2016 D1A Schedule Topology Part 2 - Round Robins

July 10, 2016

By the end of the season it will take at least four playeds in an A played B played ... chain to connect the last 291 of the 8,256 team-pairs. The 712 games ( A played B) will have turned into 1,899,566 paths with four playeds in them (plus an additional 69,444 paths four playeds long that begin and end at the same team.)

Technically, there will be 949,783 paths that can be traversed in either direction.

As you can tell from the path counts above, except for length one there are usually more than one "shortest path." For teams connected at length two, the number of paths is just the number of opponents the teams have in common. If you try to draw such a graph (try it!) you'll see you wind up with what looks like a solid mass of spaghetti. Instead of trying to take in the entire games graph at once I like to look at what I call its skeleton.

The general idea is to find the edges (games) that contribute the most to shortening the average path length. A bit of thought suggests that this will be the subset of games between teams that have no common opponents, then the ones that have just one common opponent, and so on. The reason is that each of the teams' opponents would have a pathlength > 2 to each of the teams were it not for the game, so the game contributes about 20 "length 2 shortest paths" (The sum of D1A opponents for each team not counting the other team in the pair. It is 20 if both teams play one game against a non-D1A opponent, 21 if one does, and 22 if neither does.)

To get to the "skeleton" of the games graph I begin by finding all subsets of { teams in D1A } in which every team plays every other team. There are 3,379 of these with the largest being the round-robin played by the Big 12 (10 teams.) The first step in finding the skeleton is to strip away the fat: 3,048 of those are just subsets of larger round-robins. Of the remaining 331, 161 consist of teams from the same conference - they only contribute to the extent that the constituent teams do.

That leaves us with 170 round-robins involving teams from more than one conference. These 255 games of the 712 form the skeleton of the games graph. The 132 games between teams with no common opponents form it's "spine." Start again with the 128 vertexes and connect just these edges to get a feel for how the schedule topology affects relative ratings within the (D1A) field.

Unique Interconference Round-Robins

All of the unique round-robins (including those due only to conference schedules) are available here - Unique Round-Robins.