Today we find ourselves in mid-October, with six full weeks of college football behind us. It's about this point in the season that fans start to really get a feel for where their team stands in the broader landscape. It's also about this point in the season that my Transitivity Rankings have seen enough results to start being interesting.
As you may or may not remember from last year, the Transitivity Rankings are my attempt to rank teams using only win/loss results and the transitive property (the argument that says if team A beat team B, and team B beat team C, then team A must be better than team C). The main idea is that teams are rewarded for beating strong opponents, and penalized for losing to weak ones.
More precisely, for every team we count the number of "transitive wins" they have this season, counting FBS opponents only. For example, Kansas State has beaten UTSA and Louisiana Tech this season, accounting for two transitive wins. Moreover, UTSA has beaten UTEP, and Louisiana Tech has beaten FIU and UL-Lafayette, adding three more to K-State's transitive win total. As new teams are added to the list, we continue to add these teams' wins to K-State's total, until there are no more teams to add.
Once we've counted each teams' transitive wins, we sort teams from highest to lowest. When there are ties, we break them by looking at the length of the shortest argument for each win. For example, Kansas State has a "win path" to FIU that is two games long, since K-State beat Louisiana Tech, who then beat FIU. Shorter win paths are clearly preferable to longer ones, so ties go to the team with the shorter average path length.
That's just half of the story, though. We also rank teams based on their transitive losses and loss path lengths, in an analogous way. This time, fewer transitive losses are preferred to more, and longer paths are preferred to shorter paths. By this point, each team has a ranking for both wins and losses, which are averaged to determine the final rank. What does it look like when applied to this season's results? Let's take a look (the table is sortable by clicking on column headers).
The top of the rankings is dominated by undefeated teams, which makes sense. Our top team, Utah, owns transitive wins over 89 teams, more than two-thirds of the country. The top ten coincides fairly well with this week's AP top ten, though the Transitivity Rankings are higher on Cal and Toledo than the voters are, and less impressed by AP number one Ohio State (I guess some things never change).
Kansas State is not fairing too well at the moment. The Wildcats have no impressive wins, evidenced by their paltry 79 ranking by wins only. But they don't have any bad losses either, and stand fairly high in the loss rankings at 29. Put them together, and the Cats are sitting at 53 in the combined rankings. Maybe a tad lower than I would like, but probably about right to an outsider.
Interactive Path Finder
Now I get to unveil my favorite new feature of the post, the interactive path finder! Did you come to this post wondering, I don't know, what's the fastest transitive argument for picking Clemson over Boston College this week? Simply scroll down to the map below, select Clemson for the winning team, and select Boston College for the losing team. Not only does the shortest possible path appear (a nine-game doozy including Notre Dame, Texas, Oklahoma, West Virginia, Maryland, South Florida, Syracuse, and Wake Forest), but the path is also helpfully (?) laid out on a map of the US.
Great, now let's try something else. How about Boise State to Colorado? There's a nice, simple path there, from Boise to Hawaii to Colorado. But something sure feels weird about calling that the "shortest" path, right? Hawaii, while beautiful, feels a bit out of the way. No worries, just change the "criteria" dropdown box from fewest games to shortest distance. In this mode, we still only consider paths flowing between winners and losers of individual games, but now we take the actual distance between schools into account. What we end up with still isn't the most efficient route from Boise to Boulder, but at least you don't need to fly to the middle of the Pacific to complete it.
Finally, there is a "Path Type" toggle that lets you choose whether to find a path of wins from your chosen team to an opponent, or a path of losses. Additionally, when the first team is chosen, a breakdown of who they can and cannot beat, and how quickly, is provided. Or a list of who can beat them, depending on your selected mode.
That should be about all the info you need to start playing around with it. This feature is very much still in beta, so if you find any bugs (either technical glitches or missing games results) let me know in the comments.