Note: In the text below the term "Division 1A" should be understood to mean "Division 1 Bowl Subdivision" and "Division 1AA" to mean "Division 1 Championship Subdivision" in light of the NCAA's new nomenclature. I am going to use the old terms until they announce the new name for the 1AAA classification.
There are a couple of changes for 2006 that are worth noting in the pre-season. The 12-game regular season returns, but not quite in the same shape as the 2002-03 experiment.
From an analyst's standpoint, 12 games is probably the right number for a 119-team field. When 12 games were played in 2002 and 2003, more than half of all possible team-pairs were related by no worse than an opponents' opponent relationship, and 66 (of 117) teams were so related to over half the field. With an 11-game schedule since then there's only been around 41 percent of all team pairs as closely related as an opponent's opponent.
That's important, because the more "connected" the schedule is, the more accurately we'll be able to measure relative team performance.
The minor one has to do with the Pac 10. The conference decided to use the 12th game to play a 9th conference game. Fans applaud the resulting round-robin conference schedule (no possibility of ties for the championship) but with only three non-conference games the teams contribute no more to the field's connectivity than teams that play non division 1A opponents.
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And that's the major change. With the permanent 12-game schedule teams can now count wins against 1-AA teams every year for bowl-eligibility purposes (as opposed to once evey four years). In 2003 58 teams played fewer than 11 division 1A opponents, and in 2006 73 will play fewer than 12. (Four more play a division 1AA opponent but also play an exempt 13th game.)
The extra games against division 1AA teams have the same effect as playing fewer games. In the table at the left, CI is the number of (division 1A) teams that are opponents or opponents' opponents. APL is Average Path Length. For any one team, opponents are 1 unit away, opponents' opponents who are not also opponents are 2 units, and their opponents who are not already counted are 3, and so on. APL for a given team is just the average of what the value is for its relationship to all other teams, and what we listed is the median APL for all teams.
For our purposes, the medians are probably more important than the averages, since there are always a few teams whose schedules are so atypical that they skew averages.
For the field, besides the average and median path length between any two teams, we care about the maximum. It turns out for NCAA division 1 sports the maximum pathlength between teams is 4 for baseball, basketball and division 1A football. This means no pair of teams is "farther" apart than an opponent's opponent's opponent's opponent. It turns out that's not the case for all division 1 football when 1-AA is included.
There are 7021 pairs in a 119-team field, and in 2006:Part of how each team "fits" within the field can be determined from the distance matrix, which lists the number of other teams at each pathlength. For each team we can count the number of teams at each distance and calculate the average distance. We'd expect that teams that played more 1-A opponents would be more connected, but in the 2006 matrix the teams with the most opponents' opponents have 11 opponents and the team that is least connected has 12.
677 are teams that play each other - length 1 2570 are teams that don't play each other but have at least one common opponent - length 2 3461 are three units apart (opponents' opponents' opponents) and 313 are four units apart.
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It's worth a diversion to compare division 1A football to the other major division 1 sports. There are more teams and games in basketball and baseball, so we'll look at the percentage of total team-pairs that are opponents, opponents' opponents, etc.
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Division 1AA is in even worse shape, because quite a few teams play more than one non-division one team. All 1A teams are connected to all other 1A teams by no worse than a path length of 4, but only 14 1AA teams are connected to all other 1AA teams with two or fewer intermediaries.
Since 73 of 119 1A teams play at least one of 60 (of 122 total) 1AA teams, it might be reasonable to try to connect the whole field using all of division one. The chart at the right shows, though, that this just makes things worse - the cases where the presence of a 1AA team provides a shorter path between two 1A teams don't make up for the additional pairs to be connected.
(All of division one is about twice as large as division 1A in terms of number of teams, but the number of team pairs goes up as the square of the number of teams. For twice as many teams, there are four times as many pairs to be connected. As a result, the percentage of division 1 team pairs that are opponents is less than half that for division 1A.)
We can do a little better considering only the 60 1AA teams that have at least one 1A team as an opponent with the 119 division 1A teams. This is labeled 1A* in the table, and the graph includes games between 1AA teams with 1A opponents and all games involving 1A opponents. The graph is more "connected" than either 1AA graph or the division 1 graph, but still not as connected as the 1A graph or the graphs for other division one sports.
"T#Gms" is the minimum number of games (actually opponents) required for it to be possible for all teams to be related to each other by no worse than an opponent's opponent relationship (i.e. be connected by a path of length 1 or 2). It is possible for D-1A or D-1AA to be so connected in an 11 or 12 game season. Division 1 overall can't be so connected no matter how the teams schedule. (Note that it just possible for 1AA to be connected, it is not at all likely. In order for 11 game-seasons to connect all 122 teams, there must be no common opponents among any teams' opponents, which is impossible if there are conferences.)
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I may look again at the distance matrix and its implications for rating systems and possible playoffs as the season progresses.