### Directed Games Graph -based Rating

#### November 1, 2014

In Metrics on the Directed Games Graph I defined a number of ways to represent team position in the directed games graph. Two of those - Second Order Winning Percentage and Weighted Wins, result in a linear ordering of teams. Those are based the subset of the DGG that uses the number of shortest win-chains A⇒B and B⇒A for all pairs where either or both exist. The linear orderings are based upon the count of other teams at a given distance and the number of paths at those distances.

The graph is more complicated than that. In those metrics an A⇒B chain that has one chain 2 →s long beats the B⇒A chain that's 3 →s long even if there are 5 of those. To capture the complexity I defined a metric that depends upon the longest of the shortest paths between any team-pair for all team-pairs:

 F(A,B) = ρ ∑ [ ( wpλ(A,B) − wpλ(B,A) ) × ω(λ) ] λ=1

I used F(teamA,teamB) to generate 128 different rankings of the 128 teams, and reported four different consolidations derived from the 128 based upon vote-counting methods. In response to the article Kenneth Massey asked:

Is the "Weighted Wins" the same as the F(A,B) function you describe? If so, I'd like to include that ranking on my composite page.
The answer was "no." The Weighted Wins function uses a different value of λ for A⇒B and B⇒A in it's calculation, while F(A,B) uses all paths of lengths 1 through the graph "radius" for both the F(A,B) and F(B,A) calculations. If the function Weighted Wins uses is called G, it is not true that G(A,B) = -G(B,A) and G(A,A) is not defined.

There are many ways to derive an F()-based linear ranking. Define the Weighted Win Paths metric as:

 WWP(A) = ∑ F(A,t ∈ {teams}) {teams}

Such a WWP rating is problematical in that the definition of F is not "fixed" in the sense that the ω(λ) factor is not necessarily the same for every instance of the games graph. In the article I wrote

• The ω(λ) factor keeps the large number of long-length paths from counting more than a win. Currently ω(λ) = 21-λ. The largest number of 20-"→" paths counts about one 60th win with this definition.
But with one more weekend's games, the radius dropped from 20 to 16 and the total number of paths at the radius length went from thousands to millions and the same ω(λ) that gave a pathlength-20 1/60th the value of a win gives a pathlength of 16 over 100 × the value of a head-to-head win. The increase in number of paths from week to week compared to radius isn't exponential, it is very much greater than exponential.

In my measurements of the directed games graph, ω() was essentially an ad-hoc adjustment. For a WWP rating, I would want it to be well-defined, even if there might be a different ω() for each instance of the games graph. So for a fixed definition of the exponential decay I will use

 ω(λ) = κ(1-λ) where κ = ⌈  ρ-1√max { wpρ(A,B) } ⌉
⌈ x ⌉ is the smallest integer ≥ x.
For example, after 444 games, the radius ρ is 16, and the largest number of paths with 16 edges is Mississippi State ⇒ Kent State: there are 166054. The 15th root of 166054 is about 2.23, and the next integer greater than or equal to that is 3. So instead of a 1, ½, ¼ ... sequence we use a 1, ⅓, 1/9... sequence of weights. Notice how even though more than half of all paths are at the maximum length, this weighting gives the total of those the least contribution.

#### Pathcounts and Weights by PathLength

 PathLength #Paths #SameEnds Weight SumWeighted 1 444 0 1 444 2 1179 0 3 393 3 2882 21 9 320.2222 4 6760 60 27 250.3704 5 15525 110 81 191.6667 6 35247 135 243 145.0494 7 79200 280 729 108.6420 8 176836 612 2187 80.85780 9 393845 1605 6561 60.02820 10 875756 3320 19683 44.49301 11 1945184 6710 59049 32.94186 12 4316128 14463 177147 24.36467 13 9575686 32357 531441 18.01834 14 21251741 73472 1594323 13.32963 15 47178658 160736 4782969 9.863885 16 104737193 350004 14348907 7.299315 ∑ 190592264 643885 2144.147332

#### Weighted Win Paths

A WWP ranking would look like WWP through games of 31 October.

