philosopher, cognitive scientist, and activist. He is an Institute Professor and Professor in the Department of Linguistics & Philosophy at MIT, where he has worked for over 50 years. Chomsky has been described as the "father of modern linguistics" and... |
Living | Chomsky number |
Bacon number
The Bacon number (as in the game Six Degrees of Kevin BaconSix Degrees of Kevin Bacon
Six Degrees of Kevin Bacon is a trivia game based on the concept of the small world phenomenon and rests on the assumption that any individual involved in the Hollywood, California film industry can be linked through his or her film roles to actor Kevin Bacon within six steps. The name of the game...
) is an application of the same idea to the movie industry, connecting actors that appeared in a film together to the actor Kevin Bacon
Kevin Bacon
Kevin Norwood Bacon is an American film and theater actor whose notable roles include Animal House, Diner, Footloose, Flatliners, Wild Things, A Few Good Men, JFK, Apollo 13, Mystic River, The Woodsman, Trapped, Friday the 13th, Hollow Man, Tremors, Death Sentence, Frost/Nixon, Crazy, Stupid, Love....
. 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. One example is the actress-mathematician Danica McKellar
Danica McKellar
Danica Mae McKellar is an American actress, academic, and education advocate. She is best known for her role as Winnie Cooper in the television show The Wonder Years, and later as author of the three The New York Times bestsellers, Math Doesn't Suck, Kiss My Math, and Hot X: Algebra Exposed, which...
, best known for playing Winnie Cooper on the TV series, The Wonder Years
The Wonder Years
The Wonder Years is an American television comedy-drama created by Carol Black and Neal Marlens. It ran for six seasons on ABC from 1988 through 1993. The pilot aired on January 31, 1988 after ABC's coverage of Super Bowl XXII....
.
Her Erdős number is 4 and her Bacon number is 2. The lowest known
Erdős–Bacon number is three for Daniel Kleitman
Daniel Kleitman
Daniel J. Kleitman is a professor of applied mathematics at MIT. His research interests include combinatorics, graph theory, genomics, and operations research.- Biography :...
, a mathematics professor at MIT; his Erdős number is 1 and his Bacon number is 2.
Shusaku number
The Shusaku number represents the "distance" between a GoGo (board game)
Go , is an ancient board game for two players that originated in China more than 2,000 years ago...
player and Honinbo Shusaku
Honinbo Shusaku
Honinbo Shusaku was a professional Go player and is considered by many to be the greatest player of the golden age of Go in the mid-19th century.- Biography :He was nicknamed "Invincible" after he earned a perfect score for 19 straight wins in the annual castle...
, measured in Go opponents. 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.
Berezovsky number
M.V. (Mikhail) Simkin suggests that number should be used to represent the distance between Boris Berezovsky and connecting scientists.See also
- James Parry, target of a similar number.
- Morphy numberMorphy numberThe Morphy number is a measure of how closely a chess player is connected to Paul Morphy by way of playing chess games. People who played a chess game with Morphy have a Morphy number of 1. Players who did not play Morphy but played someone with a Morphy number of 1 have a Morphy number of 2...
- Small world experimentSmall world experimentThe small world experiment comprised several experiments conducted by Stanley Milgram and other researchers examining the average path length for social networks of people in the United States. The research was groundbreaking in that it suggested that human society is a small world type network...
- Small-world networkSmall-world networkIn mathematics, physics and sociology, a small-world network is a type of mathematical graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps...
- List of people by Erdős number
Further reading
Original Spanish version in Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 23 (89) 563–582, 1999, .External links
- Jerry Grossman, The Erdös 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 Collaboration Distance. A search engine for Erdős numbers and collaboration distance between other authors. As of 18 November 2011 no special access is required.
- "Theorems for Sale" (From Science News, Vol. 165, No. 24, June 12, 2004)