Mathematicians examine the BCS? [repost]
The figure to the right represents the expected distribution of model random-walker votes cast for each NCAA Division I-A football [known as the "Football Bowl Subdivision"] team in 2001 pre-bowl-games rankings. The organization of the teams and the lines connecting them represent the community structure hierarchy, of which the conferences are one level of organization. The colors represent the expected percentage of votes cast per team at each level in the hierarchy, from individual teams up through intra-conference organization, the conferences, and the connections between conferences. The biased probability for voting for the winners of games in the data represented in this figure is p = 0.65. Details are interspersed throughout these pages.
This work grew out of a Research Experiences for Undergraduates (REU) project in Summer 2003 by Georgia Tech undergraduate Thomas Callaghan, in collaboration with postdoctoral visiting assistant professor Mason Porter and assistant professor Peter Mucha. This work was funded (to pay Thomas' summer salary) by an NSF VIGRE "vertical integration" grant, justified by the enrichment of Thomas' educational experiences and by its true vertical integration spirit of joint work between an undergrad, postdoc, and professor. Later support for Thomas was also provided by the Georgia Tech President's Undergraduate Research Award (PURA). After graduating from GT, Thomas went on to a Ph.D. program in Computational and Mathematical Engineering at Stanford.
At the outset, we want to make three things very clear:
(1) We have NOTHING to do with the official Bowl Championship Series (BCS) standings.
(2) Volumes have been written by many mathematically- and statistically-inclined football fans who have developed a multitude of different ways of ranking college football teams (see David Wilson's excellent site).
(3) We don't claim that the system described here is "better"; rather, our approach was to ask if the most naive ranking system we could dream up would do a reasonable job. So we envisioned a collection of random walkers, which you might prefer to think of as monkeys.
[This is a partially-edited repost of material from the original, pre-blog, random walker rankings site. As such, the listed post date is approximate.]