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As many mathematicians know, each person has an Erdős number (see: http://en.wikipedia.org/wiki/Erd%C5%91s_number). That is, Erdős himself has Erdős number zero, each person who published anything with Erdős (mathematical or not) has an Erdős number one, and each person who published something with someone with Erdős number 1 but not with Erdős himself will have Erdős number 2, and so on. If you have never published anything with someone with finite Erdős number, then you have Erdős number infinity.

The set of people with Erdős number 1 is very easy to track; the set with Erdős number 2 still possibly easy but more difficult, and as time goes on it becomes increasingly difficult to keep track of the set of people of Erdős number exactly $n$.

My question is, what kind of system (is this an example of a dynamical system? I am not sure) behaves like Erdős numbers? If so, can we answer any of the following questions in a meaningful way?

1) What is the expected value of Erdős numbers at time $t$?

2) What is the probability of finding someone with a large (but finite) Erdős number, as a function of the number $n$ and time $t$?

3) As $t \rightarrow \infty$, is it expected for the set of people with finite Erdős number to fill up the entire population, or at least a positive portion of the population? (That is, let $N(t)$ denote the number of people at time $t$ with finite Erdős number, and let $R(t)$ denote the set of all people at time $t$. Does the ratio $N(t)/R(t)$ approach some positive limit?)

This question is motivated by a remark made by Doron Zeilberger a few years ago at Herbert Wilf's 80th Birthday Conference, where he said that it is remarkable to find someone with a large but finite Erdős number. Further, he is not sure if anyone with an Erdős number 7 or larger (but finite) exists.

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    $\begingroup$ Shouldn't Erdös technically have an Erdös number of $2$? He hasn't published with Erdös but he has published with someone that has published with Erdös. This is similar to the 'issue' that prompted Conway to suggest that $-1$ is prime. $\endgroup$
    – mathematics2x2life
    Commented Jan 1, 2014 at 7:10
  • $\begingroup$ As people have said, this is an interesting question, both for empirical and theoretical modeling. For the empirical model of Erdos number as time increases, one fact that may well have an influence is the increasing number of multi-person papers in mathematics. Probably this phenomenon has also been studied, but I don't know if it's been (empirically) quantified, and one would certainly need to do so in order to estimate the future distribution of Erdos numbers. $\endgroup$
    – Joe Silverman
    Commented Jan 1, 2014 at 12:50

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This is not really an answer and rather an extended comment. But there is a lot of serious research on network systems that formalize or are related to collaboration networks of authors. A good starting point may be this webpage on research on collaboration in research by the Erdős Number Project. I'm no expert, but the article seems to summarize the history very well and provides lots of links to actual serious research. For example, the abstract of one article mentioned there reads:

A.L. Barabasi, H. Jeong, Z. Neda, E. Ravasz, A. Schubert, T. Vicsek, Evolution of the social network of scientific collaborations, Physica A 311, (2002), 590-614.

The co-authorship network of scientists represents a prototype of complex evolving networks. By mapping the electronic database containing all relevant journals in mathematics and neuro-science for an eight-year period (1991-98), we infer the dynamic and the structural mechanisms that govern the evolution and topology of this complex system. First, empirical measurements allow us to uncover the topological measures that characterize the network at a given moment, as well as the time evolution of these quantities. The results indicate that the network is scale-free, and that the network evolution is governed by preferential attachment, affecting both internal and external links. However, in contrast with most model predictions the average degree increases in time, and the node separation decreases. Second, we propose a simple model that captures the network's time evolution. Third, numerical simulations are used to uncover the behavior of quantities that could not be predicted analytically.

The article's preprint is available here on arXiv.

The following is a review article on serious research on network systems:

M. E. J. Newman, The structure and function of complex networks, SIAM Review 45, (2003) 167-256.

(And its shorter version on Proc. Natl. Acad. Sci. USA 98, (2001) 404-409.)

I wish I could answer your questions by writing an interesting mathematical post. Alas, all I know is that it is an interesting and serious research topic. Since you got downvoted right away, hopefully someone familiar with this kind of research will post a nice answer before this thread gets closed...

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The systems you are looking for belong to the class of small-world networks, introduced in 1998 by Watts and Strogatz. Quite generally, a small-world network is defined to be a network where the typical distance between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes in the network.

Wikipedia provides a good starting point for exploration:

Small-world networks

Small-world experiments

Specifically related to the Erdős number is this online lecture by John Barrow:

Erdős Numbers: A mathematical example of 'small world' networks

Related MO posts dealing with the dynamics of the Erdős number (and suggesting the introduction of a Mathoverflow number):

How does the distribution of Erdős number evolve over time ? How to build a model to fit the real data ?

The diameter of the Erdös component of the collaboration graph

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