College Football Transitivity Rankings & Awards - Week 9

It was a relatively calm week in the world of college football, but that doesn't mean all was peaceful in Transitive Land. This week, we see LSU rise, Minnesota tumble, and our Wildcats pull themselves into a three-way tie for fourth in the CTR. If you're new to the Transitivity Rankings, you'll want to read this introductory post to get the basics. For the rest of you, here's the quick glossary in case your memory needs jogging:

• Win path: A transitive argument for one team over another. For example, you might argue that Kansas State is better than Western Michigan this season since K-State beat Iowa State, who in turn beat Toledo, who has beaten Western Michigan.
• Path length: The number of games in a transitive argument. The path in the above example has length 3.
• Loss path: Exactly like a win path, but reversed.
• Win Transitivity Rankings (WTR): A ranking of FBS teams calculated exclusively from win paths. It roughly measures the strength of a team's wins.
• Loss Transitivity Rankings (LTR): A ranking of FBS teams calculated exclusively from loss paths. It roughly measures how bad a team's losses have been.
• Combined Transitivity Rankings (CTR): The star of these posts. It's calculated by averages a team's rankings in the WTR and LTR.

Now that everyone's (re)calibrated, let's take a look at this week's Combined Transitivity Rankings:

Combined Transitivity Rankings

Rank	Team	Change	Avg Rank	Win Rank	Loss Rank	Num Win Paths	Avg Win Path	Num Loss Paths	Avg Loss Path	FBS Wins	FBS Losses
1	Mississippi State	-	1.0	1	1	124	3.847	0	NaN	7	0
2	Auburn	1	3.0	2	4	123	3.537	1	1.000	6	1
3	LSU	11	5.0	3	7	121	3.810	2	1.000	6	2
4	Florida State	1	6.0	11	1	114	3.737	0	NaN	6	0
4	Kansas State	2	6.0	6	6	118	4.390	2	1.500	5	1
4	Mississippi	2	6.0	4	8	120	3.517	3	1.667	7	1
7	Alabama	3	7.0	5	9	119	3.882	4	2.250	7	1
8	Marshall	-	7.5	14	1	112	4.848	0	NaN	7	0
9	Notre Dame	2	8.0	12	4	112	3.884	1	1.000	6	1
10	TCU	2	9.0	7	11	117	4.325	9	3.667	5	1
11	Oklahoma	1	10.0	8	12	117	4.402	9	3.000	5	2
12	Baylor	1	11.0	10	12	117	4.504	9	3.000	5	1
13	Texas A&M	1	11.5	13	10	112	4.330	5	1.600	4	3
13	West Virginia	1	11.5	9	14	117	4.427	9	2.333	5	2
15	Louisiana Tech	2	15.5	16	15	112	5.839	10	3.300	5	2
15	Oklahoma State	1	15.5	15	16	112	4.902	11	2.545	4	3
17	Nebraska	3	19.5	21	18	111	3.811	115	9.139	6	1
18	Michigan State	-	20.0	19	21	111	3.694	115	8.200	6	1
19	Georgia	2	21.5	18	25	111	3.595	115	7.365	6	1
20	Oregon	2	22.5	17	28	111	3.477	115	7.261	6	1
21	UCLA	-	24.5	20	29	111	3.766	115	7.200	6	2
22	Clemson	1	26.0	32	20	111	4.279	115	8.226	5	2
23	Utah	6	31.0	26	36	111	4.126	115	6.548	5	1
24	Arizona	1	32.0	22	42	111	3.964	115	6.296	6	1
