There
are myriad college football rankings, and, as Scott has noted, the statistical
choices that go into each are inherently subjective. I am going to propose one more ranking to the
many that already exist, but as the below comparison demonstrates, most
existing rankings are currently converging toward what the ranking I describe
below already listed last week (Notre Dame 1, Alabama 2), providing a
compelling case in favor of my method. The
subjective choice about the relevant data to include is simple: none of
it. To create my rankings I use only
wins and losses, ignoring statistics such as margin of victory, total yards,
turnovers, etc. for two reasons. First, we are asking an enormous amount from
our data if we are generating predictions of probable team performance based
upon a limited number of observations (at this point, 10 or 11 games). Second, statistics used in existing rankings
are incomparable across games given the dramatically different contexts. If two teams both have 300 total yards of
offense, but one team played in the snow in Ann Arbor while the other played on
a sunny day in the Rose Bowl, we are essentially equating completely
incomparable numbers to generate relative rank.
Margin of victory is perhaps one of the more egregious variables, with
large numbers frequently a better indicator of poor sportsmanship, personal
grudges, or heated historical rivalries than a decisive indicator of
excellence.
By mid-season (week 7
this year) if we draw lines between teams that have played one another, every
team is connected by some degrees of separation to every other team. The below figure is the network of games
played between FBS teams following week 12.
An arrow pointing at a team signifies that team is the winner of the
game played.
As we expect, teams in
the same conference cluster together tightly given most games are
intra-conference. You can interpret
team’s proximity to one another as the degree to which the teams share a
similar schedule. Because teams are now
linked in some way to every other team, we can generate an ordering of the
quality of team wins based on how central they are in the web of wins in the
network. To determine how badly their losses are, we can do the same thing for
the web of losses. Think of it like
this: we’re playing a big game of six degrees of separation and trying to
figure out which team is Kevin Bacon, or the common denominator to which all
other teams in the FBS are connected.
The football team that reaches the most teams through their wins in the
fewest degrees of separation (and likewise the fewest teams through their
losses) is the highest rank team. As an
example, if Alabama plays 10 teams, but then those 10 teams lose all their
other games, Alabama is only connected to 10 teams through their network of wins. That’s not very good. If Alabama plays 10 teams, but those ten
teams defeat all their other opponents, Alabama is now connected by two degrees
of separation to 100 teams. That’s a lot
better.
We can add up all these
links between teams using a measure called “average reciprocal distance” (ARD),
a centrality measure in network analysis (the program Ucinet 6 by Borgatti,
Everett, and Freeman was used to generate the above illustration and the
following rankings). ARD measures how
far on average each team is from every other team in the network, which I
calculate separately for the paths of wins and losses. The higher the value, the more central or
connected a team is in the network, or the more teams it is connected to by
fewer links. For the network of wins,
this will translate to a higher rank, with a higher ARD value signifying
greater centrality. In the network of
losses, centrality to the network will result in a lower ranking. Because we can assume that the direct inverse
of a win is a loss, we simply subtract the centrality of a team to the FBS
network of wins by the centrality of a team to the FBS network of losses. For a full discussion of the underlying
concept, see Steve Borgatti’s research on the key player problem (https://sites.google.com/site/steveborgatti/research/publications).
Since we now know the
outcomes of the games played from this weekend, we’ll use last week’s rankings as
a comparison. The below table lists (1)
the ARD in the network of FBS wins, (2) the ARD in the network of FBS losses,
(3) the win ARD minus the loss ARD to generate (4) the Network Ranking. The following columns compare the Network
Ranking with the BCS, the AP Poll and Scott Albrecht’s (Hybrid) ranking for all
teams in the top 10 in at least one of the rankings
Rankings preceding the Week of November 11-17.
Team
|
ARD
Wins
|
ARD
Losses
|
Wins
– Losses
|
Network
Rank
|
BCS
|
AP
|
Albrecht
|
Notre
Dame
|
48.27
|
0
|
48.27
|
1
|
3
|
3
|
4
|
Alabama
|
45.56
|
2.58
|
42.98
|
2
|
4
|
4
|
3
|
Florida
|
44.81
|
2.28
|
42.53
|
3
|
6
|
7
|
5
|
Ohio
State
|
41.96
|
0
|
41.96
|
4
|
-
|
6
|
7
|
Oregon
|
41.53
|
0
|
41.53
|
5
|
2
|
1
|
1
|
LSU
|
43.29
|
3.33
|
39.96
|
6
|
7
|
8
|
13
|
Georgia
|
42.26
|
2.58
|
39.67
|
7
|
5
|
5
|
6
|
Kansas
State
|
39.36
|
0
|
39.36
|
8
|
1
|
2
|
2
|
Texas
A&M
|
41.36
|
3.33
|
38.02
|
9
|
8
|
9
|
8
|
South
Carolina
|
40.42
|
3.33
|
37.09
|
10
|
9
|
12
|
11
|
Oklahoma
|
34.19
|
2
|
32.19
|
12
|
12
|
13
|
9
|
Florida
State
|
34.75
|
23.74
|
11.01
|
32
|
10
|
10
|
10
|
First,
we see how the Network Ranking operates.
LSU, for example, is more central to the network of FBS wins than Ohio
State or Oregon going into the week, but its two losses, while not at all
strongly central to the network of FBS losses, drag it down to number 6 (by
comparison the ARD loss score for New Mexico State, the bottom ranked team, is
46.23).
Second,
while no one could have predicted the outcome of the Kansas State and Oregon
games, the Network Ranking is alone in suggesting that both teams were
over-ranked. Meaning, they had not yet
provided sufficient evidence demonstrating their centrality to the network of
FBS wins to merit being ranked over Notre Dame, Alabama, etc. We see a similar dissonance present with
Florida State further down the list. Prediction
(and predictive modeling inherent in most computer rankings) is exactly the
problem exposed by the Oregon and Kansas State collapses this weekend. Predicting based on such a limited number of
observations (or in practice as Kirk Herbstreit refers to it, the “look test”) will
result in correspondingly limited success.
Third,
as we would expect, the rankings are now converging. Every ranking this week has Notre Dame #1 and
Alabama #2, and the Network Ranking is no different. But, that’s what the network ranking had last
week! In other words convergence is happening, but
all other rankings are converging toward the Network Rankings.
The Network
Rankings conceptually aren’t doing anything new – they are built on the same goal of ranking teams based on wins and
losses. However, unlike alternatives,
the method underlying the Network
Rankings best corresponds with that goal, with the result being a ranking that actually
ranks based on wins and losses rather than predicted probabilities from
incomparable metrics. If we’re going to rank on wins and
losses, all we need is wins, losses, and some careful counting of links between
teams.
Assistant
Professor
International
Studies and Political Science
Virginia
Military Institute
rhameyjp@vmi.edu
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