A Different Look at Schedule Topology
Most of the time I've represented the field's connectivity from the teams' perspectives. FBS Connectivity shows the number of other teams at distance 1 (Opponents), 2 (Opponents' Opponents), 3 and 4 units apart. The connectivity measurements I use there are the percentage of the field that is either an opponent or opponents' opponent for each team, and the Average Path Length from each team to every other team.
| For those interested in more about what I called the Incidence Matrix, Dr. Massey brought to my attention Network Properties Revealed Through Matrix Functions.(PDF) The authors of the paper refer to it as the "Adjacency Matrix" which is a better term, since "Incidence Matrix" is used in other contexts with a different definition. |
It's not hard to see why the teams that contribute the most to the field's connectedness are the independents. They typically do not play more than a few games against teams from the same conference, so there is less duplication in the opponents' opponents contribution from a particular game.
That view is focused on the vertexes of the games graph. Analysts are more interested in characterizing the edges (i.e. games.)
| Order | #Subsets | — | Unique | — | Interconf |
| 2 | 674 | | 117 | | 117 |
| 3 | 934 | | 53 | | 47 |
| 4 | 909 | | 79 | | 2 |
| 5 | 665 | | 18 | | 0 |
| 6 | 396 | | 18 | | 1 |
| 7 | 180 | | 0 | | 0 |
| 8 | 57 | | 3 | | 0 |
| 9 | 11 | | 1 | | 0 |
| 10 | 1 | | 1 | | 0 |
|
First we begin by finding all subsets of the field in which each team plays every other team in the subset. There are 3,827 of those, including the 674 subsets of order 2 that represent the scheduled games.
There's a lot of duplication in that list: the one subset of size 10 (the Big "12" plays a round-robin) has 1022 proper subsets of its own that we don't need to count if we count the one of order 10.
When we eliminate all the redundant subsets we wind up with 290 unique subsets. 123 of those, though, consist of teams all from the same conference so do not include any "discretionary" games.
That leaves 167 subsets that contribute the most to "connecting" the field. If we order those by the number of ⇔2 connections added by each game we find that it is the 117 interconference matchups by teams that have no common opponents that contribute the most.
This is easy to figure out. If teams who play each other have no common opponents, each of them gets a ⇔2 path to the other's opponents. In the list below, the order-2 subsets are ordered in terms of decreasing interest.
O-Os
added | Games | O-Os added
per game | Order | | | | | | | | |
| 20 | 1 | 20 | 2 | { | 3-Oregon | 9-LSU | } | | | | |
| 21 | 1 | 21 | 2 | { | 13-Florida State | 5-Oklahoma | } | | | | |
| 21 | 1 | 21 | 2 | { | 17-Texas A&M | 12-Arkansas | } | | | | |
| 20 | 1 | 20 | 2 | { | 13-Florida State | 24-Florida | } | | | | |
| 20 | 1 | 20 | 2 | { | 29-Pittsburgh | 27-Utah | } | | | | |
| 20 | 1 | 20 | 2 | { | 29-Pittsburgh | 30-Iowa | } | | | | |