24	Georgia Tech	10	32.0	27	37	111	4.135	115	6.452	5	2
26	Arizona State	4	33.0	43	23	111	4.622	115	8.165	5	1
26	Colorado State	2	33.0	34	32	111	4.324	115	7.096	6	1
26	South Carolina	6	33.0	25	41	111	4.099	115	6.383	3	4
29	Duke	4	33.5	43	24	111	4.622	115	7.391	5	1
30	Miami (Fl)	5	36.0	33	39	111	4.288	115	6.409	4	3
31	Boise State	6	37.0	29	45	111	4.216	115	6.122	6	2
31	East Carolina	5	37.0	47	27	111	4.667	115	7.330	5	1
33	Iowa	6	38.0	24	52	111	4.054	115	5.904	4	2
34	Kentucky	3	39.0	53	25	111	4.739	115	7.365	4	3
35	Ohio State	2	40.0	23	57	111	4.045	115	5.748	6	1
35	Washington	4	40.0	49	31	111	4.703	115	7.104	4	3
37	California	4	41.0	49	33	111	4.703	115	6.957	3	4
38	Arkansas	7	43.5	66	21	111	5.180	115	8.200	3	4
39	Louisville	2	44.0	37	51	111	4.495	115	5.939	5	2
39	Minnesota	24	44.0	40	48	111	4.595	115	6.017	5	2
39	Wisconsin	28	44.0	38	50	111	4.586	115	5.957	4	2
42	Missouri	-	45.0	28	62	111	4.162	115	5.609	5	2
42	Nevada	3	45.0	41	49	111	4.604	115	6.009	3	3
44	Maryland	16	45.5	35	56	111	4.477	115	5.800	4	3
44	Stanford	3	45.5	48	43	111	4.685	115	6.217	4	3
46	Utah State	4	46.5	49	44	111	4.703	115	6.183	4	3
47	Iowa State	5	47.5	55	40	111	4.811	115	6.400	2	4
48	Florida	1	48.0	58	38	111	5.018	115	6.435	3	3
49	USC	4	51.0	36	66	111	4.486	115	5.322	5	3
50	Tennessee	1	52.0	74	30	111	5.514	115	7.113	2	5
51	Texas Tech	3	52.5	86	19	111	5.982	115	8.704	2	5
52	North Carolina	7	54.5	45	64	111	4.631	115	5.557	3	4
53	UTEP	3	56.0	95	17	111	6.468	115	9.478	4	3
53	Virginia Tech	1	56.0	30	82	111	4.252	115	4.783	3	4
55	Virginia	6	57.0	38	76	111	4.586	115	5.061	3	4
56	Rutgers	8	58.0	63	53	111	5.144	115	5.878	4	3
57	BYU	4	59.0	42	76	111	4.613	115	5.061	4	4
58	Northwestern	9	61.0	46	76	111	4.658	115	5.061	2	4
59	Penn State	3	61.5	54	69	111	4.766	115	5.243	4	3
60	Washington State	1	62.5	62	63	111	5.099	115	5.574	1	6
61	Air Force	5	63.5	56	71	111	4.955	115	5.226	4	2
61	UL Lafayette	54	63.5	92	35	111	6.369	115	6.870	3	3
63	Arkansas State	11	65.0	75	55	111	5.541	115	5.843	3	3
63	Illinois	30	65.0	57	73	111	4.982	115	5.157	3	4
65	Central Florida	-	66.0	67	65	111	5.198	115	5.391	4	2
66	Pittsburgh	8	66.5	30	103	111	4.252	115	4.148	3	4
67	Texas	3	68.0	75	61	111	5.541	115	5.617	3	5
68	Boston College	6	68.5	52	85	111	4.730	115	4.722	4	3
68	Michigan	6	68.5	77	60	111	5.550	115	5.626	3	5
70	Kansas	7	69.5	85	54	111	5.955	115	5.870	1	5
70	South Alabama	3	69.5	106	33	111	7.865	115	6.957	5	2
72	Indiana	3	70.0	60	80	111	5.072	115	4.852	2	4
73	Oregon State	1	70.5	94	47	111	6.414	115	6.035	3	3
73	Syracuse	13	70.5	83	58	111	5.901	115	5.722	2	5
75	Toledo	5	72.0	70	74	111	5.243	115	5.104	4	3
75	Wyoming	2	72.0	63	81	111	5.144	115	4.800	2	5
77	Rice	7	73.0	78	68	111	5.766	115	5.270	4	3
