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Facts about Erdös Numbers and the Collaboration Graph

[For an older page, with the corresponding facts as of May, 2000, click here. It is interesting to note that over this 4-year period, 64,000 new authors were added to the MR database, but the number of authors who have written only solo-authored papers has DECREASED, from just over 84,000 to just under 84,000. Similarly, the mean number of collaborators per author increased 14%, from 2.94 to 3.36.]

A different standard is used for collaborations here than is used in constructing our Erdös-1 and Erdös-2 lists. First, for our lists we use sources in addition to Mathematical Reviews; the conclusion on this page are based just on the MR data. Second, we generally do not count articles that are not the result of research collaboration as establishing a link. For example, if Jack and Jill wrote a joint obituary article on Humpty Dumpty when he died, that article might appear in the MR database and establish a link between Jack and Jill for the conclusions reached on this page, whereas the traditional definition of the collaboration graph would not suggest putting an edge between them just on this basis. There are also a few author identification problems in the Math Reviews database (primarily prior to 1985), which make the conclusions here only approximate.

Data on the entire collaboration graph

There are about 1.9 million authored items in the Math Reviews database, by a total of about 401,000 different authors. (This includes all books and papers in MR except those items, such as some conference proceedings, that do not have authors.) Approximately 62.4% of these items are by a single author, 27.4% by two authors, 8.0% by three authors, 1.7% by four authors, 0.4% by five authors, and 0.1% by six or more authors. The largest number of authors shown for a single item is in the 20s, but sometimes the author list includes “et al.”, whom we did not count as a real person. The fraction of items authored by just one person has steadily decreased over time, starting out above 90% in the 1940s and currently standing at under 50%.

Let B be the bipartite graph whose vertices are papers and authors, with an edge joining a paper with each author of that paper. Then B has about 2.9 million edges. The average number of authors per paper is 1.51, and the average number of papers per author is 7.21. Click here to see the distribution of the number of papers per author. The median is 2, the mean is 7.21, and the standard deviation is 18.02. It is interesting (for tenure review committees?) to note that the 60th percentile is 3 papers, the 70th percentile is 4, the 80th percentile is 8, the 90th percentile is 18, and the 95th percentile is 32. Indeed, over 42% of all authors in the database have just one paper.

There are four authors with more than 700 papers: Paul Erdös with 1416 (he actually wrote more papers than that, but these are just the ones covered by Math Reviews), Drumi Bainov with 823, SAHARON SHELAH with 760, and Leonard Carlitz with 730. Bainov’s Erdös number is 4, SHELAH’s is 1, and Carlitz’s is 2. The other mathematicians with more than 500 papers listed in MathSciNet (and their Erdös numbers) are Hari M. Srivastava (2), Lucien Godeaux (infinite — actually he wrote only one joint paper), Ravi Agarwal (3), Edoardo Ballico (3), FRANK HARARY (1), Josip E. Pecaric (2), Owa Shigeyoshi (3), and Richard Bellman (2). The most prolific authors listed in the DBLP (dealing with computer science publications) can be found on a list at their website (DBLP), which is definitely worth exploring.

The collaboration graph C has the roughly 401,000 authors as its vertices, with an edge between every pair of people who have a joint publication (with or without other coauthors — but see below for a discussion of &#147Erdös number of the second kind”, where we restrict links to just two-author papers). Click here for a picture of a small portion of this graph. The entire graph has about 676,000 edges, so the average number of collaborators per person is 3.36. (If we were to view C as a multigraph, with one edge between two vertices for each paper in which they collaborated, then there would be about 1,300,000 edges, for an average of 6.55 collaborations per person.) In C there is one large component consisting of about 268,000 vertices. Of the remaining 133,000 authors, 84,000 of them have written no joint papers (these are isolated vertices in C). The average number of collaborators for people who have collaborated is 4.25; the average number of collaborators for people in the large component is 4.73; and the average number of collaborators for people who have collaborated but are not in the large component is 1.65.

Click here to see the data on the number of collaborators per author (in other words, the numbers of coauthors mathematicians have). In graph-theoretical terms, this table shows the degrees of the vertices in C. The median is 1, the mean is 4.25, and the standard deviation is 6.61. (If we omit the isolated vertices, then the median degree is 2, and the mean is 5.37.) Recent research (see our research page) has indicated that we should expect the nonzero degrees to follow a power law: the number of vertices with degree x should be proportional to x raised to a power, where the exponent is somewhere around –2 or –3. Indeed, when we fit such a model to our data from May, 2000 (grouping the data in the tail), we find the exponent to be about –2.97, with a correlation coefficient for the model of r = 0.97. A slightly more accurate model throws in an exponential decay factor, and with this factor present, the exponent is –2.46, and r = 0.98. Apparently these models are appropriate for our data.

The five people with more than 200 coauthors are Paul Erdös (of course) with 509 (although the MR data actually show only 504, missing some coauthors of very minor works or works before 1940, when MR was started), FRANK HARARY (Erdös number 1) with 268, Yuri Alekseevich Mitropolskii (Erdös number 3) with 244, NOGA ALON (Erdös number 1) with 227, and Hari M. Srivastava (Erdös number 2) with 244.

