Wednesday, September 21, 2016

Paul Erdős and Kevin Bacon

In his famous 1967 paper, Stanley Milgram describes a study he conducted that shows that people are related to each other via a very small of number of acquaintances. This led to the phrase "six degrees of separation" being coined by John Guare in his play of the same name. Mathematicians study a similar concept called Erdős numbers. Two persons are linked if they have co-authored a mathematical paper together and a person's Erdős number is the minimum number of links between him/her and Paul Erdős. Thus Paul Erdős has Erdős number 0, A person other than Erdős who has written a paper with Erdős has Erdős number 1. A person who does not have Erdős number $\leq 1$ and has written a paper with a person with Erdős number 1 will have Erdős number 2, etc.  If you have not written a paper with anyone with a (finite) Erdős number, then your Erdős number is $\infty$.

As a consequence of a drinking game, there is a similar notion among actors, called the Bacon number. Kevin Bacon has Bacon number 0 and the authors who are his co-stars in a movie have Bacon number 1, etc.

The fascinating aspect is that the Erdős number and the Bacon number of most people whose number is finite is relatively small, which is typically referred to as the "small world effect". There are various websites that lets you type in a name and it will attempt to find the Erdős or Bacon number of this person.

There is an additional notion of an Erdős-Bacon number which is the sum of a person's Erdős number and Bacon number. As of now the lowest Erdős-Bacon number appears to be 4.

There is something unsatisfactory about the definition of the Erdős-Bacon number.  In particular, both Paul Erdős and Kevin Bacon have very large (and possibly infinite) Erdős-Bacon number.  As of today, according to this link and this link, Paul Erdős has Bacon number $\infty$ and Kevin Bacon has Erdős number $\infty$.

There is one way to remedy this injustice.  Kevin Bacon should publish a math paper with someone with Erdős number 1 (unfortunately Paul Erdős died in 1996) and starred in a movie with people who appeared in the documentary "N Is a Number: A Portrait of Paul Erdős".  Since several mathematicians in the above documentary have Erdős number 1, both these activities can be combined if  Kevin Bacon makes a documentary of how he collaborated with one (or more) of these mathematicians on a math paper. This will ensure that both  Paul Erdős and Kevin Bacon have Erdős-Bacon number 2 (the lowest possible) and the universe will be in order again.

So Mr. Bacon, if you are reading this, please make that your next project!

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