# Research on Collaboration in Research

The earliest discussion we could find on Erdös numbers are articles by Casper Goffman [“And what is your Erdos number?”, American Mathematical Monthly 76 (1969), 791] and RONALD LEWIS GRAHAM (writing under the pseudonym Tom Odda) [“On properties of a well-known graph or what is your Ramsey number?”, Topics in Graph Theory (New York, 1977) 166-172]. According to MELVIN HENRIKSEN, the idea was suggested by John Isbell at Princeton University around 1957.

Our first paper on this subject is “On a portion of the well-known collaboration graph” (1995) by Grossman and Ion (936K, 3 pages). It was presented at the 26th Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, Florida, March, 1995) and is published in Volume 108 (1995) of *Congressus Numerantium*, pages 129–131. This paper discusses the **rise of collaboration** in the mathematical sciences over the past half century in general, and the extent of collaboration among the people on our lists in particular. For example, we find that Paul Erdös’s collaborators tend to be prolific collaborators themselves. A **follow-up paper**, “Erdös number update” was published in 1998 on a CD-ROM of information about Paul Erdös and his work.

De Castro has written a paper (“Sobre el número de Erdös”) in which he shows that (almost) all Fields Medal winners have a finite Erdös number. De Castro and Grossman wrote longer paper (“Famous trails to Paul Erdös” (35 pages). It appears (somewhat abbreviated) in *The Mathematical Intelligencer*: vol. 21, no. 3 (Summer 1999), 51–63; and (in Spanish and updated) in the journal of the Colombian Academy of Sciences (Revista de la Academia Colombiana de Ciencias Exactas, Fisicas y Naturales: vol. 23, no. 89 (December, 1999), 563–582). In this paper, the authors show, among other things, that all Fields Medal winners through 1998 have Erdös numbers less than 6 and that at least 63 Nobel prize winners have Erdös numbers less than 9. Here is the primary reference list for the *Mathematical Intelligencer* paper.

Another article by Grossman, “Paul Erdös, the master of collaboration” (1996), formed the basis for a talk at the on Graph Theory, Combinatorics, Algorithms, and Applications (Kalamazoo, Michigan, June, 1996) and an updated version appears in Volume II of a wonderful two-volume collection, The Mathematics of Paul Erdös, second edition, edited by Ron Graham and Jaroslav Nesetril. This paper looks more deeply at the extent of Erdös’s collaborative efforts (which started back in 1934) and the wide range of areas in which he published. It also lists several articles about Paul himself.

Our paper on the research collaboration graph, which briefly summarizes some of the random graph approaches (see below) and contains a lot of the information on our Facts About Erdös Numbers and the Collaboration Graph page. (It appears in the Proceedings of 33rd Southeastern Conference on Combinatorics (Congressus Numerantium, Vol. 158, 2002, pp. 201-212). An abbreviated version appears in SIAM News 35:9 (November, 2002), pp. 1, 8-9.

Grossman’s article in *Geographical Analysis* discusses the mathematical collaboration landscape.

Jason Cory Brunson and others have a recent paper on the evolution of the mathematical research collaboration graph over time.

W. Roy Utz, Jr., wrote two letters to the editor of the Notices of the AMS, in 1962 and 1981, noting the rise of collaboration in papers in mathematics. For example, he observed that the fraction of jointly authored papers listed in Mathematical Reviews in the years 1940, 1950, 1960, 1970, and 1980, was 5.8%, 6.5%, 10.8%, 14.0%, and 21.4%, respectively (only MR sections 00 through 58 were included).

Mark Newman and others have studied scientific collaboration graphs as part of their ongoing research into small world networks and other large social and technological real-world graphs. Newman has written an excellent review article for SIAM Review (vol. 45, no. 2 (2003), 167-256). One article has appeared in the Proceedings of the National Academy of Sciences (volume 98 (2001), 404-409); a longer version was published in Physical Review E. Perhaps even more relevant is a preprint by A. L. Barabasi et al.

Eugene Garfield has done some thinking and writing about this subject over many years.

Fan Chung Graham and others have been looking at random graph models for massive graphs such as collaboration graphs. Preprints are available at her website. Additional researchers in this field include Bela Bollobas, as well as others whom we will be glad to add to this list if they write to us and request it.

Brian Hayes has a two-part article on massive graphs such as collaboration graphs in the 2000 volume of The American Scientist.

Lock, Pong, and Wittmond published an article entitled “Refinement of Metrics: Erdös Numbers, a Case Study”.

See also our page of Items of Interest Related to Erdös Numbers.

Sachi Sri Kantha wrote some articles on what he calls Kilo-Base Goliaths—people who have published more than 1000 papers. See “Productivity drive”, *Nature*, volume 356 (April 30, 1992), 738 (Letter); and “Clues to prolific productivity among prominent scientists”, *Medical Hypotheses*, volume 39 (1992), 159–163.

There is an article in the May 2000 issue of Social Networks (volume 22, number 2, pp. 173-186) entitled “Some Analyses of Erdös Collaboration Graph”, by Vladimir Batagelj and Andrej Mrvar, University of Ljubljana, Slovenia. More information, including a pdf file of the paper, is available from the authors.

Michael Barr of McGill University has an interesting suggestion for **rational Erdös numbers**, generalizing the idea that a person who has written p joint papers with Erdös should be assigned Erdös 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 authors 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. Professor Barr has made available a short article on this.

Valdis Krebs has an excellent page on collaboration networks, including a picture of the collaboration graph of Erdös’s collaborators.

Charles Packer has also done some interesting analysis of our data, including preparing a visual communications matrix.

Ken Herold (Director of Library Information Systems at the Burke Library, Hamilton College in New York) pointed out the following insight of Alan Turing: “The remaining form of search is what I should like to call the ‘cultural search’. As I have mentioned, the isolated man does not develop any intellectual power. It is necessary for him to be immersed in an environment of other men, whose techniques he absorbs during the first twenty years of his life. He may then perhaps do a little research of his own and make a very few discoveries which are passed on to other men. From this point of view the search for new techniques must be regarded as carried out by the human community as a whole, rather than by individuals.”

Jerry Grossman, Marc Lipman, and Eddie Cheng looked at **some questions in pure graph theory motivated** by the collaboration graph. Some results of this work were presented at the Ninth Quadrennial International Conference on Graph Theory, Combinatorics, Algorithms, and Applications (Kalamazoo, Michigan, June, 2000) and are contained in a paper on time-stamped graphs appearing in Discrete Applied Mathematics.

This page was last updated on September 16, 2020.