Note: If you find this boring, then I apologize in advance.
The six degrees of separation is a theoretical concept that says, "everyone is on average approximately six steps away, by way of introduction, from any other person on Earth, so that a chain of, "a friend of a friend" statements can be made, on average, to connect any two people in six steps or fewer. (Wikiped’d)". In other words, if you wanted to connect Nerd the Rebel to say… E. Honda, you could say Nerd the Rebel knows Person A who knows Person B who knows… who knows E. Honda, and you’d only need at most six people in the middle.
Well how about this: Given two BCS football teams A and B and the 2012 schedule, how many games does it take to connect A and B. For example, let’s take Ole Miss and Oklahoma. Well, Ole Miss plays Texas who plays Oklahoma, so the answer is two games. But what is the maximum number of games we would need to connect any two teams? I think this is a difficult question in general, but I want to share a few thoughts I have.
We’re going to model this question by using vertices to represent teams, and if two teams play each other, they will be connected by an edge. (A couple of years ago, I wrote another post with a similar construct.) As an example, here is what Ole Miss’s schedule looks like. (Central Arkansas is in red, because we really only are considering BCS teams, not FBS.)
OFF TOPIC: As a side note, I think this modeling systems makes an easy graphic to see a team's success overall. Just replace an edge with an arrow facing the winning team. Here’s Ole Miss’s 2011 schedule:
Yeah, pretty depressing, but it’s easy to see that Ole Miss went 2-10 without having to count the W and L column on a schedule. Just another example, here’s Georgia’s 2011 schedule:
And we can also model the conference’s performance as a whole. Here’s the SEC conference schedule ( I didn’t add out of conference games, but it could easily be done.) :
This idea may not be ideal for basketball and baseball as the number of edges would be much greater.
Back on Topic:
Again, we want to know what is the maximum number of games required to connect any two teams via schedules. Honestly, I don’t know the exact answer, but I’ll share some of what I have found so far.
Let conf(A) mean the conference of team A. Let dis(A,B) be the distance from A to B (the number of games connecting A to B), and let DIS(conf(A),conf(B)) be the maximum dis(C,D) for any team C in conf(A) and any team D in conf(B). Makes sense?
If team A and team B are in the same conference (conf(A)=conf(B)), then dis(A,B) is at least 1 and at most 2. For example, dis(Ole Miss, Florida) = 2 because Ole Miss plays Vanderbilt who plays Florida. Suppose A and B are not in the same conference then. Well, this can get kind of tricky. I think for any two conferences conf(A) and conf(B), DIS(conf(A),conf(B)) will be at most six, but I’m not sure.
Let’s do an example, what is DIS(SEC,PAC12)? (again, this means the maximum number of games that connects any SEC team to any PAC12 team) Well, dis(LSU, Washington) = 1 because LSU plays Washington (and also dis(Mizzou, Arizona State) = 1). So DIS(SEC, PAC12) is at least 1. And the worse case scenario can be viewed in the picture below.
Let’s figure out dis(Auburn, UCLA). Well, Auburn plays LSU who plays Washington who plays USC who plays UCLA. That is the shortest path in this process since Auburn doesn’t play Mizzou (Then the shortest path would be Auburn to Mizzou to Arizona State to UCLA). Hence dis(Auburn, UCLA) = 4. However, it may be quicker to go through a third conference. For instance dis(Alabama, UCLA) = 3 if you take a path through the Big 10.
So dis(Auburn, UCLA) might be smaller than 4, and I just am not smart enough to see the path through a third conference. It gets really confusing when you look at the WAC, Sun Belt, etc. Anyway, what I do know is that DIS( SEC, PAC 12) is at most 4 with the worst case scenario being something like the Auburn to UCLA example I showed above.
I said that I think six is the worst possible for any two teams, and the reason I said that is because not every conference plays every conference. For instance, the Big East and the Big 12 do not have a regular season meeting in 2012. The only conference the SEC avoids is MWC (and a few of the independents). If two conferences don’t play each other directly, then you’ll have to go through a third conference, and that’s very hard to spot just looking at schedules. It’s possible, but it would take some time.
If any of you are thinking about writing a mathematical paper, this might be kind of an interesting question to pursue, except you would want to answer the question regardless of what the schedule looked like. Anyway, I hope you found this interesting.