Subbing for Schedules

July 24, 2015

It's often noted that the Adjacency Matrix for college football is "sparse" - there are only 1421 games between Division 1 teams out of 31,878 team-pairs. That's about 4.5 per cent of the team-pairs that play a game. The denominator comes from the formula ½ × 253 × (253 - 1) where 253 is the number of Division 1 teams.

There are some analyses that make 1421 out of 31,878 look positively huge. Suppose instead of looking at the number of games in a category you want to look at the number of teams from different categories that play each other. Now we are looking at the number of subsets of the set of 253 teams, and there are 2²⁵³ of those. That's a 77 digit number:

14474011154664524427946373126085988481658748083205070504932198000989141204992

254 of those are uninteresting - the null set and the 253 subsets consisting of only one team. But if you start with the 1421 subsets consisting of

{ x,y | x,y ∈ D1 ∧ x plays y }
(In English, x and y are two D1 teams who play each other in 2015)

Well, your computer can. Not only are computers good at Boolean Algebra, if you look closely enough that is the only thing they do.

In everything that follows we will assume that {x | x ∈ D1 } is the universal set, which I'll refer to as D1. So there is an implied whatever ∈ D1 no matter what symbol I use for whatever. Also D1-{ some subset } means the complement of the subset: all teams that are not members.

First, we can form the 253 subsets that consist of each team's opponents:

Opp(x) = { y | y plays x }

CommonOpps(x,y) = CommonOps(y,x) = { z | z ∈ Opp(x) ∧ z ∈ Opp(y) }
= Opp(x) ∩ Opp(y)
(In English, the list of teams that played both team x and team y.)

More Fun With Subsets

For example, if X is a conference, then the subset of teams that are members of the conference is

Conf(X) = { x | x ∈ X }

This makes it simple to find the number of teams from conference A who play teams from conference B. Well, simple for computers. First we find for all conferenxes Opp(X), which is:

Opp(X) = { Opp(x₁) ∪ Opp(x₂) ∪ Opp(x₃) ... }

NC(X) = { D1-Conf(X) } ∩ Opp(X)

Now for each pair of conferences X,Y the subset of teams from conference X who play teams in conference Y is just

Conf(X) ∩ NC(Y)

NC Matchup Team Counts

Linking Teams

{ NC(X) - Conf(Y) } ∩ { NC(Y) - Conf(X) }

Here's the list for all the D1A conference pairs, showing the linking-team's conference in parntheses. 1AA teams are bold and their conference affiliation italicized.

Without showing any formulas we can look at the subset of all linking teams. 19 of them are 1AA teams and 126 are D-1A. Two D-1A teams have all of their D-1A non-conference opponents against teams from the same other conference. It should not be a surprise that those are from conferences who play only three non-conference games. Also it is no surprise at all that the teams that show up most often as linking teams are the D1A independents (every one of their games is a non-confernece game.)

Conference-Pair Link Counts by Linking Team