# 4.74 Degrees of Separation

We’ve talked before about your Erdos number. This is the number of steps between you and Paul Erdos on the collaboration graph of mathematics.  Where each vertex on the graph is a mathematician and each connecting edge between two mathematicians is a paper that they wrote together.

Unlike Kevin Bacon, Paul Erdos didn't drive a car. We can't rule out Paul Erdos dancing, though!

The Hollywood version of the Erdos number is the Bacon number.  In this case, the vertices are actors and an edge connects two actors if they were both in the same movie.  Your Bacon number is the number of steps along this graph between you and Kevin Bacon.   For example, Humphrey Bogart, Ingrid Bergman, Peter Lorre, and Claude Rains all have a Bacon number of 2*.  Ours, sadly, is infinity.  We’re still waiting for Peter Jackson to return our phone calls about OU Math Club: The Movie.

Of course, how far apart two people are on a graph depends on how connected it is.  A typical movie has lots of actors, so there is lots of connections between actors.  So you’d expect that the average distance between two actors is pretty low.   And, of course, not many people are actors, so the overall size of the graph is also pretty small.

Which brings us to the most massive network of people ever studied.  Facebook.  When the study was done there were roughly 700 million users and 69 billion friendships on Facebook (it’s already much more).  Which, by the way, puts the OU Math Club Facebook page square in the middle of this graph:

Cumulative degree distribution is the percentage of people who have that many friends or fewer.

The folks at Facebook crunched the numbers and found that the average distance between any two people connected on the Facebook graph is currently 4.74 steps (or hops as they call them).  Interestingly, in 2008 the average was 5.28 on the Facebook graph.  Here’s the data:

Another remarkable thing they found was that for 84% of people of facebook, their friends have more friends than they do.   That is, it’s likely that your friends have more friends than you.  That’s depressing!

It’s not because you’re unlovable!  It’s the nature of the friendship graph.   It’s a little counter intuitive until you ponder it for a bit.

Think about Kevin Bacon.  He’s been in many, many movies.  More than most actors.  In particular, more than most of the actors who have been in a film with Kevin Bacon.  A few of them (including Mr. Bacon) will have been in a large number of movies.  But for the large majority of the actors in Kevin Bacon’s films, if you ask them if they’ve been in a lot of movies, they’ll say “not as many as Kevin!”.   From the pool of people who have been in a film with a Kevin Bacon, the large majority of them will have been in fewer films than some of the people they have acted with.  It’s similar for the Facebook graph.  So don’t be sad.  This phenomena is discussed by Scott Feld here.

For all the details on the Facebook calculations, go here.

*  To calculate Bacon numbers, use the handy dandy Oracle of Bacon (not as delicious as it sounds).