Patterns of Collaboration in Mathematical Research

Each year mathematicians publish more than 50,000 research papers. Since 1940, Mathematical Reviews (MR; available electronically on the World Wide Web as MathSciNet [8]) has catalogued most of them, and MR’s current database contains more than one and a half million items, produced by more than a third of a million authors. By studying this wealth of data, we can discern some interesting patterns of publication, and in particular some interesting patterns of collaboration. For simplicity, we call each authored item in the MR database a “paper,” although some of them are monographs of various kinds. We ignore nonauthored items in the database, such as conference proceedings—the relevant papers in the proceedings have their own entries as authored items. In maintaining this database, and making it available to subscribers in print form and on the Internet, the MR editors and staff have taken pains to identify authors as people and not merely as name strings—strings of characters that the journal listed as an author’s name. For example, Raymond L. Johnson, Roberto Johnson, and Russell A. Johnson all published under the name string “R. Johnson,” but each of the papers by “R. Johnson” in the database is identified with exactly one of these three people. To the extent that MR has been successful in this endeavor, the data will accurately reflect the publication habits and the social network of actual individuals. (Some errors of this type remain, to be sure, but we do not think they substantially affect our results. Indeed, before they corrected the mistake in 1995, MR listed a paper by the physicist Paul Erdo′′s as being by the mathematician Paul Erdo′′ s. Now, MR denotes these two individuals Paul Erdo′′s and Paul Erdo′′s, respectively, using a convention that has become increasingly necessary. See [11] for more details.) The data used in this article cover approximately the period from 1940 to 1999, inclusive, and we have broken it down approximately by decade. (There is necessarily a lot of imprecision in dating, partly because the reviews in MR typically appear about a year after publication, which in turn is often more than a year after submission.) We thank the American Mathematical Society for providing access to this data, as well as Patrick Ion of Mathematical Reviews for helpful conversations. The cumulative data are summarized in Table 1, whose integer entries represent thousands. The left-most column includes all the data, and the remaining columns truncate the data after one or more decades. Data are given for all authors, as well as just for authors who have collaborated. The third row of Table 1 shows the average number of papers per author. Since mathematicians at all stages in their careers are included, it is hard to know exactly how to view the statistic that the mean number of papers is about 7. Of course the distribution has a long right tail, with a standard deviation of more than 15. Table 2 shows the fractions of mathematicians who have written various numbers of papers. It can be seen from this table that just slightly more than half of all publishing mathematicians have published more than one paper, that the median number of papers is just 2, and that more than two thirds of us have written fewer than five papers. At the other extreme, eight people have written more than 500 papers apiece, including the legendary Paul Erdo′′ s, with about 1500 papers. When tenure committees count publications, this kind of information might help to put things in context. Table 3 summarizes the data decade by decade, giving a better view of how things have changed over the years. Row 2 shows the explosion in the number of practicing mathematicians during the period we consider, a compounded annual growth rate of 6% per year (compared with a rate of less than 2% for the population of the world during the same period). We infer from row 3 that in the 1940s the mean number of papers per mathematician per year was thru 90s thru 80s thru 70s thru 60s thru 50s thru 40s