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In the context of some discussions we are having at my university, it has become evident that some statistical information regarding publishing practices in the various areas of mathematics would be necessary to proceed---you know, facts. In particular, I would be immensely happy to know

  • are there measurable and measured differences in the number of papers published by people working in different areas (think PDEs v. Algebraic Geometry v. Number theory v. Combinatorics; top level MSC groups, say)? Here I mean papers published by individual authors as well as collectively.

  • are there measurable and measured differences in the number of citations gotten by papers in each area?; similar question about the out-degree in the citation digraph?

  • what is the time profile of the citations to a typical math paper (ideally, broken by area again), whereby I mean: how are the citations to papers typically distributed in time?

Google has pointed to studies in which such comparisons are made between different disciplines (mathematics v. chemistry, say) but not at all between areas of mathematics.

Can anyone point to such information?

I would love to get hold of MathSciNet's raw tables (only papers, authors, subject area, citations) which would allow me to compute such things... (MathSciNet only has citation information since 2000, and I do not really know if that would make a representative sample for all-time statistics. It would be very interesting to have these kind of information diachronically, but I don't expect that data to be available)

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 Notice this is not a soft question: the ideal answer is in the form of actual data; I'm sure there's some sensible way to add more tags... There's been a meta thread at meta.mathoverflow.net/discussion/1003 – Mariano Suárez-Alvarez Apr 1 2011 at 4:08
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 Actually, MathSciNet has citation information since 1997. – Denis Serre Apr 1 2011 at 6:30
You might like to look at ams.org/notices/200603/comm-milman.pdf, ams.org/notices/201103/rtx110300434p.pdf, and ams.org/notices/200901/tx090100008p.pdf. My own feeling is that high citations might correlate to a good paper, but not the other direction. There are many good papers which are not cited much. – Yiftach Barnea Apr 1 2011 at 7:26
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 Related question: mathoverflow.net/questions/10801 ? The data there was also taken from MathSciNet. – José Figueroa-O'Farrill Apr 3 2011 at 12:36
@José: how did you get that information from MathSciNet? – Mariano Suárez-Alvarez Apr 3 2011 at 17:27
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3 Answers

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On Mariano's request I'm adding my comment on meta.MO as an answer here. This only concerns (part of) the first bullet point in the question.

There was an article Topical Bias in Generalist Mathematics Journals by Joseph F. Grcar in the december 2010 issue of the Notices of the AMS. According to the text, the statistics are based on 854,547 entries from the 2000-2009 period of the Zentralblatt database. Unfortunately the article remains silent on exactly how the data was gathered, but it might be a starting point for your own investigations.

For the convenience of the readers I take the liberty of reproducing the statistics most relevant for the present question from that article:

For more detailed information please follow this link or the ones provided above.

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Thanks. I edited a bit the first bullet. – Mariano Suárez-Alvarez Apr 1 2011 at 4:33
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 A caveat with this (or any MSC data) is that most authors self-classify their articles, or editors do it, and there is huge variation in how. Personally I narrow my work down to a few MSCs, but which ones is almost random; I never put all that apply. No two MSC topics are conceptually disjoint (e.g. operator/systems theory, Fourier/abstract harmonic analysis, CS/logic, category theory/everything), so the precise methodology in going from "MSC hits for X" to "papers/activity in X" seems more significant than the data itself. That Grcar did not address this is a huge hole in his argument. – anon Apr 1 2011 at 5:04
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 For my purposes, I actually prefer that authors pick themselves their MSC because presumably it describes their intent. – Mariano Suárez-Alvarez Apr 1 2011 at 5:32
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 Whether it describes their intent depends on how well their work fits the classification. The situation I often run into is that there are half a dozen somewhat relevant MSCs, which can be in different top-level classifications. It's impossible to make a principled choice, so I grit my teeth and just choose a couple. I think this experience is quite common, and many papers have somewhat arbitrarily chosen MSCs. That doesn't mean they aren't useful for gathering statistics, but there's a lot of noise and they don't necessarily reflect the authors' true preferences. – Henry Cohn Apr 1 2011 at 13:50
The MSC codes used on MathSciNet are, I think, mostly chosen by the reviewers, and are in some cases quite different from any that were chosen by the authors. – Mark Meckes Apr 1 2011 at 19:27
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That was a comment, but it turned out to be too long ...

This does not exactly answers the first question. The distribution provided by Theo is the product of two functions: 1- the number of researchers in each domain; 2- the number of papers that a typical member of the given domain publishes each year. You cannot extract these two factors from their product only.

The answer to the second question raised by Mariano is definitively Yes. The number of citations depends not only on the quality of the work, but also upon the size of the community. And this size depends on several factors, among which the history (the older the domain, the larger the size). I have observed this for a long time within PDEs. This domain contains several sub-domains, mostly characterized by the type of the PDEs under study (elliptic, parbolic, hyperbolic, dispersive, ...) This is justified by the fact that the techniques are very different. Historically, elliptic equations developped first, thank to Hilbert's variational approach and then to the burst of the theory of distribution. Then parabolic, and later hyperbolic ones. The citation number of the world best researcher in hyperbolic PDEs (plenary speaker at ICM 2002) is about that of good ones in elliptic theory. It is about one tenth of that of world leaders in elliptic PDEs. There must be a similar bias in other domains.

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In addition to the previous responses:

  1. Pure mathematicians are a lot worse about citing each other than applied mathematicians. Similarly, the threshold for being an author is quite high (I have seen (many) papers by X where the key lemma was proved by Y. In applied math or CS there is no question that Y would be a coauthor).

  2. In applied mathematics it is much more common for the advisor to be co-author on papers written by his/her grad students/postdocs (perhaps because of the greater influence of hard sciences, where the lab director's name appears on ALL papers published by people working in the lab. In the hard sciences the lab director's name is typically not the first, but in applied mathematics people use the pure math practice of alphabetical name order [at least more often than not]). This leads to a much greater average number of authors per paper, and so more publications and more citations. Unfortunately, the system is that a co-author of a five author paper gets full "citation credit".

If you look at the mathscinet list of top cited journals, you will see that the best pure math journals (Annals, Acta) are doing at best as well as some fairly generic applied math journals, which supports the above.

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"Pure mathematicians are a lot worse about citing each other than applied mathematicians" Would you mind explaining that a little better? I try to cite all papers I think is relevant in my papers and, by and large, I think my colleagues are citing relevant stuff. I rarely see a paper where I think that the authors have failed to cite relevant literature. Should we be citing our friends just for the sake of raising their citation index? – Felipe Voloch Apr 2 2011 at 0:01
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 I'm not sure what Igor meant, but I've seen some differences in citation patterns between pure math and theoretical CS (and I don't think I'm imagining them). My impression is that CS papers are more likely to have lengthy discussions of related work, in which they demonstrate the importance of the topic by citing many previous papers on related problems, and even papers that are only marginally related get mentioned for completeness or in the process of explaining how they differ. Pure mathematicians are more likely to cite only papers that are directly relevant or historically important. – Henry Cohn Apr 2 2011 at 0:56
@Henry: And why is this worse? – Felipe Voloch Apr 2 2011 at 4:43
Many pure mathematicians cite secondary references (books or later papers that give minor extensions of a result) and not mention the original source. I have not witnessed that in the theoretical CS literature I read. – Bill Johnson Apr 2 2011 at 10:29
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 @Bill,Igor. If I systematically trace original sources of results I use (I do number theory/algebraic geometry, btw) I will end up with a bunch of references written in languages most of my readers can't read, whose authors have been dead for decades. – Felipe Voloch Apr 2 2011 at 20:17
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