Click here for information on publication habits over time (1940 to 1999). It is clear from these data that collaboration has increased over the past 60-odd years, especially so recently. By 2000, less than half of all mathematics papers were by a single author, about a third were by two authors, about an eighth by three authors, and 3% by four or more authors. The table also indicates that the average number of papers per author in a decade has slowly increased over time, now standing at about 5 (although the variance is very large, and the median is only 2).

The radius of the large component of C (as it existed in Mathematical Reviews data as of July, 2004) is 12, and its diameter is 23. There are exactly two vertices with eccentricity 12 — Izrail M. Gelfand (Rutgers University) and Yakov Sinai (Princeton University), both of whom have Erdös number 3 — but not including Paul Erdös! (In other words, there is no one with Gelfand number or Sinai number greater than 12, whereas the maximum Erdös number is 13. In all, 1220 people have eccentricity 13.) Erdös does have the distinction of having the smallest mean distance to the other vertices, though: 4.65. There are four other people with means less than 5. In order of increasing mean, they are RONALD GRAHAM, ANDREW ODLYZKO, NOGA ALON, Larry Shepp, and FRANK HARARY. All of them have eccentricity 14 and Erdös number 1 except for Shepp, whose eccentricity is 13 and whose Erdös number is 2. The means for Gelfand and Sinai are slightly higher than 5.

Based on a sample of 100 pairs of vertices in this component, the average distance between two vertices is around 7.64 (between 7.41 and 7.87 with 95% confidence), with a standard deviation of about 1.19. The median of the sample was 7, with the quartiles at 6 and 8. The smallest and largest distances in the sample were 4 and 11, respectively. The appropriate phrase for C, then, is perhaps “eight degrees of separation”, if we wish to account for three quarters of all pairs of mathematicians.

The “clustering coefficient” of a graph is equal to the fraction of ordered triples of vertices a,b,c in which edges ab and bc are present that have edge ac present. (In other words, how often are two neighbors of a vertex adjacent to each other?) The clustering coefficient of the collaboration graph of the first kind is 1308045/9125801 = 0.14. The high value of this figure, together with the fact that average path lengths are small, indicates that this graph is a “small world” graph (as defined by Duncan Watts — see our pages on research on collaboration and related concepts).

We also have some data on the portion of the collaboration graph outside the &#147Erdös component” (the one giant component). We are ignoring here the 84,000 isolated vertices and looking only at those authors who have collaborated but do not have a finite Erdös number. There are about 50,000 such vertices. There are about 41,000 edges in these components, so the average degree of these vertices is 1.65. In other words, a person who has collaborated but does not find herself in the Erdös component of C has on the average collaborated with only one or two people. In contrast, the average degree of vertices in the Erdös component is 4.73 (there are about 634,000 edges and 268,000 vertices). Click here for the distribution of component sizes. As would be expected, most of these roughly 18,000 other components are isolated edges (64% of them, in fact). The largest component has 32 vertices. Its most collaborating author is Yu. A. Shevlyakov (Department of Applied Mathematics, Simferopol State University, Crimea, Ukraine), who has 13 coauthors. The person outside the Erdös component with the most coauthors is Gholam Reza Jahanshahloo (Department of Mathematics, University for Teacher Education, Tehran, Iran), who is in a component with 23 vertices (he has collaborated with all but two of them).

Smaller collaboration graphs

The distribution of Erdös numbers

      Erdös number  0  ---      1 person
      Erdös number  1  ---    504 people
      Erdös number  2  ---   6593 people
      Erdös number  3  ---  33605 people
      Erdös number  4  ---  83642 people
      Erdös number  5  ---  87760 people
      Erdös number  6  ---  40014 people
      Erdös number  7  ---  11591 people
      Erdös number  8  ---   3146 people
      Erdös number  9  ---    819 people
      Erdös number 10  ---    244 people
      Erdös number 11  ---     68 people
      Erdös number 12  ---     23 people
      Erdös number 13  ---      5 people

One of the five people with the largest finite Erdös number is Arturo Robles, and one shortest path goes like this (year of joint work in parenthese): Erdös to Daniel D. Bonar (1977) to Charles L. Belna (1979) to S. A. Obaid (1983) to Wadie A. Bassali (1981) to Ibrahim H. M. el-Sirafy (1976) to Konstantin Chernous (1977) to Jose Valdes (1980) to B. Dugnol (1980) to P. Suarez Rodriguez (1995) to A. E. Alvarez Vigil (1995) to C. Gonzalez Nicieza (1992) to Jose Angel Huidobro (1986) to Robles (1990).

Since Paul Erdös collaborated with so many people, one would expect this distribution for him to be shifted downward from that of a random mathematician. For example, “Jerry Grossman numbers” have a median of 6, a mean of 5.71 (standard deviation = 1.22), and range as high as 15; and “Arturo Robles numbers” have a median of 15, a mean of 15.06 (standard deviation = 1.21). It turns out that the standard deviation is almost exactly the same for almost everyone in the large component.

Erdös numbers of the second kind