Got Math? Part Two: The Consequences

In an earlier post, we explored the remarkable similarities between our RWFL rankings posted on Kenneth Massey's comparisons page, using our selected p=0.75 bias value, and Eugene Potemkin's E-Ratings. Through completely independent rationalizations, we ended up at equivalent linear algebra problems that we each solved to reach our rankings.

Well, this didn't seem to be adding much value to the comparisons, so Kenneth nicely asked me if we would do something to make sure ours were unique. So we're going to tweak our algorithm used to bring you weekly rankings, though we're going to do so in a logically consistent way. From now on, we're going to bring you the RWFL rankings as obtained by running the algorithm on the full set of 716 connected college football teams that include the FBS (that is, including all the FCS and DivII schools that play against the FBS, and all the schools who play them, etc.), and we'll report the ordered results from the FBS. This isn't actually "new" per se for us, as these are the rankings we've been using for our bowl predictions the past two years, because we think in principle they should be better. We just didn't want to spring a change without a compelling reason; needing to do something distinct from the E-Ratings is certainly a good enough reason.

If you decide you liked the old RWFL run on the FBS plus a single made-up catch-all non-FBS team, don't worry: you can still see those as the E-Ratings in Massey's comparisons. Indeed, comparing and contrasting the two should be interesting, in that the difference is all because of the treatment of the non-FBS teams, emphasizing the follow-on indirect effects present in the rankings.

An interesting part about this switch has to do with the only other change we've ever made in our rankings. Back in the original days of the Random Walker rankings, when all of us involved were all still at Georgia Tech, our "RW" rankings were just the linear algebra problem described in our manuscripts (which you can reach quickly from the sidebar), describing walkers with first-place votes. As noted at the end of our American Mathematical Monthly paper, there were a lot of reasons to expect improvement using this along with a second set of walkers, with last-place votes. For years, we've simply subtracted these second vote counts from the first to give the RWFL rankings ("Random Walkers First-Last").

But on the whole connected network of 716 teams, very little total weight of those last-place votes ends up in the FBS at all, so the rankings of the FBS teams are only very slightly modified by the last-place piece. One might argue that it would be more interesting to look at ratios instead of differences between the first-place and last-place votes, but that's not something we're going to do without some mathematical and computational investigation first.

Without further ado, as a means of comparison, let's back up to the beginning of the week (not just so we can relive the Carolina victory over Virginia Tech). The rankings listed below with the full connected set of teams definitely differs in some places from the old, simpler setting.

