The Ulitimate Strength of Schedule

© Copyright 2006, Paul Kislanko

Usually we've considered the games graph to be defined as teams A and B are "connected" if they play each other, or each played a team that the other played. But we could also treat the games graph as a directed graph. Instead of A↔B meaning teams A and B played, we only note A→B when A wins a game vs B.

We can't talk about the "connectivity" of the directed graph, because obviously any undefeated team cannot be "connected to" by any other team, and a winless team does not "connect to" any other team. There is an equivalent to the "diameter" of the games graph, though, and it's not "the longest path" between any two teams.

Define the diameter of the graph as "the largest number of steps required to connect every team to be all the teams that it can be connected to by a victory chain." Note that this will be lower than the longest chain, since step N can add chains longer than N.

• Step 1 is all the A→B pairs, where A's are all winners, length 1
• Step 2 adds the B→C pairs to the A→B giving A→B→C, giving paths of length 2 or 3
• Step 3 adds the C→D pairs plus any D→C′→B′→A′, giving paths of up to length 6
• step 4 can result in paths of length 12 (6 steps to get to a team added in step 3, plus up to 6 steps for its longest chain from step 3)
• ...generally, in every step from the third on, the longest possible chain is double that of the previous step

In general if there's an A→…→Z chain there's also a Z→…→A chain. You can always find a victory chain connecting two teams except:

• You can't find a path to an undefeated team, or to a team that's only lost to undefeated teams
  (except from the undefeated team(s) that beat them, of course)
• You can't find a path from a winless team, or from one whose only wins are against winless teams

Folly, or Method?

2006-10-28 @ Temple                  28    Bowling Green           14          
2006-10-14 @ Bowling Green           24    E Michigan              21          
2006-10-21 @ E Michigan              17    Toledo                  13          
2006-09-15 @ Toledo                  37    Kansas                  31          
2006-10-28 @ Kansas                  20    Colorado                15          
2006-10-14 @ Colorado                30    Texas Tech               6          
2006-11-04 @ Texas Tech              55    Baylor                  21          
2006-09-30 @ Baylor                  17    Kansas St                3          
2006-11-11 @ Kansas St               45    Texas                   42          
2006-10-07   Texas                   28    Oklahoma                10
2006-09-09 @ Oklahoma                37    Washington              20          
2006-09-23 @ Washington              29    UCLA                    19          
2006-11-11 @ UCLA                    25    Oregon St                7          
2006-10-28 @ Oregon St               33    USC                     31          
2006-09-02   USC                     50  @ Arkansas                14          
Transitivity proves that Temple is better than Arkansas.
This 15 game conquering path predicts: Temple over Arkansas by 208 points.

2006-09-16   Arkansas                21  @ Vanderbilt              19          
2006-09-30 @ Vanderbilt              43    Temple                  14          
Transitivity proves that Arkansas is better than Temple.
This 2 game conquering path predicts: Arkansas over Temple by 31 points.

If you could look at every such chain (there were 42,523 of them last time I looked) you could come up with a way to rank all 119 teams, just as the computer ratings do (they are better at making thousands of comparisons than we are.)

First, you'd want to a way to choose between chains like Temple→Vanderbilt and Vanderbilt→Temple, so we can say A→…→Z is stronger than Z→…→A if:

• the victory chain A→…→Z is shorter than Z→…→A or
• the chains are the same length but there are more A→Z chains than Z→A chains
  If the chains are the same length and there are an equal number of chains of that length in each direction, we count A→…→Z and also Z→…→A.

That cuts the number of chains down considerably, but you could still have more than 10,000 to turn into a ranking. The next step is to find a way to summarize all that data by team.

Summarizing Victory Chains

NT

is the Number of other Teams to which the team has a stronger (or equal strength) win-chain to than have a stonger (or equal) win chain to them

Span

is the largest number of →s required to connect the team to the NT other teams to which it has a win-chain

APL

is the Average Path Length, defined as in Calculating the Distance Matrix, lower numbers are better
NTi (where i is { 1, 2,…Span }) is the number of teams for which the team has a victory chain of i →s

Current values for Division 1-A is an index to details for the win chains from every team to every other team.

The "current values" table is ordered by WeightedWins, which combines the number and length of the shortest win chains between teams. For a pair of teams A and B, if the win path from A to B is shorter than or equal to that from B to A, the contribution of B to A's weighted wins is:

#PathsA→B	-	#PathsB→A
2PL(A→B)-1		2PL(B→A)-1

So if A's shortest win path to B is 2, each such path is worth 1/2, if it's A→C→D→B each is worth 1/4, and so on. Naturally, a head to head win is worth 1.