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This site claims that the diameter of the Erdös component of the collaboration graph in 2004 was 23. What is it now? Is it increasing or decreasing with time? Recall that the vertices of the collaboration graph are mathematicians and two vertices are connected if the mathematicians co-author a paper. MathSci allows one to find the collaborative distance between any two mathematicians. So in principle one can find the diameter just by using MathSci. In general (and more seriously) is there a mathematical theory which describes the growth of "real life" networks like the collaborative graph?

Update 1 A process that I had in mind is something like this. At every step one of the following things can happen.

  1. A new vertex $a$ is born with some probability $p$. Usually that vertex is a "student" of some other vertex $x$. We can connect $a$ and $x$ by an edge.

  2. A randomly chosen vertex $y$ gets connected with a randomly chosen vertex $z$. But the probability for choosing $z$ is not uniform. Those vertices that are closer to $y$ have more chances getting connected to $y$.

  3. A randomly chosen vertex "dies" (with some probability $q$) meaning it does not participate in new edges any longer.

Update 2. Many thanks to Balazs and Joseph for their answers. But the first question still remains: what is the diameter of the Erdös component now?

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@Steve: Thank you! I meant something like that, but it is not quite the same as the collaborative graph. I explain what kind of process I mean in the Update to the question, because there is not enough space in a comment. –  Mark Sapir Nov 10 '10 at 19:10

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up vote 7 down vote accepted

There is a very large literature on this, written by people doing "network science". One of the names you might want to look up is that of Mark Newman, see for example his papers The structure of scientific collaboration networks or Who is the best connected scientist? A study of scientific coauthorship networks as well as Clustering and preferential attachment in growing networks. Another important player in this field is Albert-Laszlo Barabasi, whose group had a model (for the web graph) which is somewhat similar to what you suggest, Emergence of scaling in random networks. Some of this work is reviewed nicely in Statistical Mechanics of Complex Networks. The model by Barabasi et al is studied in mathematical terms by Bollobas-Riordan-Spencer-Tusnady's The degree sequence of a scale-free random graph process.

To answer your specific question on the diameter, I would expect it to be quite small ("six degress of separation"); whether it decreases in time must depend on the relative rate of birth of new vertices versus new edges (collaborations). If you take a finite graph and add completely random links, then already a few links lead to a diameter which is logarithmic in the size of the network. I think this remains the case also if you weigh your edge creation mechanism by the distance of the vertices; I once played with a model where the only new edges were created between 2-step neighbours, which still lead to small diameter.

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Thanks! This might be exactly what I wanted. –  Mark Sapir Nov 10 '10 at 23:25

The paper "Some analyses of Erdös collaboration graph" by Vladimir Batagelj and Andrej Mrvar (Social Networks Volume 22, Issue 2, May 2000, Pages 173-186) is filled with fascinating data and analysis as of 2000. (Because this is a decade old, it does not answer your question.) Here is their figure of "cliques of the main core," with many recognizable names (if you can read them at this resolution!).
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I see many (63) later papers that cite this one, but none that directly update it for this specific Erdös graph.

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@Joseph: Thanks! I did not know about this article. Do they consider the diameter also? I wonder if it was smaller or larger than 23 in 2000. I did not find it in the paper yet. –  Mark Sapir Nov 10 '10 at 20:37
@Mark: There were 17 connected components (after removing the Erdös-root!). One of them was large: 6045 authors. The diameter of that component was 12. They do not seem to report the diameter of the connected graph, but it might be implied by data in their tables. –  Joseph O'Rourke Nov 10 '10 at 20:46
@Joseph: Thanks for sending me the paper by email. I thought that 12 in the paper was the diameter of the Erdös component. In principle, it could be that for appropriately chosen parameters in "my" model (described in the question) one can get similar results. Then we would be able to predict the future structure of the mathematics community. –  Mark Sapir Nov 10 '10 at 22:57
@Mark, p.176: "The diameter of the large component in graph $E'$ is 12 with three diametric pairs of vertices..." Not that it matters. Re predicting the future of the math community: I assume you are kidding! Regardless, collaboration may be stronger in Erdös-fields than in other areas of mathematics. –  Joseph O'Rourke Nov 10 '10 at 23:47
Laszlo Lovasz (in the clique above) just received the Kyoto Prize! –  Joseph O'Rourke Nov 11 '10 at 20:24

In the long run, the diameter will grow. Let k be an upper bound to the number of years between a mathematician's first and last publication. Then nk years after Erdos' last publication, all new nodes in the Erdos component will have Erdos number at least n, so the diameter will be at least n.

(I am assuming here that there will be new nodes, which is not true for very large values of n. Also, I am assuming that k is constant, but only to keep the estimate "nk" simple.)

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Of course, for my question to be non-trivial, one should consider the max. distance between "live" vertices only. –  Mark Sapir May 21 '11 at 20:28

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