My experience with linearizations of the DGG metrics is that they can change very dramatically until and unless there are winpaths from every team with at least one win to every team with at least one loss. That may never happen - for instance even an undefeated team may have no win-paths to a team whose only loss(es) is(are) to undefeated teams or teams whose only losses are to undefeated teams.

For instance, as of this writing there are 2,763 paths that are "missing" from the DGG. So the WPP is a "fuzzier" metric than most of the advanced systems that are more sophisticated. In the spirit of ratings analysis, I'll publish it for the rest of the season.

#### Update: Week 10 games effect on the DGG

There were 50 1A↔1A games on November 1st, and the radius of the DGG dropped from 16 to 14. There were about 10 million fewer total paths 1-14 edges long, but more paths 14 edges long than there were paths 16 edges long before the additional edges were added. The largest number of 14-edge paths is now Auburn⇒Kent State at 132946, so κ remained at 3. You can compare the graph above to its new version to get a feel for how the graph contracts as more games are played.

#### Pathcounts and Weights by PathLength

 PathLength #Paths #SameEnds Weight Tot 1 494 0 1 494 2 1461 0 3 487 3 4005 54 9 445 4 10496 100 27 388.741 5 26910 205 81 332.222 6 68289 384 243 281.025 7 172028 749 729 235.978 8 432803 1876 2187 197.898 9 1087166 4797 6561 165.701 10 2728692 11535 19683 138.632 11 6849099 26631 59049 115.990 12 17192688 65764 177147 97.0532 13 43165571 165945 531441 81.2236 14 108393626 412615 1594323 67.9872 ∑ 180133328 690655 3528.45

After the November 1st games, the results are WWP week 10.

#### Update: Week 11 games effect on the DGG

The 48 1A vs 1A games in week 11 had a major impact, with the radius expanding to 15 and the total number of paths increasing by nearly an order of magnitude.

 PathLength #Paths #SameEnds Weight Tot 1 542 0 1 542 2 1761 0 3 587 3 5343 78 9 593.6667 4 15520 156 27 574.8148 5 43906 345 81 542.0494 6 122501 720 243 504.1193 7 338461 1393 729 464.2812 8 931693 3764 2187 426.0142 9 2558142 10311 6561 389.9012 10 7011584 27135 19683 356.2254 11 19201361 69333 59049 325.1767 12 52562605 185076 177147 296.7174 13 143870968 503256 531441 270.7186 14 393778763 1366967 1594323 246.9881 15 1077761481 3713253 4782969 225.3332 ∑ 1698204631 5881787 6345.0062

Although not every team with at least one win can be connected to every team with at least one loss, that will become possible as soon as an SEC West team besides Arkansas loses to a non-SEC West team (or to Arkansas.) The pseudo-Smith Set is down to eight:

• The three undefeated teams { Mississippi State, Florida State and Marshall }
• The five SEC West teams that only Mississippi State has win-paths to (besides themselves) { Alabama, Auburn, Mississippi, LSU and Texas A&M }

#### Affects of Week 12 games

Sure enough, all of the SEC teams who played teams outside the set lost, so the only "missing" paths are now those teams whose only wins are over winless teams.
 PathLength #Paths #SameEnds Weight Tot Max 1 592 0 1 592 0 2 2141 0 4 535.25 4 3 7268 135 16 454.25 12 4 23913 308 64 373.64 31 5 77187 680 256 301.51 77 6 246165 1821 1024 240.40 212 7 778598 4718 4096 190.09 698 8 2452838 13316 16384 149.71 2323 9 7707835 39348 65536 117.61 7369 10 24177361 116690 262144 92.229 23412 11 75747361 343442 1048576 72.238 74484 12 237134727 1021661 4194304 56.537 233992 13 742003130 3097289 16777216 44.227 731599 14 2320773923 9443000 67108864 34.582 2300892 ∑ 3411133039 14082408 3254.3