Serious Fun

© Copyright 2009, Paul Kislanko

In my last few essays I've analyzed the schedule graph using various tools. From a rating system perspective, though, we're more interested in the games graph. Instead of ab meaning "a plays b" rating systems depend upon "a played b" and the games graph changes with every completed game.
Date #Games CI% APL #Paths Paths/Pair #Paths/Game Diameter #Pairs at D
23-Sep 138 4.75 8.22 12258 1.70 88.83 19 3
30-Sep 189 8.47 4.67 15828 2.19 83.75 9 1
7-Oct 239 13.17 3.74 18279 2.53 76.48 7 1
14-Oct 290 18.49 3.28 22600 3.12 77.93 5 394
21-Oct 343 23.67 3.05 26266 3.63 76.58 5 91
28-Oct 399 28.74 2.88 29286 4.05 73.40 5 15
4-Nov 453 33.04 2.75 31251 4.31 68.99 5 1
11-Nov 506 36.39 2.66 31997 4.41 63.24 4 712
18-Nov 563 39.85 2.58 31316 4.31 55.62 4 478
25-Nov 614 42.58 2.54 31524 4.33 51.34 4 350
2-Dec 662 45.49 2.49 31543 4.33 47.65 4 255
9-Dec 674 46.23 2.48 31441 4.31 46.65 4 238
16-Dec 675 46.30 2.48 31549 4.32 46.74 4 237

is the last day of the "college football week" running Thursday-Wednesday; 23-Sep includes the first three weekends
is the number of games scheduled to be played through Date
is the percentage of all 7140 team-pairs that are connected by paths of length one or two (they have either played each other or at least one common opponent.)
is the Average Path Length between two teams
is the number of unique paths that are the shortest between any pair of teams
is the average number of unique paths that connect any two teams
is the average number of unique paths games contribute to
is the length of the longest path required to connect two teams
#Pairs at D
is the number of team-pairs for which "diameter"-length paths are required to connect the teams

There are "milestones" that have to be achieved for various categories of rating systems to provide meaningful results, and the serious part of this analysis has to do with finding those for the different categories. To begin, note that there are 7140 team-pairs xy in the FBS. Here I use ⇔ to mean the shortest connecting paths

where λ is distance(x,y). Also note that except for head-to-head (xy) there are more than one of these for each pair.

Advanced rating systems depend upon these connecting paths to form their ratings. After week 3 the average game's results are used to compare nearly 88 other pairs of teams - varying from 436 other pairs being ranked based upon the results of Army vs Eastern Michigan down to 6 games whose results can be used only to relate the pair represented by each team's single other opponent. In general, the fewer teams whose relative ratings depend upon a single game's outcome, the more "connected" the field and accurate an advanced system's results:


On the other hand, the more unique paths that connect each pair, the more data the advanced systems have to work with and again, the more accurate their results:



The table above begins after weekend three because until the field is connected almost no advanced rating can produce results based only upon this year's results. After the second weekend there are still four teams who've played no games against an FBS opponent (one FCS game and an off week.)
An interesting side note is that weekend three is the earliest the field has become connected since I began making these measurements. Despite the profusion of games vs FCS opponents, there's more inter-regional FBS games than in prior years.
In week 8 (games through October 28th): Different ratings may depend upon different features of the games graph, and at least some consider one or more of these significant. Last year, Jeff Sagarin's ratings became "unbiased" (by prior year results) when the games graph connected 25% of the field by no worse than an O-O relationship. This doesn't imply that's the criterion Jeff uses, but even if a coincidence this date is a "milestone."

In week 10 (games through November 11th):

It is at this point that scheduled games make the games graph "denser" by shortening xy paths without creating new paths - same-conference xy's become xy eliminating all the λ= 4 paths between x and y but adding shorter paths for their opponents who play one of but not both x and y (oxxyoy). After week 10 most advanced ratings are "stable", and more games just make their rating more precise.