2009 Random Walker Rankings (RWFL, p=0.75)
Games through Saturday October 24th:
1. Iowa (8-0) [1.5817]
2. Florida (7-0) [1.5259]
3. Alabama (8-0) [1.5150]
4. Boise St (7-0) [1.1477]
5. TCU (7-0) [1.1042]
6. Southern Cal (6-1) [1.0839]
7. Oregon (6-1) [1.0780]
8. Texas (7-0) [1.0615]
9. LSU (6-1) [1.0351]
10. Georgia Tech (7-1) [1.0196]
11. Cincinnati (7-0) [0.9606]
12. Virginia Tech (5-2) [0.8658]
13. Arizona (5-2) [0.8354]
14. Penn State (7-1) [0.7979]
15. Miami FL (5-2) [0.7636]
16. Notre Dame (5-2) [0.7396]
17. South Carolina (6-2) [0.7216]
18. Pittsburgh (7-1) [0.7082]
19. Houston (6-1) [0.7080]
20. Oklahoma St (6-1) [0.6849]
21. Ohio State (6-2) [0.6686]
22. West Virginia (6-1) [0.6671]
23. Wisconsin (5-2) [0.6599]
24. California (5-2) [0.6102]
25. Utah (6-1) [0.5987]
26. Clemson (4-3) [0.5983]
27. Washington (3-5) [0.5971]
28. Georgia (4-3) [0.5816]
29. Kentucky (4-3) [0.5785]
30. Central Michigan (7-1) [0.5483]
31. Stanford (5-3) [0.5317]
32. Auburn (5-3) [0.5246]
33. Mississippi (5-2) [0.5227]
34. Michigan (5-3) [0.5090]
35. Oregon St (4-3) [0.5085]
36. Brigham Young (6-2) [0.4973]
37. Arkansas (3-4) [0.4728]
38. Kansas (5-2) [0.4617]
39. Navy (6-2) [0.4476]
40. Tennessee (3-4) [0.4473]
41. Boston College (5-3) [0.4472]
42. Troy (5-2) [0.4423]
43. Idaho (6-2) [0.4359]
44. Michigan St (4-4) [0.4309]
45. South Florida (5-2) [0.4255]
46. Oklahoma (4-3) [0.4157]
47. UCLA (3-4) [0.4119]
48. Arizona St (4-3) [0.4097]
49. Fresno St (4-3) [0.4068]
50. Iowa St (5-3) [0.3971]
51. Minnesota (4-4) [0.3863]
52. Nebraska (4-3) [0.3834]
53. Florida St (3-4) [0.3770]
54. Texas Tech (5-3) [0.3711]
55. Kansas St (5-3) [0.3708]
56. Marshall (5-3) [0.3688]
57. Missouri (4-3) [0.3631]
58. Virginia (3-4) [0.3422]
59. Wake Forest (4-4) [0.3391]
60. Rutgers (5-2) [0.3358]
61. Temple (5-2) [0.3357]
62. Mississippi St (3-5) [0.3356]
63. UTEP (3-4) [0.3356]
64. Nevada (4-3) [0.3286]
65. North Carolina (4-3) [0.3278]
66. Connecticut (4-3) [0.3232]
67. Purdue (3-5) [0.3231]
68. Louisiana-Monroe (4-3) [0.3146]
69. Duke (4-3) [0.3062]
70. SMU (3-4) [0.2991]
71. Texas A&M (4-3) [0.2976]
72. East Carolina (4-3) [0.2973]
73. North Carolina St (3-4) [0.2907]
74. Louisiana-Lafayette (4-3) [0.2842]
75. Colorado St (3-5) [0.2837]
76. Northern Illinois (4-3) [0.2814]
77. Southern Miss (5-3) [0.2748]
78. Wyoming (4-3) [0.2716]
79. Ohio U. (5-3) [0.2709]
80. Colorado (2-5) [0.2703]
81. Northwestern (5-3) [0.2684]
82. Air Force (4-4) [0.2677]
83. Bowling Green (3-5) [0.2592]
84. Syracuse (3-4) [0.2571]
85. Middle Tennessee St (4-3) [0.2457]
86. Toledo (4-4) [0.2452]
87. Western Michigan (4-4) [0.2379]
88. Indiana (4-4) [0.2353]
89. Tulsa (4-3) [0.2344]
90. Washington St (1-6) [0.2296]
91. Baylor (3-4) [0.2288]
92. Louisville (2-5) [0.2238]
93. Central Florida (4-3) [0.2203]
94. San Jose St (1-5) [0.2175]
95. San Diego St (3-4) [0.2165]
96. Arkansas St (2-4) [0.2080]
97. Buffalo (3-5) [0.2024]
98. Florida Atlantic (2-4) [0.2010]
99. Maryland (2-6) [0.1958]
100. Hawai`i (2-5) [0.1819]
101. Kent St (4-4) [0.1814]
102. UNLV (3-5) [0.1725]
103. Louisiana Tech (3-4) [0.1615]
104. Alabama-Birmingham (2-5) [0.1571]
105. Tulane (2-5) [0.1545]
106. Vanderbilt (2-6) [0.1516]
107. Utah St (2-5) [0.1453]
108. New Mexico St (3-5) [0.1423]
109. Memphis (2-5) [0.1413]
110. Illinois (1-6) [0.1366]
111. North Texas (1-6) [0.1264]
112. Florida Int'l (1-6) [0.1256]
113. Army (3-5) [0.1185]
114. Miami OH (0-8) [0.1078]
115. Akron (1-6) [0.1038]
116. Ball St (1-7) [0.0379]
117. Rice (0-8) [0.0368]
118. New Mexico (0-7) [0.0185]
119. Western Kentucky (0-7) [0.0139]
120. Eastern Michigan (0-7) [0.0083]
Conference Rankings (Average Per Team):
SEC 0.7010
Pac10 0.6296
Big10 0.5452
ACC 0.4894
BigEast 0.4877
Big12 0.4422
FBSInd 0.4353
MWC 0.3812
WAC 0.3519
CUSA 0.2690
SunBelt 0.2180
MAC 0.2169
Non-FBS -0.0867

Rankings through October 24th

Many of the top teams won easily yesterday, while both Iowa and Alabama maintained their undefeated records through end-of-game heroics. But to these rankings, a win is a win, period.

Comparing with last week's RWFL ranking, there has been significant shakeup in the ordering of the teams immediately following the top 4. GT and VT both fall a few spots because of Miami's loss to Clemson (the intertwined nature of these three teams was discussed last week). In contrast, USC jumped a number of spots, presumably due to similar secondary effects from victories obtained by teams they previously beat (e.g., Ohio State and Notre Dame). We note the very close net vote percentages in square brackets, from #5 USC [1.6882] down to #9 Boise State [1.6228], with the difference between #5 and #7 in the fourth digit after the decimal point (under rounding). That's really close, so future secondary effects could continue to shake up these rankings, while we await more losses among the top 10 to hopefully decide things.

