Pathcounts and Weights by PathLength
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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
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.⌈ x ⌉ is the smallest integer ≥ x.
ω(λ) = κ(1-λ) where κ = ⌈ ρ-1√max { wpρ(A,B) } ⌉
Pathcounts and Weights by PathLength
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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.
Pathcounts and Weights by PathLength
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After the November 1st games, the results are WWP week 10.
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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:
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© Copyright 2014, Paul Kislanko