Fifty+ Years of Game Graphs

© Copyright 2012, Paul Kislanko

My interest in the mathematics of sports ratings began with Boyd Nation's article on the Distance Matrix. Since I read that some years after it was published, I have used metrics based upon the "incidence matrix" to analyze the data available to computer rating systems.

Kenneth Massey shares my concern (or more appropriately I share his) that the trend toward fewer interconference games and more FCS non-conference opponents makes it more difficult to compare FBS teams that haven't played each other. Dr. Massey made available to me gamescores for all "big college" games from 1960 through 2011. I used Dr. Massey's historical gamescore data to calculate two of what I believe to be important measures of the games graph. %O+OO is the percentage of the field that is "connected" by no worse than an Opponent's Opponent relationship (higher is better - ideally this should be greater than 50%) and APL is Average Path Length between any two teams (lower is better, ideally it would be 2.)

connectivity% average path length

It is not hard to explain why the field was so weakly connected in the 1960s and 1970s:

We see improvements in both metrics in the 1980s because
  1. The number of games allowed was increased to 11
  2. The sport became profitable enough to pay for cross-country travel
  3. The field became smaller with the split into "1A" and "1AA" subdivisions
Of course, 3 significantly contributed to 2.

What was that?

FBS connectivity reached its peak in 2002 and 2003 for one simple reason: due to calendrical coincidence, there were 12 regular-season games those years. For a 117-team field, that provided enough scheduling opportunities to connect more than 50% of the 6786 team-pairs. We went back to 11 games for the 2004 and 2005 seasons, and that's not enough.

For the 2006 season and beyond 12-game schedules were instituted on a permanent basis (we'd like to think because they noticed what we analysts did, but we analysts suspect it was more for the extra money.)

So why did the connectivity metric only rebound to the 1990s level? Or for that matter why for the intervening two 11-game years did it drop back to the 10-game 1980s level? Before I answer that question, it is profitable to look at not just how many steps along the path it takes to connect the field, but how many different paths by which each team-pair is connected.

Click on the picture for a larger version with a legend. The main thing to notice is that the number of team pairs connected by less than a tenth the number of paths between the most-connected pair is much larger for 2011 than 2002.

Simple. At about the same time, the rules were changed to allow a win over a non-FBS opponent to count towards bowl-eligibility every year instead of once every rolling four-year period. In 2002, about a quarter of FBS teams had an FCS opponent, in 2011 over two thirds did.

A 12-game season is about the right number of games to connect a field that includes ~120 teams, but not if most teams don't play 12 games against the field.