2009 Random Walker Rankings (RWFL, p=0.75)
Games through Saturday October 24th:
1. Florida (7-0) [2.5831]
2. Iowa (8-0) [2.5206]
3. Alabama (8-0) [2.4836]
4. Texas (7-0) [1.8001]
5. Southern Cal (6-1) [1.6882]
6. TCU (7-0) [1.6881]
7. Oregon (6-1) [1.6879]
8. LSU (6-1) [1.6597]
9. Boise St (7-0) [1.6228]
10. Cincinnati (7-0) [1.5428]
11. Georgia Tech (7-1) [1.5167]
12. Virginia Tech (5-2) [1.2926]
13. Arizona (5-2) [1.0958]
14. Notre Dame (5-2) [1.0319]
15. Miami FL (5-2) [0.9884]
16. Penn State (7-1) [0.9727]
17. Pittsburgh (7-1) [0.9532]
18. Houston (6-1) [0.9350]
19. South Carolina (6-2) [0.8567]
20. Oklahoma St (6-1) [0.8559]
21. Wisconsin (5-2) [0.8480]
22. Ohio State (6-2) [0.8447]
23. Utah (6-1) [0.8052]
24. West Virginia (6-1) [0.7966]
25. California (5-2) [0.6695]
26. Kentucky (4-3) [0.6415]
27. Washington (3-5) [0.6179]
28. Georgia (4-3) [0.6150]
29. Clemson (4-3) [0.5506]
30. Brigham Young (6-2) [0.5400]
31. Oregon St (4-3) [0.5221]
32. Stanford (5-3) [0.5134]
33. Mississippi (5-2) [0.4995]
34. Central Michigan (7-1) [0.4918]
35. Auburn (5-3) [0.4483]
36. Kansas (5-2) [0.3811]
37. Idaho (6-2) [0.3730]
38. Boston College (5-3) [0.3716]
39. Oklahoma (4-3) [0.3697]
40. Michigan (5-3) [0.3167]
41. Arkansas (3-4) [0.2987]
42. Navy (6-2) [0.2916]
43. Troy (5-2) [0.2619]
44. Arizona St (4-3) [0.2591]
45. UCLA (3-4) [0.2513]
46. Minnesota (4-4) [0.2418]
47. Iowa St (5-3) [0.2333]
48. Nebraska (4-3) [0.2251]
49. Michigan St (4-4) [0.2172]
50. South Florida (5-2) [0.2126]
51. Missouri (4-3) [0.1572]
52. Tennessee (3-4) [0.1469]
53. Fresno St (4-3) [0.1418]
54. Rutgers (5-2) [0.1200]
55. Nevada (4-3) [0.0854]
56. Connecticut (4-3) [0.0675]
57. Texas Tech (5-3) [0.0539]
58. Florida St (3-4) [0.0486]
59. Louisiana-Monroe (4-3) [0.0448]
60. Marshall (5-3) [0.0350]
61. Mississippi St (3-5) [0.0117]
62. Kansas St (5-3) [0.0113]
63. North Carolina (4-3) [-0.0818]
64. Wake Forest (4-4) [-0.0973]
65. Texas A&M (4-3) [-0.1199]
66. Air Force (4-4) [-0.1264]
67. Purdue (3-5) [-0.1306]
68. Northern Illinois (4-3) [-0.1443]
69. Colorado St (3-5) [-0.1773]
70. Southern Miss (5-3) [-0.1884]
71. East Carolina (4-3) [-0.2288]
72. UTEP (3-4) [-0.2325]
73. Baylor (3-4) [-0.2581]
74. Wyoming (4-3) [-0.2635]
75. Louisiana-Lafayette (4-3) [-0.2675]
76. Northwestern (5-3) [-0.2768]
77. Syracuse (3-4) [-0.2855]
78. North Carolina St (3-4) [-0.3074]
79. Louisville (2-5) [-0.3100]
80. SMU (3-4) [-0.3102]
81. San Diego St (3-4) [-0.3315]
82. Duke (4-3) [-0.3404]
83. Central Florida (4-3) [-0.3514]
84. Middle Tennessee St (4-3) [-0.3598]
85. Ohio U. (5-3) [-0.3613]
86. Virginia (3-4) [-0.3724]
87. Colorado (2-5) [-0.3881]
88. Tulsa (4-3) [-0.3909]
89. Temple (5-2) [-0.4241]
90. Western Michigan (4-4) [-0.4333]
91. Indiana (4-4) [-0.4431]
92. Bowling Green (3-5) [-0.4488]
93. Arkansas St (2-4) [-0.4608]
94. Toledo (4-4) [-0.4756]
95. Florida Atlantic (2-4) [-0.5493]
96. UNLV (3-5) [-0.5680]
97. Louisiana Tech (3-4) [-0.5888]
98. San Jose St (1-5) [-0.6078]
99. Kent St (4-4) [-0.6440]
100. Washington St (1-6) [-0.6731]
101. Buffalo (3-5) [-0.6979]
102. Hawai`i (2-5) [-0.7916]
103. Utah St (2-5) [-0.7968]
104. Tulane (2-5) [-0.8196]
105. Alabama-Birmingham (2-5) [-0.8352]
106. Memphis (2-5) [-0.8717]
107. New Mexico St (3-5) [-0.8827]
108. Maryland (2-6) [-0.9288]
109. Vanderbilt (2-6) [-0.9366]
110. Illinois (1-6) [-1.0920]
111. Army (3-5) [-1.1913]
112. Florida Int'l (1-6) [-1.3427]
113. North Texas (1-6) [-1.4819]
114. Akron (1-6) [-1.4836]
115. FCS teams (XXX-XXX) [-1.6343]
116. Miami OH (0-8) [-1.7733]
117. Rice (0-8) [-2.3124]
118. New Mexico (0-7) [-2.3608]
119. Ball St (1-7) [-2.7930]
120. Western Kentucky (0-7) [-2.8963]
121. Eastern Michigan (0-7) [-3.4677]
Conference Rankings (Average Per Team):
SEC 0.7757
Pac10 0.6632
BigEast 0.3871
Big10 0.3654
Big12 0.2768
ACC 0.2200
FBSInd 0.0441
MWC -0.0882
WAC -0.1605
CUSA -0.4643
SunBelt -0.7835
MAC -0.9735
Non-FBS -1.6343

Got Math?

On a day full of exciting action, including a last-second blocked FG attempt that may turn out to have serious BCS implications, it may seem rather pedestrian to ask a math question. Then again, that's essentially what we do here. So while we're watching the rest of the games, I have a question, brought to my attention by another football ranking fan, Martien Maas.

