mesmerism! presidential assassinations! circuses on fire!
Football is not transitive. What do I mean? Simple. If Team A beats Team B, and then Team B beats Team C, it does not mean that Team A will beat Team C. We all know this.
Proof of non-transitivity this year comes in the Big Ten. Let's look at the Big Ten Graph. The graph is simple to understand: each team is a node (circle), and there is an line connecting each team that played another team. The line is actually an arrow, making this a directed graph, in the obvious form: if there is an arrow from Team A's node to Team B's node, it means Team A beat Team B. Here is the graph:
The Victory Graph (Click on it for full size)
There are lots of fun cycles to find in the graph. For example, Minnesota beat Iowa, who beat Michigan State, who beat Minnesota. See how many of these three-node cycles you can find (there are plenty). Or not, depends how bored you are at work. There are bigger ones too: for example, Michigan beat Indiana who beat Purdue who beat Minnesota who beat Iowa who beat Michigan State who beat Wisconsin who beat Michigan. And it goes on.
The most amazing fact from the graph, thanks to Indiana finally getting a win, is that the graph is strongly connected. In graph terminology, this means you can get from any node in the graph to any other node, simply by following arrows, for all pairs of nodes. This really shows how non-transitive football is: you can use this graph to say any team "transitively beat" any other team, at least in the Big Ten this past year. For example, Indiana beat Purdue, who beat Minnesota, who beat Iowa, who beat Michigan State, who beat Wisconsin, who beat OSU. If football were transitive, Indiana "beat" OSU! Except when they played, of course.
One interesting metric for each pair of teams (A, B) is the shortest path to victory for A over B. Some of these "shortest paths to victory" are easy to find: for example, it is unfortunately the case that there is a short and quite direct path from OSU (at the top) to Michigan. Some are harder to see: for example, see if you can find the path where Michigan "transitively" beats OSU. This "shortest path" is actually long: 6 steps (the answer is at bottom).
We can then use this graph to order the teams a different way: what is the shortest path between a team and every other team in the Big Ten? Lower is better here: a path of length 1 means Team A directly beat Team B, whereas a path of length 2 between Team A and Team B means that Team A beat Team C who in turn beat Team B. Here is the full summary of the shortest paths between all pairs of Big Ten teams:
You can then use these to create a new ranking among teams, based on their average shortest path to victory:
This ranking kind of makes sense, too. If you beat a lot of teams directly, then you will have an average near 1 (note that even undefeated teams will average higher than 1, because teams don't all play each other). If you only beat bad teams, who in turn only beat other bad teams, your average will be higher. Thus, Michigan does poorly in this comparison; Minnesota does better because they beat Iowa, who actually beat some good teams (like MSU). Only Indiana fares worse than our boys in Blue.
You can also prune the graph to arrive at some interesting findings. For example, let's say we remove all edges where one team didn't resoundingly beat the other team. I will arbitrarily deem a win as a "strong" win when one team beats the other by more than 10 points. The graph now looks like this:
The Strong Victory Graph (Click on it for full size)
Wow, that is a much different graph! The first thing that stands out: there are no cycles in this graph. That means that if Team A "strongly beat" Team B, and Team B "strongly beat" Team C, that Team C didn't "strongly beat" Team A. There are no cycles here my friends.
We can also then use the "Strong Win" Graph to compute a new ranking. For each strong win, you get a +1, and for each strong loss, you get -1. Here are the teams, ranked by this new "Strong Win" scoring system:
This is actually a pretty reasonable ranking I think. Wisconsin is on top, because they beat the tar out of everyone (almost). Michigan State doesn't fare nearly as well as Wisconsin and OSU, because they had many close wins and one game where they were trounced (Iowa). Michigan ends up behind Illinois and Penn State in this ranking, because those two teams had a number of big wins, where Michigan only had one (Purdue, and barely "strong" at that).
Anyhow, that's a short look at how graphs can help us rank teams in different ways. And if you didn't like it, well, remember that I Hate Everything too.
[EDIT: Some people asked how I generated the graphs. All automated, given an input of games and scores. Some python code to compute shortest paths between nodes (there are some fairly standard algorithms for doing this) and then Graphviz to layout the graphs automatically. It would be easy to do this for any set of games.
One other note: the real point of the "Strong Win" graph is how silly it is that score differential is ignored in current computer rankings. A big score difference is a useful metric, and one that I think is better than many other simple ways of comparing teams. One could likely come up with a slightly more nuanced "Strong Win" definition (say, win by 10 and outgain the other team by some threshold number of yards); this was just a simple and easy way to start.]
The path for "transitive victory" of Michigan over OSU: Michigan beat Illinois who beat Northwestern who beat Iowa who beat Michigan State who beat Wisconsin who beat OSU. Ugh, it is really hard for us to beat OSU, apparently.
I do not doubt that Mike Barwis is a good Strength & Conditioning coach. How much better is he, than at other elite programs?
Michigan this year was not noticeably better conditioned than its opponents. I don’t think better conditioning was the obvious difference in any game. I don’t recall a game in which the Wolverines ground down their opponent in the fourth quarter.