| 20 | 1 | 20 | 2 | { | 27-West Virginia | 9-LSU | } | | | | |
| 21 | 1 | 21 | 2 | { | 3-Oregon | 26-Nevada | } | | | | |
| 20 | 1 | 20 | 2 | { | 19-Missouri | 32-Arizona State | } | | | | |
| 21 | 1 | 21 | 2 | { | 33-Miami-Florida | 7-Ohio State | } | | | | |
| 21 | 1 | 21 | 2 | { | 33-Georgia | 4-Boise State | } | | | | |
| 20 | 1 | 20 | 2 | { | 38-Maryland | 27-West Virginia | } | | | | |
| 22 | 1 | 22 | 2 | { | 37-Tulsa | 4-Boise State | } | | | | |
| 20 | 1 | 20 | 2 | { | 39-BYU | 6-TCU | } | | | | |
| 20 | 1 | 20 | 2 | { | 12-Wisconsin | 41-Oregon State | } | | | | |
| 20 | 1 | 20 | 2 | { | 44-Illinois | 32-Arizona State | } | | | | |
| 20 | 1 | 20 | 2 | { | 46-UCF | 39-BYU | } | | | | |
| 20 | 1 | 20 | 2 | { | 43-Penn State | 3-Alabama | } | | | | |
| 20 | 1 | 20 | 2 | { | 12-Wisconsin | 44-Northern Illinois | } | | | | |
| 21 | 1 | 21 | 2 | { | 47-Texas | 39-BYU | } | | | | |
| 20 | 1 | 20 | 2 | { | 18-Nebraska | 45-Washington | } | | | | |
| 21 | 1 | 21 | 2 | { | 49-Michigan | 43-San Diego State | } | | | | |
| 20 | 1 | 20 | 2 | { | 33-Miami-Florida | 50-South Florida | } | | | | |
| 20 | 1 | 20 | 2 | { | 52-Boston College | 46-UCF | } | | | | |
| 20 | 1 | 20 | 2 | { | 33-Miami-Florida | 58-Kansas State | } | | | | |
O-Os
added | Games | O-Os added
per game | Order | | | | | | | | |
| 20 | 1 | 20 | 2 | { | 60-Baylor | 6-TCU | } | | | | |
| 20 | 1 | 20 | 2 | { | 61-Georgia Tech | 33-Georgia | } | | | | |
| 20 | 1 | 20 | 2 | { | 61-Louisville | 61-Kentucky | } | | | | |
| 21 | 1 | 21 | 2 | { | 65-Syracuse | 26-Southern California | } | | | | |
| 20 | 1 | 20 | 2 | { | 52-Boston College | 65-Northwestern | } | | | | |
| 19 | 1 | 19 | 2 | { | 30-North Carolina State | 66-Cincinnati | } | | | | |
| 23 | 1 | 23 | 2 | { | 7-Ohio State | 67-Colorado | } | | | | |
| 20 | 1 | 20 | 2 | { | 66-Cincinnati | 49-Tennessee | } | | | | |
| 22 | 1 | 22 | 2 | { | 47-Texas | 67-UCLA | } | | | | |
| 21 | 1 | 21 | 2 | { | 18-Nebraska | 69-Fresno State | } | | | | |
| 20 | 1 | 20 | 2 | { | 71-SMU | 6-TCU | } | | | | |
| 21 | 1 | 21 | 2 | { | 50-California | 69-Fresno State | } | | | | |
| 21 | 1 | 21 | 2 | { | 17-Texas A&M | 71-SMU | } | | | | |
| 21 | 1 | 21 | 2 | { | 72-Toledo | 4-Boise State | } | | | | |
| 21 | 1 | 21 | 2 | { | 7-Ohio State | 72-Toledo | } | | | | |
| 20 | 1 | 20 | 2 | { | 71-Mississippi | 39-BYU | } | | | | |
| 20 | 1 | 20 | 2 | { | 30-Iowa | 72-Iowa State | } | | | | |
| 20 | 1 | 20 | 2 | { | 38-Maryland | 72-Temple | } | | | | |
| 20 | 1 | 20 | 2 | { | 43-Penn State | 72-Temple | } | | | | |
| 21 | 1 | 21 | 2 | { | 67-UCLA | 70-Houston | } | | | | |
| 20 | 1 | 20 | 2 | { | 51-Connecticut | 72-Iowa State | } | | | | |
| 21 | 1 | 21 | 2 | { | 19-Missouri | 75-Miami-Ohio | } | | | | |
| 20 | 1 | 20 | 2 | { | 65-Syracuse | 72-Toledo | } | | | | |
| 21 | 1 | 21 | 2 | { | 12-Arkansas | 79-Troy | } | | | | |
| 20 | 1 | 20 | 2 | { | 51-Connecticut | 77-Western Michigan | } | | | | |
O-Os
added | Games | O-Os added
per game | Order | | | | | | | | |
