Connecting the Dots

© Copyright 2006, Paul Kislanko

The Distance Matrix shows how well the field is connected, and which teams contribute the most to connecting the field. For many rating systems though, it matters just as much how many paths connect two teams as what the shortest path is.

Heading into Week 9 of the 2006 season, the Games Graph for Division 1-A looks like this:
• The graph diameter is 5 (13 pairs)
• 406 of 7021 team-pairs (5.78%) have played
• 1883 pairs (26.82%) have at least one common opponent
• Average # common opponents = 1.28
• 194 pairs (2.76%) pairs are Opponents with no common opponents

So, even after eight weeks, less than 30 percent of the field can be compared by head-to-head matchups or results versus common opponents. That should improve considerably in week nine, as we shall see.

The 194 team pairs that are opponents with no COs and the team-pairs that haven't played but have exactly one common opponent have contributed most to connecting the field thus far, but those numbers will shrink drastically as conference play progresses. For instance, Auburn and Arkansas have no common opponents, despite being in the same division of the SEC. Slightly less surprising is that Michigan and Michigan State only have Notre Dame in common.

Just as having too few connections by common opponents can make rating systems less "accurate", having too many common opponents causes troubles. Because connections (games) are used to connect teams that are already connected, they can't be used to connect that set to the rest of the field. (I think of these as "knots in the spaghetti.") The more common opponents there are, the better the relative measurement of those specific teams, but the fuzzier the measurement of either compared to the rest of the field.

What happens when there are common opponents who also play each other is that a team begins to contribute to its own rating. Most systems do a good job of keeping a team from being counted as its own opponent's opponent, but if they must look beyond O-Os to O-O-Os you have A plays B and C, and once B and C play, A becomes its own O-O-O (and in some systems such as the RPI, B and C also become O-O-Os - all three teams' winning percentages are counted in each team's OOWP.)

What will happen is that as more conference games cause more "knots" in the graph, every team involved in the knot will benefit from those teams that have the "one team no one else played" that strengthens the connections between knots, and shrinks the diameter of the graph so that fewer of those O-O-O-O... connections are required to compare any two teams. In Division 1-A football (and Division 1 in other sports) by the end of the year only about 25 percent of the field will have really fuzzy numbers.

But as I noted back in August, the ratings for 2006 will be far fuzzier than in the last five years. Just as playing only teams that all of your opponents play doesn't contribute to the ability to compare teams, so does playing games against teams outside the field. Games vs 1-AA teams are necessary and even desirable to support the game, but too many of them are problematical. At the end of the 2006 season for the first time since at least 2002, fewer than half of the team-pairs in Division 1-A will be connected by no worse than an O-O relationship.

I think 1-AA games should not count towards bowl-eligibility more often than they count towards BCS-bowl eligibility. If there is not to be a playoff (and I am not particularly for one) there must be a reliable way to rank teams who haven't played each other.

A secondary but significant effect in 2006 is the shape of the interconference schedule graph. The trend towards fewer non-confernece games (in the case of the Pac 10) and all conference teams scheduling OOC games against teams from the same conference (Big 10-MAC, SEC-CUSA, Pac 10-WAC) contributes to the "knottedness" of the graph. Forget how good it is for the fans, the 2002-2003 practice of scheduling inter-regional home-and-home games (see Auburn/Southern Cal) is necessary for the game. We need more of those, not fewer.