Martien Maas' Rating System also appears on Kenneth Massey's College Football Ranking Comparison page. Perhaps in part because we ended up very close to each other in the comparisons this week, Martien noted that the RWFL rank order this week is precisely the same as that from Eugene Potemkin's E-Rating System (see also his more detailed discussion). Indeed, the two are nearly the same every week (except for some examples from last year, including here, here, and here). And there are clearly some philosophical similarities between the two rankings. But I haven't sat down to try to work out whether we're mathematically equivalent, so I'd be happy if someone could tell me if they have an expert opinion here. My gut instinct is that our p=0.75 bias value choice happens to set our rankings to the same linear algebra problem, with perhaps the small differences in the past due to details about how non-FBS teams are handled. But, like I said, I haven't looked at it sufficiently yet. Nevertheless, I thought it was worth mentioning...

----

Addition (October 25): Of course, while I tried to leave this puzzle for others, I couldn't let it go myself. I can never resist a good puzzle. It's probably a good thing that I get to solve puzzles for a living. Plus I received an email from Eugene Potemkin responding to a query I sent him directly.

Eugene and I had a wonderfully pleasant exchange of emails back and forth today, wherein he shared some of the details of his E-Rating implementation for college football, adding further mathematical details, including: (1) Where he uses ratios of "ratings" and "anti-ratings" to obtain scores in other sports, he uses a difference for American college football (this is the same as the "First-minus-Last" part in RWFL). (2) Like us, he usually treats the collection of all non-FBS teams as effectively one team. (3) To get around the singular nature of random walks on the fully directed graph---sorry for the lingo here but be thankful I'm not using it to launch into an entire discussion of how this relates to the original PageRank algorithm!---he doesn't treat a win as a full win; rather he equates a win as effectively 3 wins and 1 loss. This is perfectly identical to the "bias value" p=0.75 choice that we've espoused here, which is nice for a variety of reasons. So it appears that the minor differences must be small round-off or tie-breaking differences, and the RWFL(p=0.75) and E-Ratings are completely identical.

Again, a huge thanks to both Martien and Eugene. It's been nice emailing with both of them.

Going forward, we still have value to add, don't worry. For instance, we should spend a lot more time in future posts looking at the plots I post every week that show the top rankings across different choices of this infamous "bias value" p, because those plots hold a lot of utility in being a proxy for various kinds of ranking choices.

Rankings through October 17th

Some quick, probably insufficiently thought out comments about the new rankings... It seems fairly typical (in an unscientifically sampled way) to see algorithmic rankings start to make more sense here in the middle part of the season, as there is more information available and, in particular, as the number of undefeateds dwindles. The big end-of-season controversies usually don't start to make themselves clearer until later, because there are so many games left to play with so many different possible outcomes between now and then. Still, if you want, you can definitely start to guess at possible controversies to come, if the game outcomes align certain ways, particularly as more of the remaining undefeateds eventually lose.

Looking at the one-loss teams in the top 10 or top 12, along with two-loss Virginia Tech, one could certainly quibble over ordering; but each team's appearance there seems reasonable enough at this stage. LSU lost to Florida. Oregon's only loss was to Boise State (and helps to make the Broncos look subsequently better). But how does two-loss VT stay ranked so high? It's all about who they lost to, and who those teams lost to. Taken as a 3-team unit, ignoring their games against each other, the GT-VT-Miami triangle have only one loss: VT's loss to Alabama. The other three losses on their combined schedules are the three times one of them beat another. So in the "but my team beat your team" arguments, there are a lot of victories drawing some votes towards these teams, only the one loss to Alabama draining them away, and a lot of votes cycling around the triangle made up of these three teams.

And just in case you think there's an ACC bias here (there isn't), take a look at the conference rankings at the bottom of this post (told you).

