## Monday, November 25, 2013

### Collaboration Distance and the Math Genealogy Project

The American Mathematical Society has a fun site on "collaboration distance" between various mathematicians. The idea is simple: If, for example, I wrote with X, and X wrote with Z, then my collaboration distance to Z is two. There's a good description here, and the actual calculator is here.

You can track your collaboration distance not only to Erdos (of course), but also to all-time giants like Gauss or Laplace. The calculator reveals, for example, that my collaboration distance to Gauss is just eight:

I co-authored with Marc Nerlove
Marc Nerlove co-authored with Kenneth J. Arrow
Kenneth J. Arrow co-authored with Theodore E. Harris
Theodore E. Harris co-authored with Richard E. Bellman
Richard E. Bellman co-authored with Ernst G. Straus
Ernst G. Straus co-authored with Albert Einstein
Albert Einstein co-authored with Hermann Minkowski
Hermann Minkowski co-authored with Carl Friedrich Gauss.

Wow -- and some great company along the way, quite apart from the origin at old Carl Friedrich!

Of course I understand the "small-world" network phenomenon, but it's nevertheless hard not to be astounded at first.

So how truly astounding is my eight-step connection to Gauss? Let's do a back-of-the-envelope calculation. For a benchmark Erdos-Renyi network we have:

$$max \approx \frac{\ln N}{\ln \mu},$$
where $$max$$ is the maximum collaboration distance, $$N$$ is the number of authors in the network, and $$\mu$$ is the mean number of co-authors. Suppose there are 1,000,000 authors ($$N=1,000,000$$), each with 5 co-authors (so, trivially, $$\mu=5$$). Then we have $$max \approx 9$$.

Hmmm...I'm no longer feeling so special.