78	Northern Illinois	1	73.5	65	82	111	5.171	115	4.783	5	2
79	Memphis	4	75.5	82	69	111	5.883	115	5.243	3	3
80	Middle Tennessee State	13	76.5	81	72	111	5.856	115	5.217	4	3
81	Colorado	-	77.5	96	59	111	6.505	115	5.643	2	6
82	Georgia Southern	1	79.0	112	46	111	8.748	115	6.078	5	2
82	Purdue	13	79.0	69	89	111	5.225	115	4.626	2	5
84	Cincinnati	2	79.5	84	75	111	5.937	115	5.096	4	3
85	Western Michigan	2	81.0	72	90	111	5.306	115	4.565	4	3
86	Bowling Green	4	82.0	68	96	111	5.216	115	4.409	4	3
87	Central Michigan	1	82.5	58	107	111	5.018	115	4.035	4	4
88	North Carolina State	1	84.0	101	67	111	7.081	115	5.313	3	4
89	Akron	11	84.5	61	108	111	5.090	115	4.017	3	4
90	San Jose State	35	86.0	87	85	111	6.027	115	4.722	2	4
90	Western Kentucky	5	86.0	70	102	111	5.243	115	4.270	3	4
92	Hawaii	1	88.5	89	88	111	6.099	115	4.652	1	5
93	Florida Atlantic	4	90.0	79	101	111	5.811	115	4.339	3	5
94	UAB	-	90.5	88	93	111	6.045	115	4.470	3	4
95	Ball State	6	91.0	73	109	111	5.333	115	3.800	2	4
95	Houston	4	91.0	91	91	111	6.360	115	4.539	3	3
97	San Diego State	1	92.0	97	87	111	6.568	115	4.696	3	3
98	Ohio	1	92.5	79	106	111	5.811	115	4.078	3	5
99	Navy	10	95.0	93	97	111	6.378	115	4.400	3	4
100	UL Monroe	25	95.5	107	84	111	7.946	115	4.765	3	4
101	Vanderbilt	1	96.0	113	79	111	9.045	115	4.913	1	6
102	Old Dominion	11	97.0	99	95	111	6.649	115	4.461	2	5
103	Wake Forest	20	97.5	102	93	111	7.207	115	4.470	1	6
104	UTSA	4	100.0	100	100	111	6.892	115	4.374	2	6
105	Army	2	100.5	90	111	111	6.252	115	3.548	2	4
106	New Mexico	2	101.0	105	97	111	7.784	115	4.400	2	5
107	Florida International	5	101.5	98	105	111	6.640	115	4.087	2	4
108	Temple	2	103.0	115	91	111	9.703	115	4.539	3	3
109	Fresno State	4	104.0	104	104	111	7.261	115	4.139	2	5
110	Texas State	8	105.0	111	99	111	8.649	115	4.391	3	3
111	Kent State	2	109.0	103	115	111	7.225	115	3.296	1	7
112	Miami (Oh)	-	110.0	108	112	111	8.135	115	3.530	2	6
112	UNLV	4	110.0	110	110	111	8.243	115	3.670	1	6
114	Massachusetts	3	112.0	108	116	111	8.135	115	3.252	2	7
115	Buffalo	2	114.0	114	114	111	9.108	115	3.400	1	5
116	Eastern Michigan	2	114.5	116	113	111	10.081	115	3.452	1	6
117	Southern Miss	1	117.0	117	117	6	2.167	116	5.241	2	5
118	South Florida	1	118.0	118	118	3	1.333	116	4.716	2	5
119	Appalachian State	-	119.5	119	120	3	2.000	117	4.897	1	4
119	Idaho	3	119.5	120	119	2	1.500	116	3.879	1	6
121	Tulsa	2	121.0	120	122	2	1.500	117	3.897	1	6
122	North Texas	2	122.0	123	121	1	1.000	117	4.162	1	6
123	Troy	3	122.5	120	125	2	1.500	118	4.178	1	6
124	Tulane	1	123.5	123	124	1	1.000	118	4.381	1	5
125	SMU	1	124.5	126	123	0	NaN	118	4.390	0	7
126	New Mexico State	-	125.0	123	127	1	1.000	120	3.567	1	6
127	Connecticut	-	126.0	126	126	0	NaN	119	4.168	0	6
128	Georgia State	-	127.0	126	128	0	NaN	121	3.876	0	7