In Michigan’s seven wins, excluding Bowling Green, the Wolverines were actually outscored (modestly) in the fourth quarter. (I exclude BG because the starters were no longer playing at the end of the game.) In their five losses, Michigan did outscore the opponents in the fourth quarter, but only by a total of 9 points—not a huge difference.
Michigan’s players certainly did not seem to be any less injury-prone than their opponents; if anything, they were more injury prone.
Although Barwis has clearly improved Michigan’s conditioning over the latter years of the Carr program, I do not see the evidence that he has made it a distinct advantage. All that one can say, is that it is no longer a liability.
For those of you that have been waiting, below is an official copy of the Prohibition on Coaching Change Posts Act of 2010. For those of you that are predisposed to comply with the terms of the Act, I suggest printing the below and posting it in your cubicle, office, dorm room, or some other location where it can easily be referenced. This act was codified on November 30, 2010 but is always subject to further revision. Please let me know if you see any glaring issues that need to be addressed. Thank you in advance.
The Prohibition on Coaching Change Posts Act of 2010.
Article 1 - Notices
(a) Notices of MGoInfraction shall be "filed" by Profitgoblue as soon as practicable after a violation of this Act.
(b) Notices shall be in writing and shall specifically (i) explain the violation, (ii) inform the offender of the fine levied, and (iii) inform the offender of their opportunity for appeal.
Article 2- Infractions, Defined
(a) Thou shalt not, under any circumstances, discuss Jim Harbaugh in connection with the Michigan head coaching job unless and until Rich Rodriguez has officially been relieved of his current contractual obligations.
(b) Thou shalt not, under any circumstances, lobby for or otherwise argue in support of the firing of Rich Rodriguez until at least the end of 2011.
(c) Thou shalt not, under any circumstances, speculate, opine, or otherwise comment on Dave Brandon's decision timeframe with respect to the head coaching and/or defensive coordinator job(s).
(d) Thou shalt not, under any circumstances, post a personal opinion with respect to whether or not Rich Rodriguez will be fired.
(e) Catch-all provision - additional infractions can be identified at a later date by Profitgoblue in his sole discretion.
Article 3- Penalties
(a) All violations shall carry a minimum fine of 1 MGoPoint, issued by Profitgoblue, with a maximum fine to be determined by the MGoCommunity at large.
(b) Fines are subject to waivers as more fully described below.
Article 4- Waivers
(a) Waivers of Application of the Act - posts that would otherwise be deemed violations of the Act can be excused by formal, written waiver issued by Profitgoblue in his sole discretion.
(b) Waivers of Levy of Penalty - posts that are deemed violations of the Act shall be cited by penalties can be excused by formal, written waiver issued by Profitgoblue.
Article 5- Appeals
(a) All MGoInfractions will be final determinations unless a formal, written appeal is "filed" within twenty-four (24) hours of the initial notice of the infraction.
(b) All appeals must be "file" by posting a reply to the initial Notice.
(c) The merits of all properly-filed appeals, as well as all amicus briefs filed on the offender's behalf, will be considered by Profitgoblue in his sole discretion.
(d) If successful on appeal, the MGoPoints deducted by Profitgoblue will be restored.
(e) If unsuccessful on appeal, the MGoPoints deducted by Profitgoblue will be lost forever. However, a mandatory restoration of MGoPoints shall occur if the ruling on appeal is sufficiently negged.
Henri, the Otter of Ennui, is mad as hell and he is not going to take it any more.
In Boca Raton, FL, two days before the Wisconsin game, he attacked and bit a dog. The day after the game, two people were attacked and bitten. Three days after the Ohio State debacle, Henri struck with a vengeance and the young man in the video was attacked.
Animal control has fielded over 100 calls from residents in and around Boca of an out-of-control otter - suspected to be the same one.
David Hitzig, executive director of Busch Wildlife Sanctuary in Jupiter, says, "It is a mystery as to what's going on and why we've had these attacks recently." "The behavior we've seen in these incidents does not point to a diseased animal." Experts say if the otter was rabid, it would have likely have died by now.
Reports are uncertain as to whether Henri's behavior is related to Michigan's football coaching situation, area recruits, or existential angst within the MGoBlogosphere.
A speedy recovery to those upon whom Henri's pent up wrath was unleashed. A pox on those who provoked him.
Hello MGobloggers. A couple of us last night decided it would be kind of fun to set up a college football bowl pick'em pool. I got the go ahead from Brian to post. This pool is open to the MGoblog family.
Go to ESPN.com
Look under Fantasy and minigames. Pick College Bowl Mania
Choose join a group.
Our pool is obviously called mgoblog
The password is Denarded.
Create your entry
No cash prize, just for bragging rights, although if you get the best on ESPN I think you win a Best Buy $2,000 gift card.
If there are any questions, don't hesitate to post them.
You can start picking in 3 days. Let the games begin, please. I will post this again before bowl season starts since it becomes locked once bowls begin.
Does anyone know when they will annoucements the teams for each bowl? I am thinking it should be this sunday but not sure!