Tuesday, July 24, 2012

Erdős, the Kevin Bacon of the Math world

If you hang out with math nerds for an extended period of time, they will inevitably bring up Erdős numbers.

So what is an Erdős number? And what does this have to do with Kevin Bacon?

Most of us are familiar with “Six Degrees of Kevin Bacon”. In this game, you try to connect any actor or actress with Kevin Bacon in the fewest number of steps.

For example, Tilda Swinton was in “Constantine” with Pruitt Taylor Vince who was in “Trapped” with Kevin Bacon. Thus, Tilda Swinton is separated from Bacon by two degrees.
(To play this game online, go to http://oracleofbacon.org. It’s actually quite hard to get a high number).

Back to our friend, Paul Erdős (http://en.wikipedia.org/wiki/Paul_Erd%C5%91s). He was a famous Hungarian mathematician known for publishing the most papers (~1525) of any mathematician (among other things).

 A photo of Paul Erdős (from Wikipedia)

An Erdős numbers is the degree of separation a mathematician has from Erdős based on co-authorship of publications.

Say Person A published a paper with Person B who published a paper with Erdős. Person A’s Erdős number would be two, and Person B’s would be one. (Erdős’ Erdős number is zero).  (For a list of small Erdős numbers, see: http://en.wikipedia.org/wiki/List_of_people_by_Erd%C5%91s_number/)

Just like with actors and Kevin Bacon, it’s astonishingly hard to find mathematicians with really high Erdős numbers (say over 10).

Side note: Interestingly, this will not be true in a few hundred years. Because Erdős is dead, nobody new will ever have an Erdős number of 1. In time, nobody will have an Erdős number smaller than 10!

This phenomenon of small degrees of separation is called the “small-world effect”, named after encounters in real life when you meet a friend of a friend and exclaim “what a small world”.

The small world effect isn’t just a series of coincidences. As long as there are some random connections in a network, it becomes relatively easy to connect any two people in the network.

An example of an un-random network would be if you could only be friends with someone if they lived within ten houses of you. In this case, it would be really hard to pass a message via friends between two people who lived on opposite sides of the country.

In the real world, we can become friends with people who live anywhere. Thus, you are connected to almost everyone in the world by a surprisingly small number of steps.

To read more about network theory, I highly recommend Duncan Watt’s pop-sci book “Six Degrees: The Science of a Connected Age” (http://www.amazon.com/Six-Degrees-The-Science-Connected/dp/0393041425).

He’s also a good author to practice reading research articles, particularly the ones he co-wrote with Steven Strogatz (http://www.stevenstrogatz.com/). (Full disclosure: Duncan Watts is one of my science crushes.)

So now you are fully armed if this subject comes up with a group of mathematicians. And if they get too uppity, remind them that they will never be able to have Erdős numbers smaller than two.

Bonus fact: Erdős referred to children as “epsilons”. 

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