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Let $\Gamma$ be the graph defined as follows:

  • the vertices are mathematicians who have published papers
  • there is an edge between any two mathematicians who have authored publications together.

Graph $\Gamma$ is certainly not connected, as there are mathematicians who have only published papers without collaborators. Obviously, the largest connected component is the one containing Paul Erdős, since almost all mathematicians today belong to this component. My question is:

Does anyone have an idea what the second-largest connected component is?

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  • $\begingroup$ Is this matter of interest ? $\endgroup$
    – Jean Marie
    Jun 10, 2017 at 11:26
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    $\begingroup$ I am curious, so yes. Isn’t that how most questions arise? $\endgroup$
    – Gaussler
    Jun 10, 2017 at 11:32
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    $\begingroup$ @AnthonyHernandez no, he is simply well-known for having many collaborators, as reflected in the use Erdős numbers. The fact that most mathematicians have an Erdős number less than infinity proves that almost all of mathematical society is connected. $\endgroup$
    – Gaussler
    Jun 10, 2017 at 13:46
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    $\begingroup$ @AnthonyHernandez it actually is true according to the data on the page linked to. It's somewhat tangential to this discussion though; the stress in the question is misplaced. I'd venture to bet that removing Erdős (or anyone for that matter) would not change the graph substantially. It would always have one "giant" component. $\endgroup$
    – quid
    Jun 10, 2017 at 15:20
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    $\begingroup$ Probably the answer depends on how you define "mathematicians". I guess that the second component is in a very applied area or one far away from mainstream mathematics. $\endgroup$ Jun 10, 2017 at 15:38

1 Answer 1

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At https://oakland.edu/enp/trivia/ you can read that

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 but are not in the large component is 1.65.

If the second largest component had two vertices that number would be just $1$.

Commenters below have found the size of that component (at various historical moments.)

You can download the data from that site and write a program to find out who is in that component.

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    $\begingroup$ From that page, it seems the answer to the question should be one click away, but the link for "the distribution of component sizes" doesn't work. I have emailed Dr. Grossman to see if he has the data. But if you only make edges for papers written with two people, then the largest component has 175902 people, and second-largest component has size merely 28. $\endgroup$
    – Mark S.
    Jun 10, 2017 at 14:19
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    $\begingroup$ @MarkS: Archive.org to the rescue! Apparently, at the time the table was compiled, the second largest component contained 32 people (and the third largest had 31, and the fourth largest 28). $\endgroup$ Jun 10, 2017 at 17:38
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    $\begingroup$ Those 32 authors may need to work extra hard to prevent their special connectivity component from getting gulped up by the mass. $\endgroup$ Jun 10, 2017 at 19:02
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    $\begingroup$ @Jyrki Lahtonen I like you humor ... $\endgroup$
    – Jean Marie
    Jun 10, 2017 at 21:23
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    $\begingroup$ @IlmariKaronen Dr. Grossman has fixed the link to point to here, but the numbers are the same as in that archive page. $\endgroup$
    – Mark S.
    Jun 13, 2017 at 0:53

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