Graphs and College Football
https://myteamisbetterthanyourteam.com/
This website is an interesting recreational example of applying graph theory to the relatively silly application of arguments between college football fans about whose team is better. Sports fans often apply the “transitive property” of winning to argue that their team, Team X is better than another team (Y) if X beat a third team Z and that third team Z beat Y. This website takes this principle to its logical extreme. For instance, this chain of inference shows that Cornell Big Red was better than Alabama Crimson Tide in the 2017 college football season:
What this website really uses to store this information is a directed graph with the vertices corresponding to college teams, and the directed edges corresponding to vertices (while I couldn’t find direct justification, the extensive use of the word “path” at https://myteamisbetterthanyourteam.com/about appears to indicate this). The website’s interface makes it easy to calculate the paths between any two college football teams in this graph. An interesting ramification of the structure of college football leagues is that this graph is a set of tightly connected clusters corresponding to NCAA divisions, and each subcluster has clusters corresponding to conferences and leagues, and a few trials of mine justified this fact.
The website runs something called the TPNC (Transitive Property National Poll), which ranks college teams by the length of the shortest path through the NCAA Victory graph from the team to the number one college football team (the University of Alabama, in the 2017 season, where I got these results).The TPNC for the victory graph at the end of last year’s season is telling (https://myteamisbetterthanyourteam.com/tpnc;pollDate=01-09-2018). In the context of Networks, the fact that this graph has both a giant strongly connected component and a relatively small radius is interesting.
First of all, the furthest team from Alabama (Martin Luther College in Minnessota) is 37 hops away from Alabama. Even though Martin Luther College is a small D3 college that is not great at football, since the NCAA league victory graph has a giant connected component (which is why the TPNC is even a meaningful metric for finding “the worst” team), this means even fans of this small college’s football team can argue they could beat Crimson Tide some day. What’s more exciting, is that within the Division 1 FBS (the top level of NCAA football), one never has to go more than 10 hops to beat Alabama, and never has to go more than 17 hops to beat Alabama from within the Division 1 FCS (the division Cornell plays in).
This implies that this graph has a relatively small radius. The Division 1 FCS graph’s radius is no more than 20 (using the relatively simple bound of creating a path between any two teams through Alabama,) although it is likely closer to 10. This means that this who-beat-who graph is pretty tight for any active league, and still relatively tight, even between D3 and D1 teams.
A silly application like the who-beat-who graph of college football may not have huge ramifications for the social or economic lives of people, but this does have huge ramifications for people who are interested in making sports leagues fun and interesting to watch. Decreasing the radius and increasing the connectivity of the who-beats-who graph for a sports league ensures the fans always have thrilling matchups to watch and talk about, since there’s always a doubt as to which team is better than the other, just as the NCAA has done.
One more image, an ego graph of Alabama (generated by the website) and all the teams they cold have beaten in 5 hops (with Cornell in orange):
One last interesting thing to note is the way some matchups between colleges in different divisions of the NCAA that exist for various traditional reasons function as bridges in this graph between the “components” corresponding to leagues. The coloring of colleges corresponding to their NCAA division makes this evident.