Imagine that each team is a point and for each game played draw an arrow from the winner's point to the loser's. With 302 teams and thousands of games you wind up with something really, really complicated. This is the raw material for any advanced rating system, so it is worth some attention.

In *Too Early* Ratings I described a "second order winning percentage" that is partially based upon the
DGG. Partial because it considers only the *shortest* path(s) from one team to the other and defines "winning paths" based upon
*series* results. Now we're going to consider every game and **all** paths between teams.

Well, not literally *all* - you can always find more paths between teams that are longer than those most-recently counted. I chose
to stop counting at pathlength *n* if all the team-pairs that *can* be connected by an *A beat B beat …* chain
**are**. As of this writing that occurs for the last team pair at 10 "beats."

The 2182 edges of the graph (games) form about 40.5 *billion*^{†} paths of up to 10 "beats" long, so it's pretty much impossible
to visualize the entire graph, but there are lots of ways to look at parts of it. To begin with, we'll try to
see how a vertex (team) looks from the perspective of all teams and how the collection of teams looks from the perspective of a vertex.
We can summarize those several ways too.

Having counted the number of paths we just make 302 lists using #paths *A⇒B* − #paths *A⇐B* for each team A, listing all
teams B in ascending order. Treating these as "votes" we list the teams based upon the average ranking by all teams (actually, the "Borda
Count" which is a little better measure than the mean), noting along the way the rank for which at least 152 teams rank the team at
least that highly (the "Bucklin rank.")

Of more interest is the rank each team assigns to itself. Since A⇒A − A⇐A is zero, this tells us how many teams have a better position in the graph than the team, as viewed from the team's perspective.

The *Directed Games Graph Analysis* includes the (Borda) rank, Borda count, best, Bucklin, and worst ranks, the rank the team assigns
itself, and the count of #1 ranks, #2 ranks, and so on through #302 ranks. The team name links to a page for each team that
includes the list of teams voting the team each rank, along with how the team ranked all the teams.

The team's "ballot" shows the (*#paths A⇒B* − *#paths A⇐B*) value used to assign the ranking, along with the counts for
each component. Then the individual counts in each direction for **P**ath**L**engths from 1 to the radius of the graph are shown.

These are not *ratings* in the usual sense, but as the season progresses the comparisons between teams provide insight into how
advanced ratings like the ISR are affected by game results.

In memory of