2009 Random Walker Rankings (RWFL, p=0.75)
Games through Saturday October 17th:
1. Florida (6-0) [2.7889]
2. Alabama (7-0) [2.5812]
3. Iowa (7-0) [2.3490]
4. Texas (6-0) [1.9274]
5. Cincinnati (6-0) [1.7367]
6. Boise St (6-0) [1.7106]
7. LSU (5-1) [1.6075]
8. Georgia Tech (6-1) [1.5798]
9. Virginia Tech (5-2) [1.4777]
10. TCU (6-0) [1.4662]
11. Miami FL (5-1) [1.4501]
12. Oregon (5-1) [1.3937]
13. Southern Cal (5-1) [1.3706]
14. Arizona (4-2) [0.9424]
15. Houston (5-1) [0.8916]
16. Pittsburgh (6-1) [0.8087]
17. Wisconsin (5-2) [0.7842]
18. South Carolina (5-2) [0.7816]
19. Notre Dame (4-2) [0.7693]
20. Oklahoma St (5-1) [0.7482]
21. Penn State (6-1) [0.7235]
22. West Virginia (5-1) [0.7054]
23. Ohio State (5-2) [0.6803]
24. Georgia (4-3) [0.6789]
25. Washington (3-4) [0.6633]
26. Oregon St (4-2) [0.6617]
27. Kansas (5-1) [0.6232]
28. Utah (5-1) [0.6163]
29. Idaho (6-1) [0.6050]
30. California (4-2) [0.5798]
31. Brigham Young (6-1) [0.5688]
32. Arizona St (4-2) [0.5329]
33. Kentucky (3-3) [0.5328]
34. Auburn (5-2) [0.5222]
35. Arkansas (3-3) [0.5151]
36. Nebraska (4-2) [0.5019]
37. Boston College (5-2) [0.4647]
38. Michigan (5-2) [0.4579]
39. Central Michigan (6-1) [0.4220]
40. South Florida (5-1) [0.3967]
41. Stanford (4-3) [0.3564]
42. Texas Tech (5-2) [0.3335]
43. Troy (4-2) [0.2902]
44. UCLA (3-3) [0.2691]
45. Minnesota (4-3) [0.2675]
46. Mississippi (4-2) [0.2595]
47. Connecticut (4-2) [0.2587]
48. Clemson (3-3) [0.2447]
49. Tennessee (3-3) [0.2421]
50. Oklahoma (3-3) [0.2386]
51. Louisiana-Monroe (4-2) [0.2129]
52. North Carolina (4-2) [0.2005]
53. Michigan St (4-3) [0.1703]
54. Missouri (4-2) [0.1654]
55. Navy (5-2) [0.1533]
56. Rutgers (4-2) [0.1225]
57. Fresno St (3-3) [0.0799]
58. Louisiana-Lafayette (4-2) [0.0288]
59. Iowa St (4-3) [0.0141]
60. Wake Forest (4-3) [0.0076]
61. Colorado St (3-4) [0.0069]
62. Marshall (4-3) [-0.0084]
63. Mississippi St (3-4) [-0.0512]
64. Ohio U. (5-2) [-0.0572]
65. Florida St (2-4) [-0.0702]
66. Colorado (2-4) [-0.1013]
67. Air Force (4-3) [-0.1065]
68. Kansas St (4-3) [-0.1351]
69. Tulsa (4-2) [-0.1379]
70. Northern Illinois (3-3) [-0.1729]
71. East Carolina (4-3) [-0.1880]
72. Nevada (3-3) [-0.1923]
73. Baylor (3-3) [-0.2273]
74. Wyoming (4-3) [-0.2399]
75. Indiana (4-3) [-0.2487]
76. Southern Miss (4-3) [-0.2644]
77. SMU (3-3) [-0.2732]
78. Purdue (2-5) [-0.2974]
79. Central Florida (3-3) [-0.3155]
80. Virginia (3-3) [-0.3174]
81. Toledo (4-3) [-0.3289]
82. North Carolina St (3-4) [-0.3388]
83. Louisville (2-4) [-0.3594]
84. Middle Tennessee St (3-3) [-0.3620]
85. Louisiana Tech (3-3) [-0.3830]
86. Syracuse (2-4) [-0.3991]
87. Bowling Green (3-4) [-0.4010]
88. Texas A&M (3-3) [-0.4198]
89. UTEP (2-4) [-0.4266]
90. Northwestern (4-3) [-0.4289]
91. Duke (3-3) [-0.4764]
92. San Diego St (2-4) [-0.5287]
93. Western Michigan (3-4) [-0.5348]
94. Buffalo (3-4) [-0.5741]
95. Arkansas St (1-4) [-0.5934]
96. UNLV (2-5) [-0.6491]
97. Tulane (2-4) [-0.6502]
98. Washington St (1-5) [-0.6666]
99. San Jose St (1-5) [-0.6679]
100. Temple (4-2) [-0.6850]
101. Maryland (2-5) [-0.7816]
102. Hawai`i (2-4) [-0.8064]
103. Alabama-Birmingham (2-4) [-0.8168]
104. Kent St (3-4) [-0.8451]
105. Florida Atlantic (1-4) [-0.8665]
106. New Mexico St (3-4) [-0.8706]
107. Memphis (2-5) [-0.9384]
108. Illinois (1-5) [-0.9454]
109. Vanderbilt (2-5) [-0.9860]
110. Florida Int'l (1-5) [-1.0750]
111. Army (3-4) [-1.2312]
112. Akron (1-5) [-1.2534]
113. Utah St (1-5) [-1.2864]
114. North Texas (1-5) [-1.5233]
115. FCS teams (XXX-XXX) [-1.6214]
116. Miami OH (0-7) [-1.8190]
117. New Mexico (0-6) [-2.2315]
118. Rice (0-7) [-2.2360]
119. Eastern Michigan (0-6) [-2.7336]
120. Western Kentucky (0-6) [-2.7543]
121. Ball St (0-7) [-3.6397]
Conference Rankings (Average Per Team):
SEC 0.7894
Pac10 0.6103
BigEast 0.4088
Big10 0.3193
Big12 0.3057
ACC 0.2867
FBSInd -0.1029
MWC -0.1219
WAC -0.2012
CUSA -0.4470
SunBelt -0.7381
MAC -0.9710
Non-FBS -1.6214

