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

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.

**NT**- is the
**N**umber of other**T**eams 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
**A**verage**P**ath**L**ength, 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.

#Paths_{A→B} | - | #Paths_{B→A} |

2^{PL(A→B)-1} | 2^{PL(B→A)-1} |