Graph Theory (or, alternate ways to rank the Big Ten this year)

The strong victory graph provides a nice objective ranking and passes the eyeball test. The first graph appears to give too much credit for a big upset. Minnesota is obviously not that strong of a team.

Now THIS is interesting. I am curious how this would look for all FBS teams. I smell a future BlogPoll entry.

It is fascinating.  Now, did anyone in the Big Ten have a common opponent with Stan . . . (third rail) . . . oh, never mind.

it is kind of like an objective power ranking of the big ten. only downfall of it is if a crappy team somehow pulls one upset out over a dominant team, their shortest path average jumps to 2 (the minnesota effect). i dont know how to tease out those sort of details, but the 'strong victory' seems like a good start

For a simple measure, it produces a plausible ranking. A few years ago I went through and compared a bunch of "power" programs' historical records in close games (8 points or less) and blowouts (20 points or more) and found that with the exception of OSU they won barely 50% of the close games, but something like 90% of the blowouts. The cliche is that clutch performance separates good teams from bad teams, but the evidence suggests that the mark of a good team is the ability to take other teams to the woodshed.

Most interesting new diary I've read in quite a while.

Was it a coincidence with shapes from Visio or did you use a mapping package?

I love to see the things I learned in my computer science and math classes applied to football. I think graph theory should be used more and it would be interesting to see larger graphs relating the conferences this season.

I'm interested to know if you automated this process?  I think the use of the graph in the college football sense.  I think in order for you to do it for all college football you would have to have an automated procedure to avoid mistakes and obviously save time.  I know there are a lot of computer scientists on this site. I think this could really be an interesting computerized metric in college football. 

This is one of the most interesting diaries filled with graphs? Graphs. that I have read in quite a while.

Good work on this diary, I appreciate it and it's given me another fascinating metric to chew on while we await the next word(s) from Dave Brandon. Also, I like how you placed and related the two graphs together, and the picture that the "strong" graph seems to paint. Again, excellent work and thank you for contributing it to this board.   

Since only 8 teams are directly faced from the conference each season and the shortest path to success is to actually play a team and beat them, then I think what you are suggesting is that if we only added one more team to the conference, then divided the conference into two divisions, then we could make sure everyone in their division played everyone else every year, plus 3 teams in the other division, then we could always exactly pick to two best teams to play a final championship game at the end of the year that would be the undeniable champion and end all arguing for-ever!?!

Shit, that just might work. 

Nah, we would still find some way to claim the wins Michigan got weren't really good wins, and while the offense seems exciting, really sucks because Denard drops the ball and Tate throws it up for grabs all the time, and shifting to a 3-3-5 confused all of the 5 star talent on the defense so they performed like 2 star talent that can't even tie the shoes of the Indiana players girlfriends, and our special teams suck, not because we're using up all the redshirt freshman on Defense instead of special teams where they normally play, but because we're using true freshman and an assortment of players who suck but show up to practice and workouts all year long anyway because this athletic scholarship was the only way they could even dream of getting a degree from a Univeristy like Michigan, or probably any University at all.

yep, we should probably just keep whining.  Cause that's what we learned during out time getting a degree at Michigan.

dang, we suck too.

I had a professor who was working on generalized graph theory and ways of bounding the size (and number of) cycles in arbitrarily sized tournaments. It's cool stuff.

I'm curious about the relationship between average path length and the threshold you used (margin of victory). The graph is strongly connected, sure, but it relies heavily on two paths (Minnesota over Iowa and Indiana over Purdue) to get this quality (and both of these paths rely on a 3 point margin of victory). Considering that this isn't a tournament, removing these paths might punish some teams arbitrarily simply because the path between two teams that didn't play may now be much longer (but this is not a metric of their relative quality!).

That notwithstanding, that's a pretty strong ranking. It's objective, sure, and you can poke some holes in it (as I have attempted to do), but from a subjective point of view it certainly passes the eyeball test. In the quarter break I may try to write software to examine an expanded graph; that could be pretty interesting.

I have been thinking about something related to this graph theory for a while, but to see this article eventually, I have to say it's NICE WORK! Well, to the very least the circles are fun to search for and also we got another theory on how bad we suck this year.

A way to incorporate the idea of "strong wins" into your first set of rankings: give the edges weights inversely proportional to margin of victory (so blowout wins are "shorter" than close wins) and then look for the shortest path.

I am a nerdy engineer so I enjoy these kinds of things. Especially when applied to Michigan Football. 

I like it.  Front page material?

So, what you have proven is the saying, "Nobody goes through the B10 schedule undefeated?"

I like the way you present and explain the information.

Another interesting thing about this method is that it doesn't give you the traditional problems that you would get if two teams played each other twice (once in a CG or Bowl) and both won one game. This would give both teams the benefit of the doubt and decrease their path length to 1. It means less in a conference ranking, but in on a national scale, it could be huge for the team that lost the first matchup but won the second one.

I've heard about stuff like this before but haven't ever seen anyone explain it in a way I could follow. Nice job.

it tells me what i knew which is that in the Big Ten you don't get unexpected lop sided losses.  

Out here where I live in Pac ten country it seems to happen all the time.  Check out a string from this year

USC 48 - Cal 14, Cal 50 - ASU 17, ASU 42-WSU 0, WSU 31 - OSU 14, OSU 36-USC 7

every one is a strong victory and in just 5 games USC beats USC by 155 points!   here is another

Wash 24 UCLA 7, UCLA 42 WSU 28, WSU 31 OSU 14, OSU 35 Cal 7, Cal 50 ASU 17, ASU 24 Wash 14.  

This kind of thing just doesn't happen in the Big Ten (as you showed)