Some thoughts:

• LSU is the big winner of the week, jumping up 11 spots to number 3 following their upset victory over Ole Miss. You may balk at a two-loss team so high in the rankings, and I don't necessarily disagree. But their only losses have come to Mississippi State and Auburn (CTR numbers 1 and 2), and they own perhaps the best victory of the season so far.
• Minnesota was hurt badly by their loss to Illinois, dropping 24 spots from number 15 last week to this week's 39. This is actually pretty explainable when considering how the CTR works. More on this later.
• Our biggest jumper of the week is Louisiana-Lafayette, whose win over Arkansas State had more impact than you may have anticipated. The victory moved the Ragin' Cajuns up from 115 to 61, a 54 spot jump. We'll come back to them as well.
• The biggest drop was suffered by San Jose State, who fell 35 spots after a loss to Navy. The former Jekyll and Hyde Award winner is now much more Hyde, ranking 87 and 85 in the WTR and LTR, respectively.

CLUSTER WATCH

We're starting a new segment this week, which I'm calling the Cluster Watch. We're going to try to identify games that could have a large effect on the CTR, and along the way learn why the CTR has teams like Texas A&M and Oklahoma State ranked so highly.

Now, take another look at the rankings, and scroll somewhere towards the middle. See in the 7th and 9th columns, where every team displays the numbers 111 and 115? What's going on here? It's telling us that all of these teams have win paths to 111 different teams, and loss paths to 115 different teams. The teams in this group create what graph theorists call a strongly connected component. That is, choose any two teams in this group, and you will be able to find a win path from either team to the other. We'll call this group of teams the cluster.

Not every team is in the cluster, of course. The teams at the top of the rankings have more win paths and fewer loss paths than the cluster teams, and the opposite is true for teams at the bottom of the list. Then we can categorize every team into one of three groups: The cluster teams, the sub-cluster teams (teams with fewer win paths than the cluster teams), and the super-cluster teams (teams with more win paths than the cluster teams). In this weeks rankings, the super-cluster teams are the ones with rankings 1 through 15, while the sub-cluster teams are ranked 117 through 128.

A defining characteristic of the super-cluster teams is that they have beaten at least one cluster team, while no cluster team has beaten them. This means ultimately that every super-cluster team will always be ranked above the cluster teams. But, suppose a super-cluster team (call them Team A) loses to a cluster team (Team B). Then, not only does Team B earn a win path to Team A, but so does every other cluster team (since they all have paths to Team B, and Team B beat Team A). The result is that Team A (and any super-cluster team that has lost to Team A) is pulled into the cluster.

This phenomenon is responsible for a few different things. First, it tells us why Texas A&M and Oklahoma State are ranked so highly, even with three losses. Each of these losses came to a fellow super-cluster team, meaning they are safe from loss paths to cluster teams. Plus, they each own victories over every cluster team, so they also have more win paths than the cluster teams.

This also helps us understand why Minnesota fell so far. Their only loss prior to Saturday was to TCU, a fellow super-cluster team, so the Golden Gophers sat safely atop the cluster teams in the rankings. Once they lost to Illinois, they were thrust into the cluster where average path lengths are the only factor for separating teams. Their wins to date are decent but not excellent, so many stronger cluster teams jumped them in the rankings, explaining the large drop. Minnesota was the only super-cluster team to drop into the cluster this week.

A similar story explains Louisiana-Lafayette's rise. Last week they were the highest-ranked sub-cluster team. A victory over Arkansas State pushed them into the cluster, and above many of the lower-ranked cluster teams. Texas State similarly made the jump from sub-cluster to cluster this week. Incidentally, UL Lafayette beat the Bobcats earlier in the year, so the Cajuns would have left the sub-cluster with or without a victory on Saturday.