Rankings through October 10th

2009 Random Walker Rankings (RWFL, p=0.75)
Games through Saturday October 10th:
1. Alabama (6-0) [2.8485]
2. Florida (5-0) [2.6827]
3. Virginia Tech (5-1) [2.2702]
4. Iowa (6-0) [2.0390]
5. Boise St (5-0) [1.6920]
6. LSU (5-1) [1.6258]
7. Cincinnati (5-0) [1.5538]
8. Texas (5-0) [1.3964]
9. Miami FL (4-1) [1.3367]
10. Oregon (5-1) [1.3116]
11. Kansas (5-0) [1.2830]
12. TCU (5-0) [1.2791]
13. Southern Cal (4-1) [1.2467]
14. Nebraska (4-1) [1.1471]
15. Ohio State (5-1) [1.1221]
16. Notre Dame (4-1) [1.0609]
17. Wisconsin (5-1) [1.0124]
18. Washington (3-3) [0.9582]
19. Georgia Tech (5-1) [0.9528]
20. South Carolina (5-1) [0.8006]
21. Idaho (5-1) [0.7461]
22. Auburn (5-1) [0.7218]
23. Stanford (4-2) [0.7102]
24. Arizona (3-2) [0.6976]
25. Arkansas (3-2) [0.6553]
26. Utah (4-1) [0.6315]
27. Houston (4-1) [0.6219]
28. Georgia (3-3) [0.5921]
29. Oregon St (4-2) [0.5826]
30. Brigham Young (5-1) [0.5434]
31. Pittsburgh (5-1) [0.5422]
32. Michigan (4-2) [0.5398]
33. South Florida (5-0) [0.5376]
34. Missouri (4-1) [0.5340]
35. Oklahoma St (4-1) [0.5112]
36. UCLA (3-2) [0.5055]
37. Penn State (5-1) [0.5033]
38. Boston College (4-2) [0.4687]
39. West Virginia (4-1) [0.4296]
40. Troy (3-2) [0.3954]
41. California (3-2) [0.3941]
42. Wake Forest (4-2) [0.3808]
43. Minnesota (4-2) [0.3324]
44. Rutgers (4-1) [0.3309]
45. Oklahoma (3-2) [0.3057]
46. Central Michigan (5-1) [0.2853]
47. Marshall (4-2) [0.2571]
48. Tennessee (3-3) [0.2478]
49. Mississippi (3-2) [0.2379]
50. Louisiana-Lafayette (3-2) [0.2371]
51. Kentucky (2-3) [0.2229]
52. Michigan St (3-3) [0.1393]
53. Northern Illinois (3-2) [0.1218]
54. Connecticut (3-2) [0.1204]
55. Arizona St (3-2) [0.0818]
56. Clemson (2-3) [0.0669]
57. Baylor (3-2) [0.0500]
58. North Carolina (4-2) [0.0485]
59. Ohio U. (4-2) [-0.0222]
60. Middle Tennessee St (3-2) [-0.0295]
61. Colorado St (3-3) [-0.0413]
62. Fresno St (2-3) [-0.0631]
63. Navy (4-2) [-0.0905]
64. SMU (3-2) [-0.0962]
65. Texas Tech (4-2) [-0.0999]
66. Louisiana-Monroe (3-2) [-0.1121]
67. Iowa St (3-3) [-0.1201]
68. Florida St (2-4) [-0.1366]
69. Tulsa (4-1) [-0.1541]
70. East Carolina (3-3) [-0.1733]
71. Bowling Green (2-4) [-0.1905]
72. Texas A&M (3-2) [-0.1974]
73. Nevada (2-3) [-0.2029]
74. North Carolina St (3-3) [-0.2071]
75. Wyoming (4-2) [-0.2754]
76. Duke (3-3) [-0.2805]
77. Mississippi St (2-4) [-0.3006]
78. Kansas St (3-3) [-0.3077]
79. Louisville (2-3) [-0.3211]
80. Arkansas St (1-3) [-0.3220]
81. Air Force (3-3) [-0.3373]
82. Southern Miss (3-3) [-0.3419]
83. Indiana (3-3) [-0.3512]
84. Northwestern (4-2) [-0.3527]
85. Central Florida (3-2) [-0.3544]
86. San Jose St (1-4) [-0.4218]
87. Syracuse (2-4) [-0.4240]
88. San Diego St (2-3) [-0.4299]
89. UTEP (2-4) [-0.4442]
90. Western Michigan (3-3) [-0.4619]
91. Maryland (2-4) [-0.4661]
92. Tulane (2-3) [-0.5388]
93. Louisiana Tech (2-3) [-0.6053]
94. Purdue (1-5) [-0.6489]
95. Illinois (1-4) [-0.6538]
96. Virginia (2-3) [-0.6539]
97. Alabama-Birmingham (2-3) [-0.6564]
98. New Mexico St (3-3) [-0.6630]
99. Toledo (3-3) [-0.6763]
100. UNLV (2-4) [-0.6777]
101. Memphis (2-4) [-0.6936]
102. Washington St (1-5) [-0.7090]
103. Colorado (1-4) [-0.7794]
104. Kent St (2-4) [-0.8684]
105. Buffalo (2-4) [-0.8856]
106. Hawai`i (2-3) [-0.9227]
107. Temple (3-2) [-0.9663]
108. Army (3-3) [-0.9929]
109. Vanderbilt (2-4) [-0.9958]
110. North Texas (1-4) [-1.0296]
111. Utah St (1-4) [-1.0645]
112. Akron (1-4) [-1.0808]
113. Florida Int'l (1-4) [-1.1920]
114. Florida Atlantic (0-4) [-1.4301]
115. FCS teams (XXX-XXX) [-1.6184]
116. Miami OH (0-6) [-1.8676]
117. New Mexico (0-6) [-2.2432]
118. Rice (0-6) [-2.3721]
119. Eastern Michigan (0-5) [-2.5733]
120. Western Kentucky (0-5) [-3.0848]
121. Ball St (0-6) [-3.6768]
Conference Rankings (Average Per Team):
SEC 0.7783
Pac10 0.5779
BigEast 0.3462
Big10 0.3347
ACC 0.3150
Big12 0.3102
FBSInd -0.0075
WAC -0.1672
MWC -0.1723
CUSA -0.4122
SunBelt -0.7297
MAC -0.9894
Non-FBS -1.6184

