Sunday, August 9, 2009

Our manuscripts about college football

In addition to the rants on this collection of web pages, we have written a pair of scholarly, academic articles about ranking with random walkers. The now somewhat amusingly misnamed "Division I-A Football" article (it was properly named when it was submitted; and the Michigan Wolverines hadn't famously lost to Appalachian State Mountaineers, an FCS née I-AA program) discusses a number of issues in greater depth than covered on this website, including the community structure of the football matchups network and its influence on rankings, ideas about choosing a good p value, and the improved properties of the RWFL ranking system.

''Random Walker Ranking for NCAA Division I-A Football,''
T. Callaghan, P. J. Mucha and M. A. Porter,
American Mathematical Monthly, 114, 761-777 (2007)
[originally made available as].
Abstract: Each December, college football fans and pundits across America debate which two teams should meet in the NCAA Division I-A National Championship game. The Bowl Championship Series (BCS) standings employed to select the teams invited to this game are intended to provide an unequivocal #1 v. #2 game for the championship; however, this selection process has itself been highly controversial in four of the past six years. The computer algorithms that constitute one part of the BCS standings often act as lightning rods for the controversy, in part because they are inadequately explained to the public. We present an alternative algorithm that is simply explained yet remains effective at ranking the best teams. We define a ranking in terms of biased random walkers on the graph formed by the schedule of games played, with two teams (vertices) connected by an edge if they played each other. Each random walker moves from team to team by selecting a game and "voting" for its winner with probability p, tracing out a never-ending path motivated by the "my team beat your team" argument. We study the statistical properties of a collection of such walkers, relate the rankings to the community structure of the underlying network, and compare these rankings for recent NCAA Division I-A seasons. We also discuss the algorithm's asymptotic behavior, illustrated with some analytically tractable cases for round-robin tournaments, and discuss possible generalizations.

''The Bowl Championship Series: A Mathematical Review,''
T. Callaghan, P. J. Mucha and M. A. Porter,
Notices of the American Mathematical Society 51, 887-893 (2004).
Abstract: We discuss individual components of the college football Bowl Championship Series. Comparing with a simple algorithm defined by random walks on a biased graph, we attempt to predict whether the proposed changes will truly lead to increased BCS bowl access for non-BCS schools. We conclude by arguing that the true problem with the BCS Standings lies not in the computer rankings, but rather in misguided addition.