More Discardable Fun With Charts and Stuff

by SMQ on Nov 15, 2006 8:15 PM EST in News

Promise, super double dog promise, this makes sense. It only looks like math.

One of SMQ's favorite football sites is the impressive victory chain linker, which highlights in the most non-smartass fashion possible the absurdity inherent in "Team A beat Team B, Team B beat Team C, so Team A would beat Team C" logic.

Regular commenter, computer defender and math-oriented person Paul Kislanko, though, mails today with a slightly more useful chain linker project from his own data-laden site.

What Paul's done is link every team to every other team through victories to date this season - and every team is eventually linked to every other team (except undefeateds and teams beaten only by undefeateds) through "Team A beat Team B beat Team C beat Team D..." methodology; the only difference is the number of steps it takes to get from one team to any other given team.

As mentioned above, this is not particularly useful, as Temple at some point is going to come out on top of Arkansas. The element Paul's added, though, is to compare the number of steps it takes for Team A to come out on top of Team Z to the number of steps it takes for Team Z to prevail over Team A; the mere existence of a "victory chain" from one team to another may say little, but determining the shorter chain of the two, it turns out, is pretty representative of the more impressive path. For example:

From Kenneth Massey's Game Graph Connections tool we find:

Temple 28 vs. Bowling Green 14
Bowling Green 24 vs. E. Michigan 21
E. Michigan 17 vs. Toledo 13
Toledo 37 vs. Kansas 31
Kansas 20 vs. Colorado 15
Colorado 30 vs. Texas Tech 6
Texas Tech 55 vs. Baylor 21
Baylor 17 vs. Kansas State 3
Kansas State 45 vs. Texas 42
Texas 28 vs. Oklahoma 10
Oklahoma 37 vs. Washington 20
Washington 29 vs. UCLA 19
UCLA 25 vs. Oregon State 7
Oregon State 33 vs. USC 31
USC 50 at Arkansas 14

Transitivity proves that Temple is better than Arkansas.
This 15 game conquering path predicts: Temple over Arkansas by 208 points.

But wait:

Arkansas 21 at Vanderbilt 19
Vanderbilt 43 vs. Temple 14

Transitivity proves that Arkansas is better than Temple.
This 2 game conquering path predicts: Arkansas over Temple by 31 points.

If you could look at every such chain (there were 42,523 of them last time I looked) you could come up with a way to rank all 119 teams, just as the computer ratings do (they are better at making thousands of comparisons than we are).

First, you'd want to a way to choose between chains like Temple>Vanderbilt and Vanderbilt>Temple, so we can say A>...>Z is stronger than Z>...>A if:

* the victory chain A>...>Z is shorter than Z>...>A, or
* the chains are the same length but there are more A>Z chains than Z>A chains

Meaning, Paul assures SMQ, the shorter of the two chains between two teams is the "stronger" chain.

So, something to look at: a neat graph Paul put together ranking teams according to the number of "stronger" chains compared to the rest of the country. To read it, first click here, then, since you won't understand that, try to think of it as a game of "Six Degrees of Kevin Bacon," where the goal between any two teams is to have the shortest number of degrees to victory (as Arkansas does above over Temple).

The number under "NT" on the far left represents teams "beaten" via a stronger chain, or fewer degrees - that is, the number of teams with a longer victory path back to the team in question than the team in question has to them. On the other side, NT1 is simply the number of straight-up wins for each given team (one degree). The number in the NT2 field represents the number of teams "defeated" through two degrees, NT3 the number of teams "defeated" on the third degree, etc. Using the specific example above, Temple would be represented in Arkansas' NT2 field, but Arkansas would not be represented in Temple's chart at all. So the numbers listed only represent teams with "weaker" (longer) chains back to the team in question, and that sum is on the far left.

Fine, just look at the chart and it will make sense. Once you're there, click on a team name to view the specific chains relative to every other team. Again, the "NT" field on the far left, representing the number of teams with weaker chains back to the team in question, is what's being measured.

Turns out, Kevin Bacon's just two steps removed from a win over New Mexico State

There's more useful data out there, but this is interesting for strength of schedule purposes - especially if you're a Rutgers fan.

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Comments

Nice explanation

In case it's not obvious, the individual teams' pages show all possible A>Z chains for team A (well, one example of the shortest path from A to Z, you can usually find more than one for degrees higher than 3), not just the "winning" ones counted in the chart. Every team that has a win chain at all has one to at least 109 teams, except UL-Monroe whose only 1-A win is against winless FIU, so you can't get to degree 2 for the Warhawks.

The 9 teams that all the others can't get to are those that are undefeated plus those who have only lost to one or more of the other 8. These are
{ Boise St, Louisville, Michigan, Notre Dame, Ohio State, Penn State, Rutgers, West Virginia, Wisconsin }.

The winner of Ohio State/Michigan will have win-chains to 114 teams, more if one of the other undefeateds loses.

Although the tool by itself isn't that useful (but it's a fun time-waster), watching changes in it can help understand the effects of a game on ranking systems that are based upon iterative processes or "infinite" WP, OWP, OOWP, OOOWP, ... formulas.

by JPK on Nov 15, 2006 9:56 PM EST 0 recs