Comparing Conferences: So much movement, so little reason

2009 Random Walker Rankings (RWFL, p=0.75)
Games through Saturday October 3rd
Conference Rankings (Average Per Team):
SEC 0.7928
Pac10 0.6051
Big10 0.4282
ACC 0.3528
Big12 0.3359
BigEast 0.3310
WAC -0.1242
FBSInd -0.1543
MWC -0.2016
CUSA -0.4783
SunBelt -0.8022
MAC -1.0214
Non-FBS -1.5533

Only a week ago, the rankings made it look like the ACC was hands down the weakest of the so-called major conferences. Sure, VT and Miami were highly ranked; but on average, the squads in the ACC garnered fewer net RWFL votes per team. But here we are only a week later, and the same methodology puts the ACC very slightly ahead but in essentially a dead heat with the Big 12 and the Big East (varying the bias value p in the plot at the bottom of this post does change things, but not as vigorously as last week).

How is this big change in one week possible?

First, it would seem easy to go to my favorite rationalization: it's still early in the season. The problem with that argument in this case is that last weekend resulted in very little new information about the ACC's strength relative to the other conferences, with 10 of the 12 teams playing against each other.

Okay, so it must be those two interconference games? Sure, Georgia Tech beat Mississippi State, and of course the big win was Miami over Oklahoma. Those two ACC victories over SEC teams certainly move up the ACC rankings, especially the win over a highly rated Oklahoma team (starting QB or no). Such apparent sensitivity of rankings to a few interconference games only highlights the difficulty in ranking teams from the limited information that the BCS Standings allow.

Did anything else happen to cause this change in the rankings? Very possibly. The ACC was likely also helped this week by intraconference outcomes changing the rankings inside the conferences. For instance, Florida State's intraconference loss to Boston College further suppressed their RWFL rating, thereby decreasing the newly-assessed value of South Florida's interconference win over Florida State the week before. Reshuffled comparisons like this are happening all throughout the season, potentially changing the relative rankings of conferences even in the absence of direct matchups. The potential importance of such indirect effects make attempts to rank teams both interesting and maddening.

Rankings through October 3rd

There are obviously still a lot of unknowns this early in the season (there's a reason the official BCS Standings don't come out this early), but rankings do start to make a little more sense with another week of games on the books. Obviously, all eyes are on the upcoming Florida-LSU game. Meanwhile, there are only two potential BCS busters left after Houston's loss.

2009 Random Walker Rankings (RWFL, p=0.75)
Games through Saturday October 3rd:
1. Alabama (5-0) [2.5017]
2. LSU (5-0) [2.4968]
3. Iowa (5-0) [2.0009]
4. Virginia Tech (4-1) [1.8947]
5. Wisconsin (5-0) [1.5998]
6. Miami FL (3-1) [1.5444]
7. Auburn (5-0) [1.5160]
8. Texas (4-0) [1.4734]
9. Florida (4-0) [1.4694]
10. Boise St (5-0) [1.4400]
11. Cincinnati (5-0) [1.3715]
12. TCU (4-0) [1.2569]
13. Stanford (4-1) [1.2250]
14. Kansas (4-0) [1.2161]
15. Southern Cal (4-1) [1.1354]
16. Georgia Tech (4-1) [1.1309]
17. Oregon (4-1) [1.0211]
18. Notre Dame (4-1) [1.0181]
19. Arizona (3-1) [1.0069]
20. Ohio State (4-1) [0.8263]
21. Washington (2-3) [0.8211]
22. Missouri (4-0) [0.8186]
23. Georgia (3-2) [0.8182]
24. UCLA (3-1) [0.7916]
25. Boston College (4-1) [0.7564]
26. Nebraska (3-1) [0.7227]
27. Idaho (4-1) [0.6980]
28. South Florida (5-0) [0.6580]
29. Michigan (4-1) [0.6452]
30. Penn State (4-1) [0.6292]
31. Baylor (3-1) [0.6022]
32. Connecticut (3-1) [0.6013]
33. Wake Forest (3-2) [0.5748]
34. Brigham Young (4-1) [0.5423]
35. South Carolina (4-1) [0.5347]
36. West Virginia (3-1) [0.4952]
37. Central Michigan (4-1) [0.4842]
38. Houston (3-1) [0.4083]
39. California (3-2) [0.3957]
40. Utah (3-1) [0.3867]
41. Oklahoma St (3-1) [0.3446]
42. Rutgers (3-1) [0.3301]
43. Minnesota (3-2) [0.3293]
44. Louisiana-Lafayette (2-2) [0.2786]
45. Middle Tennessee St (3-1) [0.2491]
46. Clemson (2-3) [0.2491]
47. Arkansas (2-2) [0.2421]
48. Pittsburgh (4-1) [0.2379]
49. Oregon St (3-2) [0.2180]
50. Kentucky (2-2) [0.1927]
51. Northern Illinois (3-2) [0.1752]
52. Michigan St (2-3) [0.1551]
53. Mississippi (3-1) [0.1403]
54. Mississippi St (2-3) [0.1342]
55. North Carolina (3-2) [0.0998]
56. Oklahoma (2-2) [0.0922]
57. Florida St (2-3) [0.0567]
58. Colorado St (3-2) [0.0383]
59. North Carolina St (3-2) [0.0314]
60. Indiana (3-2) [0.0182]
61. Marshall (3-2) [0.0152]
62. East Carolina (3-2) [0.0144]
63. Navy (3-2) [0.0051]
64. Kansas St (3-2) [-0.0052]
65. Iowa St (3-2) [-0.0111]
66. Texas A&M (3-1) [-0.0771]
67. Tennessee (2-3) [-0.0811]
68. Arizona St (2-2) [-0.1049]
69. Ohio U. (3-2) [-0.1323]
70. Louisiana Tech (2-2) [-0.1732]
71. Southern Miss (3-2) [-0.1735]
72. Troy (2-2) [-0.1832]
73. Tulsa (4-1) [-0.1996]
74. Louisiana-Monroe (3-2) [-0.2057]
75. Fresno St (1-3) [-0.2100]
76. UTEP (2-3) [-0.2418]
77. Maryland (2-3) [-0.2482]
78. Syracuse (2-3) [-0.2810]
79. Wyoming (3-2) [-0.2888]
80. Air Force (3-2) [-0.3169]
81. San Jose St (1-3) [-0.3252]
82. Texas Tech (3-2) [-0.3597]
83. SMU (2-2) [-0.3628]
84. Nevada (1-3) [-0.3841]
85. Central Florida (3-2) [-0.3879]
86. San Diego St (2-3) [-0.3948]
87. Illinois (1-3) [-0.4138]
88. Vanderbilt (2-3) [-0.4519]
89. Washington St (1-4) [-0.4590]
90. Northwestern (3-2) [-0.4649]
91. Tulane (2-2) [-0.4788]
92. Toledo (3-2) [-0.4989]
93. Kent St (2-3) [-0.5253]
94. Arkansas St (1-3) [-0.5730]
95. Bowling Green (1-4) [-0.5739]
96. Western Michigan (2-3) [-0.5815]
97. Purdue (1-4) [-0.6155]
98. Hawai`i (2-2) [-0.6160]
99. Akron (1-3) [-0.6825]
100. UNLV (2-3) [-0.6936]
101. Utah St (1-3) [-0.6950]
102. Duke (2-3) [-0.7184]
103. Louisville (1-3) [-0.7645]
104. Alabama-Birmingham (2-3) [-0.7741]
105. Colorado (1-3) [-0.7861]
106. New Mexico St (2-3) [-0.8521]
107. North Texas (1-3) [-0.9751]
108. Temple (2-2) [-1.0962]
109. Memphis (1-4) [-1.1092]
110. Virginia (1-3) [-1.1385]
111. Buffalo (1-4) [-1.1989]
112. Army (2-3) [-1.4860]
113. FCS teams (XXX-XXX) [-1.5533]
114. Florida Atlantic (0-4) [-1.6109]
115. Florida Int'l (0-4) [-1.6260]
116. Miami OH (0-5) [-1.6788]
117. New Mexico (0-5) [-2.3440]
118. Rice (0-5) [-2.4494]
119. Western Kentucky (0-4) [-2.5740]
120. Eastern Michigan (0-4) [-3.3076]
121. Ball St (0-5) [-3.6623]

