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 a↔b 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 x⇔y in the FBS. Here I use ⇔ to mean the shortest connecting paths↔a1↔a2↔...aλ-1↔ where λ is distance(x,y). Also note that except for head-to-head (x↔y) 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."
- more than a quarter of all team-pairs are related by no worse than an Opponent's opponent relationship
- the average path length becomes ≤ 3
- the average number of unique paths per pair becomes ≥ 4
In week 10 (games through November 11th):
It is at this point that scheduled games make the games graph "denser" by shortening x⇔y paths without creating new paths - same-conference x⇔y's become x↔y 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 (ox↔x↔y↔oy). After week 10 most advanced ratings are "stable", and more games just make their rating more precise.
- the diameter of the games graph becomes as small as it will ever be
- the average number of paths connecting team-pairs is as large as it will ever be