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In this 2005 Notices article, Jerold Grossman tracks the proportion of papers in Math Reviews with 1, 2, 3, and >3 authors over time. His data set ends in 1999. I seem to recall reading that in 200k, for some value of k, the number of collaborative papers in math exceeded the number of single-authored papers for the first time. But I can't find any record of this. Does anyone know what k is, what a reliable source for this kind of data is, and what the proportion of collaborative papers is now?

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I'd also like to know the answer to the "author's perspective" version of this question: On average or median, what proportion of a given mathematician's papers are collaborative? The answer would allow one to assess "Do I collaborate more or less than the average / median mathematician?" –  Andrew Critch Nov 15 '09 at 21:48
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3 Answers

According to the article, the original data was provided by the AMS. I don't think that the AMS leaves this sort of data lying around on laptops on trains, so do to it again you'd have to go and ask them. I suspect that, quite reasonably, the AMS likes to know what uses their data is put to.

On the other hand, data can be mined very easily from the arXiv via the API. I don't know if arXiv data would suffice for you. If so, a little scripting showed the following data for the month of October:

math.CO: 36, 38, 11, 7, 2, 1
math.CA: 9, 11, 5, 1, 1
math.CT: 4, 4, 3
math.GN: 6, 7, 2
math.AT: 18, 9, 2
math.AC: 6, 9, 4, 2
math.CV: 24, 16, 1
math.OC: 6, 11, 5, 3
math.MG: 7, 7, 4, , 1
math.HO: 14
math.DG: 43, 48, 16, 3, 1
math.LO: 9, 2, 2
math.RA: 12, 11, 7, , , 2
math.ST: 3, 14, 2
math.PR: 43, 45, 25, 7
math.GT: 29, 22, 4, 2
math.SG: 13, 4, 2
math.GM: 8
math.SP: 7, , 4, 1
math.FA: 22, 18, 9, 4
math.OA: 9, 6, 4, 3, 1
math-ph: 52, 53, 15, 2, 1
math.DS: 23, 17, 9, 2, 1
math.QA: 13, 13, 3
math.KT: 3, 1
math.GR: 17, 27, 3, 2
math.NA: 4, 13, 9
math.RT: 28, 14, 3
math.NT: 48, 31, 8, , , 1
math.AP: 49, 59, 29, 4
math.AG: 69, 34, 11, 1, 2
Total: 634, 544, 202, 44, 10, 4
Average: 0.44, 0.37, 0.14, 0.03, 0, 0

The ordering is by number-of-authors. So for math.KT there were 4 papers, of which 3 were single authored and 1 with 2 authors. Missing entries are 0s (so in math.NT there was a 6-author paper but none with 4 or 5). So collaborations outweigh single-author papers by a little bit (technical term).

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Wow, this is terrific! I would LOVE to see this added up for, say, all of 2009, and split up by arXiv subject area as you've done. Would it be easy? P.S. One of the six authors on that number theory paper is my graduate student! –  JSE Nov 16 '09 at 14:52
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Wouldn't be too difficult. I'd need to figure out a better search query as in the interests of speed I got the whole dataset for October (over 10Mb!) and that included all of the other areas covered by the arXiv. Once I've put that in, the datasets might be more reasonable. –  Andrew Stacey Nov 16 '09 at 15:08
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The problem with arXiv data is that, anecdotally, younger mathematicians are both more likely to collaborate and more likely to put their papers on the arXiv. –  Michael Lugo Nov 16 '09 at 16:09
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There's a simpler reason to expect the arxiv to skew towards collaboration: when a mathematician who doesn't post to the arxiv collaborates with one who does the resulting paper is more likely than not to be posted to the arxiv. Nonetheless it's cool data, nice work! –  Noah Snyder Nov 16 '09 at 17:37
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@Noah: Just checked my script, it does only count primary category. @Michael: Good grief! What did you expect for 5 minutes' work? However, I would imagine that that can be corrected for by doing a small-scale study. IANAS, though, so I merely collect the data. It's up to the rest of you to interpret it. –  Andrew Stacey Nov 16 '09 at 18:56
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I found this article from the Information Processing and Management, which may be related:

"Trends in transforming scholarly communication and their implications" (2002)

Unfortunately, I can't access the actual text, and am unwilling to pay for it.

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Great find -- I have access through my library. Doesn't have the answer to my question, but has lots of interesting tidbits. The mean number of authors for papers in Physical Review Letters went from 1.7 in 1951 to 3.8 in 1991! And a 1993 article in the New England Journal of Medicine had 972 authors. –  JSE Nov 16 '09 at 14:55
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Clearly, at least in medicine, one needs to use the median instead of the mean. –  Michael Lugo Nov 16 '09 at 16:09
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I imagine you'll find more collaborative papers in applied math than in pure math. In fact, I'd speculate that the number of authors increases steadily as you move along the continuum from most pure (e.g. category theory) to most applied (e.g. applied statistics).

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If you provide me with an arXiv-classification-to-purity formula then I could add that in to my script to check your hypothesis. –  Andrew Stacey Nov 16 '09 at 19:36
    
Not sure it's accurate to call category theory "most pure". The application of category theory to computer science (semantics of programming languages etc) is a big deal. A large proportion of the world's category theorists work in computer science departments, and can be described as applied mathematicians. –  Tom Leinster Nov 16 '09 at 20:04
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A certain XKCD cartoon springs to mind! I suspect that the correct codomain for the purity function is not a straight line. Circle, perhaps? Or mabe it's fractal! –  Andrew Stacey Nov 16 '09 at 20:43
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My advisor used to say applied mathematics is an attitude, not a subject classification. But it's probably correlated with subject classification. Some category theorists are interested in applications, and some statisticians (even "applied" statisticians) are not, but these are exceptions. –  John D. Cook Nov 16 '09 at 21:14
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