In the coming weeks, we'll use this space to track which teams have been absorbed into the cluster. We will also highlight games in the coming week whose results have a chance to relegate super-cluster teams to the cluster. This week, those games are:

• Mississippi State vs. Arkansas
• Florida State at Louisville
• Notre Dame at Navy
• Oklahoma at Iowa State
• Baylor vs. Kansas
• Texas A&M vs. Louisiana-Monroe
• Louisiana Tech vs. Western Kentucky

Of course, these games aren't played in a vacuum, and the results of one game can send multiple teams down to the cluster. The most extreme example: Should Mississippi State lose it's game with Arkansas, the super-cluster will shrink from the current 16 teams to three! This is because the Bulldogs own win paths to every SEC school in the cluster (directly to Auburn, LSU, and TAMU, to Mississippi through LSU, and to Alabama through Mississippi), as well as to each Big 12 school through Auburn's win over K-State.

AWARDS

I spent a little more time than I wanted on that last section, so let's hustle through these awards.

Jekyll and Hyde Award

Description:
For the team whose performances in the WTR and LTR are most divergent. This one's for the team which has experienced the highest highs and the lowest lows. Or perhaps the highest lows and the lowest highs.

Measure:
Maximum length shortest win path.

Winner:
UTEP

Notes:
The Miners take this award for the second week running. Their WTR of 95 and LTR of 17 make for a difference of 78 spots. This was just ahead of South Alabama's 73 spot difference between a WTR of 106 and LTR of 33.

Are We There Yet Award

Description:
Not all shortest paths are created equal. Sometimes, teams have to work very hard to claim a win path, meandering tirelessly among the season's results before the path is earned. This one goes to a pair of teams for whom the shortest win path between them is longer than all other shortest paths.

Measure:
Maximum length shortest win path.

Winners:
Temple, UTEP

Notes:
Both Eastern Michigan and Temple have shortest win paths to UTEP of length 16, longer than any other shortest paths this week. Temple's path takes the award by virtue of my questionably accurate tiebreaking system, which estimate's Temple's path at 7834 miles long, whereas Eastern Michigan's path was only 6669 miles. A representative path looks like this:

Temple > Vanderbilt > Massachusetts > Kent State > Army > Ball State > Central Michigan > Purdue > Western Michigan > Bowling Green > Indiana > Missouri > South Carolina > Georgia > Arkansas > Texas Tech > UTEP

Frequent Flyer Award

Description:
If you took all of one team's shortest win paths, and made them actual travel itineraries, for which team would you be making the most stops? To win this one, you need to have a lot of win paths, but you don't want them to be too short.

Measure:
Largest sum of shortest win path lengths.

Winner:
Eastern Michigan

Notes:
While the Eagles just missed out on the last award, they can rejoice in knowing that they've won the Frequent Flyer Award for the second week running. This week, their 111 paths total 1119 games. I'd be remiss if i didn't tie this back to our earlier discussion of cluster teams: we could probably redefine this award as going to the cluster team with the highest average win path length. In fact, just for fun, let's look at Eastern Michigan's win path diagram. We can see why their paths are so long, as we have to go seven layers down before getting to a level with more than two teams. Compare this with K-State's win path diagram (at the bottom of the article) to get a visual understanding of the difference between a good team and a not-so-good team, transitively speaking.

jeffp171/Bring on the Cats

Conference Atlas & Coattail-Rider Awards

Description:
There are two awards here, one for the team which does the most to carry its conference's reputation on its shoulders (Atlas), and another for the conference black sheep who most disgraces his comrades (Coattail-Rider).

Measure:
The team whose CTR is highest above/farthest below the average among teams in its conference.

Winner:
Marshall (Atlas), Vanderbilt (Coattails)

Notes:
The Atlas trophy is starting to get comfortable in Huntington, where it's resided for the last three weeks. The Coattails trophy is starting to gather cobwebs in Nashville, where it's been since the first award was given in week 4. In the breakdown table, we see that the SEC has extended its lead on the Big 12 for smallest average conference ranking.

