UPDATE: This application has been gone for quite a while. There are other "explore your network" applications on Facebook, if you so desire.
So, there is another Facebook application that lets you play with your network, 6 degrees of separation, by Karl Bunyan. This one goes a little further than the Facebook Friends wheel that I mentioned a few weeks ago.
Upon installation, the first thing the application does is calculate the number of hops between you and the developer, Karl Bunyan. I'm four hops. And it gives you some other tidbits, like the average number of hops is 6.03 and the maximum is 14.
There are a couple things to play with in this application. The first is to explore your own network. While doing that, if you click on enough of the people you know, it will show you how they are connected as well. For example, the screen grab here shows Luis Suarez front and center and all of the Facebook connections we share in common.
The second thing is that you can find how far you are from other Facebook users with the same calculation that Bunyan uses to get a distance between he and you. The odd thing with this feature is that most of the Facebook users I know about are already my direct connections. I tried Carl Kassel (of NPR) and am 3 degrees away through several paths. I'm not connected to Fred Flintstone, sadly.
As the application says, it is an experiment. Play, if you feel like you have the time and inclination.