Virginia Tech, Really?

The midweek games have already come and gone, and I'm still a little puzzled about our RWFL rankings previously posted. Again, it's far too early in the season to expect good performance out of a computer ranking system that, like ours, ignores margin of victory, dates of games, and the previous season. But Virginia Tech (3-1) edging Iowa (4-0) for 2nd place?!? Now, sure, from a ranking violations standpoint, that's fine, since the loss came at the hands of #1 Alabama. But with so many undefeateds (including three potential BCS busters!), I found this result surprising.

The surprise disappears when we dig just a little further into the rankings. As we've done in this space for previous years, our tabulated results are for the specific bias value p=0.75 for the random walker algorithm. What this means is that each walker, considering a game between two teams, will decide that the winner is the better team 75% of the time. Why 75%? Seriously, essentially because it's halfway between 50% (ignoring the outcome altogether) and 100% (complete certainty that the outcome represents the better team). Okay, there's very slightly more to it than that: we tested the rankings across different p values and found that the middle of the range, around 75%, typically corresponds to the low values of rankings violations and the best values to predict bowl game outcomes in historical comparisons. And if you really press me for some other mathematical reasons, it turns out that RWFL rankings of round-robin tournaments appear (in numerical exploration) to agree perfectly with the resulting standings provided p is less than a value somewhere roughly around 0.75.

So, after all that mumbo jumbo, let's vary p and see what happens. You can see in the figure below that Virginia Tech and Miami both do well on the left (p closer to 0.5), but they fall quickly from these high perches as p increases moving to the right in the figure (VT and Miami are represented by the two curves moving quickly upwards towards worse rankings as p increases from left to right). Loosely speaking, this corresponds to the algorithm assigning an on-average stronger schedule to these teams on the left, while penalizing them for their losses on the right. At this point, the balance happens to be working out one way for them; but this high ranking is clearly tenuous at best.
We close today's post by briefly noting that the upcoming conference play might drastically change the above plot against Virginia Tech and Miami, even if they win, simply because the ACC is, on average, not ranked highly by this algorithm. In the plot below, we plot the average numbers of net RWFL votes (expressed as percentages) per team for each FBS conference (grouping the independents together). The way to read this plot is to look at vertical slices (fixed p values), wherein higher values correspond to greater numbers of net votes per team. So far, the ACC appears to be the weakest of the so-called major conferences at most p values, and indeed, it ranks weaker than some of the so-called mid-majors at higher values of p! No hate mail about this please; I'm just the messenger.