Conference	Avg Conference Rank	Atlas	Atlas Rank	Atlas Difference	Coattail Hanger	Coattail Rank	Difference
AAC	97.91	East Carolina	31	66.91	Connecticut	127	29.09
ACC	50.43	Florida State	4	46.43	Wake Forest	103	52.57
Big 12	30.00	Kansas State	4	26.00	Kansas	70	40.00
Big Ten	48.79	Nebraska	17	31.79	Purdue	82	33.21
CUSA	81.69	Marshall	8	73.69	North Texas	122	40.31
MAC	97.00	Toledo	75	22.00	Eastern Michigan	116	19.00
MWC	73.92	Colorado State	26	47.92	UNLV	112	38.08
Pac-12	41.08	Oregon	20	21.08	Colorado	81	39.92
SEC	27.71	Mississippi State	1	26.71	Vanderbilt	101	73.29
Sun Belt	100.09	UL Lafayette	61	39.09	Georgia State	128	27.91

We Are Not Impressed Award

Description:
This award goes to the AP top 25 team whose reputation is most at odds with their transitive resume. Whatever human pollsters see in this team, the CTR is not buying the hype.

Measure:
The AP top 25 team whose ranking drops the most when compared to CTR.

Winner:
Ohio State

Notes:
The Buckeyes have been within range for this award for most of the season, and have finally broken through. Their CTR rank of 35 is 22 spots below their AP ranking of 13. Their closest competition here is Oregon, whose CTR rank of 20 is 15 spots off of their number 5 AP rank. I imagine Ohio State may well retain this award should they beat Illinois next week, but their game in two weeks against Michigan State may do a lot to bring their AP and CTR rankings in line.

Desserpmi Ton Era Ew Award

Description:
Like the We Are Not Impressed Award, but opposite. Indeed, the name of this award is "we are not impressed" backwards.

Measure:
The highest CTR among teams receiving no votes in the AP poll.

Winner:
Texas A&M

Notes:
I think we explained this one well enough in the Cluster Watch section. The number 13 Aggies take this award in front of fellow super-cluster member Louisiana Tech (15). The only other CTR top 25 team without a vote in the AP is Georgia Tech (24).

KANSAS STATE WIN PATHS

The Wildcats didn't pick up win paths to any new teams this week, but their average path length did improve from last week's 4.585 to this week's 4.390. Below, you will find the customary win path diagram, along with the handy depth guide. Have fun with it, and I'll see you next week.

jeffp171/Bring on the Cats

Level 1: Iowa State, Oklahoma, Texas, Texas Tech, UTEP

Level 2: Iowa, Kansas, Louisiana Tech, New Mexico, New Mexico State, North Texas, Old Dominion, Tennessee, Toledo, Tulsa, UTSA, West Virginia

Level 3: Arkansas State, Ball State, Baylor, Central Michigan, Eastern Michigan, Florida International, Georgia State, Houston, Indiana, Maryland, Massachusetts, Oklahoma State, Pittsburgh, Purdue, Rice, SMU, Southern Miss, Tulane, UL Lafayette, Utah State, Western Michigan

Level 4: Air Force, Akron, Appalachian State, Army, BYU, Boston College, Bowling Green, Buffalo, Connecticut, Florida Atlantic, Hawaii, Idaho, Illinois, Kent State, Memphis, Missouri, Northern Illinois, Ohio, South Florida, Syracuse, TCU, Temple, Texas State, UAB, UL Monroe, UNLV, Virginia Tech, Wake Forest

Level 5: Boise State, Central Florida, Cincinnati, Florida, Fresno State, Miami (Oh), Middle Tennessee State, Minnesota, Navy, North Carolina, North Carolina State, Northwestern, Ohio State, South Carolina, Troy, USC, Vanderbilt, Virginia, Western Kentucky, Wyoming

Level 6: Arizona, Colorado, Colorado State, East Carolina, Georgia, Georgia Southern, Georgia Tech, Kentucky, Louisville, Michigan, Nevada, Oregon State, Penn State, Rutgers, San Diego State, San Jose State, Stanford, Wisconsin

Level 7: Arkansas, California, Clemson, Miami (Fl), Oregon, South Alabama, Washington, Washington State

Level 8: Duke, Michigan State, UCLA, Utah

Level 9: Arizona State, Nebraska

No Path: Alabama, Auburn, Florida State, LSU, Marshall, Mississippi, Mississippi State, Notre Dame, Texas A&M