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Let $E(n,t)$ be the number of mathematicians with finite positive Erdős number $n$ at time $t$. As old mathematicians leave, and new mathematicians come, how does $E(n,t)$ change over time ?

We can consider only the cooperations between two mathematicians, and assume that every year, the number of new articles and new mathematicians are both constant. We can fix the length of career for every mathematician. Feel free to add any other assumptions.

Some obvious facts are

  • for every $n>0$, there exists a time $T(n)$, such that $E(n,t)$ is constant for $t>T(n)$.
  • the average Erdős number will increase over time, but is it linear ?
  • ...

It would be nice to find a model for the growth of network that can fit the real data. But since we are not far from Erdős ($T(1)\leq Y1996$), no data is available for large $t$, unfortunately. The distribution seems to be converging towards low Erdős numbers at this time.

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I would reformulate the question like this: how to build the model such that it will fit real data ? (It is difficult but theoretically possible to get actual value of E(n,t) from real world. It would be nice then to compare it with theory). – Alexander Chervov Jun 20 '12 at 9:13
agree. updated. – Hao Chen Jun 20 '12 at 9:23
PS Good question. I somewhat envy that it did not come to my mind :):) (Joking). I always keep in my mind the question what kind mathemetically precise questions we ask about social networks ? That is one of them. – Alexander Chervov Jun 20 '12 at 9:53
We can ask the same about "mathoverflow number" (and the answer can be much more easy to check in practice). I mean let say two users of MO are "coauthors" in they contribute to the same question. Let us select one user (some one like Erdos e.g. some one who have many coauthors) and form the "mathoverflow number" as distance to this user. Then we can ask the same question and actually more - does the evolution depends on the initial user ? Does the distribution depends on the initial user ? How much statistics we need to get stable results ? – Alexander Chervov Jun 20 '12 at 10:07
A related MO question, with references: "The diameter of the Erdös component of the collaboration graph" – Joseph O'Rourke Jun 20 '12 at 10:20

You can probably find some interesting information on the Erdös number project, here.

They have distribution below listed, but it is quite old and definitely incomplete. I just emailed them that V. Arnolds Erdös number 6 should be reduced to 5, due to a new publication in 2008.

  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
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