| 20 | 1 | 20 | 2 | { | 77-Virginia | 57-Southern Miss | } | | | | |
| 21 | 1 | 21 | 2 | { | 40-Clemson | 79-Troy | } | | | | |
| 21 | 1 | 21 | 2 | { | 82-Minnesota | 26-Southern California | } | | | | |
| 21 | 1 | 21 | 2 | { | 85-Duke | 5-Stanford | } | | | | |
| 21 | 1 | 21 | 2 | { | 46-UCF | 82-Florida Intl | } | | | | |
| 21 | 1 | 21 | 2 | { | 61-Louisville | 82-Florida Intl | } | | | | |
| 21 | 1 | 21 | 2 | { | 17-Texas A&M | 87-Idaho | } | | | | |
| 20 | 1 | 20 | 2 | { | 57-Southern Miss | 83-Louisiana Tech | } | | | | |
| 20 | 1 | 20 | 2 | { | 70-Houston | 83-Louisiana Tech | } | | | | |
| 21 | 1 | 21 | 2 | { | 82-Minnesota | 75-Miami-Ohio | } | | | | |
| 20 | 1 | 20 | 2 | { | 85-Wake Forest | 65-Syracuse | } | | | | |
| 20 | 1 | 20 | 2 | { | 65-Northwestern | 86-Army | } | | | | |
| 20 | 1 | 20 | 2 | { | 91-Kansas | 44-Northern Illinois | } | | | | |
| 20 | 1 | 20 | 2 | { | 27-Michigan State | 93-Central Michigan | } | | | | |
| 19 | 1 | 19 | 2 | { | 30-North Carolina State | 93-Central Michigan | } | | | | |
| 21 | 1 | 21 | 2 | { | 85-Duke | 82-Florida Intl | } | | | | |
| 20 | 1 | 20 | 2 | { | 77-Virginia | 87-Idaho | } | | | | |
| 20 | 1 | 20 | 2 | { | 61-Georgia Tech | 91-Kansas | } | | | | |
| 19 | 1 | 19 | 2 | { | 18-Nebraska | 96-Wyoming | } | | | | |
| 20 | 1 | 20 | 2 | { | 77-Virginia | 88-Indiana | } | | | | |
| 20 | 1 | 20 | 2 | { | 3-Alabama | 98-Kent State | } | | | | |
| 20 | 1 | 20 | 2 | { | 8-Auburn | 98-Utah State | } | | | | |
| 20 | 1 | 20 | 2 | { | 86-Rutgers | 87-Ohio | } | | | | |
| 20 | 1 | 20 | 2 | { | 61-Kentucky | 93-Central Michigan | } | | | | |
| 20 | 1 | 20 | 2 | { | 51-Connecticut | 95-Vanderbilt | } | | | | |
O-Os
added | Games | O-Os added
per game | Order | | | | | | | | |
| 20 | 1 | 20 | 2 | { | 13-Virginia Tech | 99-Arkansas State | } | | | | |
| 20 | 1 | 20 | 2 | { | 50-South Florida | 96-UTEP | } | | | | |
| 20 | 1 | 20 | 2 | { | 6-TCU | 100-UL Monroe | } | | | | |
| 20 | 1 | 20 | 2 | { | 13-Florida State | 100-UL Monroe | } | | | | |
| 20 | 1 | 20 | 2 | { | 30-Iowa | 100-UL Monroe | } | | | | |
| 20 | 1 | 20 | 2 | { | 44-Illinois | 99-Arkansas State | } | | | | |
| 21 | 1 | 21 | 2 | { | 49-Tennessee | 99-Middle Tenn State | } | | | | |
| 20 | 1 | 20 | 2 | { | 58-Kansas State | 98-Kent State | } | | | | |
| 21 | 1 | 21 | 2 | { | 61-Georgia Tech | 99-Middle Tenn State | } | | | | |
| 21 | 1 | 21 | 2 | { | 93-Marshall | 87-Ohio | } | | | | |
| 21 | 1 | 21 | 2 | { | 65-Northwestern | 99-Rice | } | | | | |
| 20 | 1 | 20 | 2 | { | 85-Wake Forest | 95-Vanderbilt | } | | | | |
| 20 | 1 | 20 | 2 | { | 12-Wisconsin | 105-UNLV | } | | | | |
| 20 | 1 | 20 | 2 | { | 95-Vanderbilt | 86-Army | } | | | | |
| 21 | 1 | 21 | 2 | { | 80-Purdue | 99-Middle Tenn State | } | | | | |
| 21 | 1 | 21 | 2 | { | 80-Purdue | 99-Rice | } | | | | |
| 22 | 1 | 22 | 2 | { | 5-Oklahoma | 108-Ball State | } | | | | |
| 22 | 1 | 22 | 2 | { | 67-Colorado | 104-Colorado State | } | | | | |
| 22 | 1 | 22 | 2 | { | 107-Tulane | 50-Hawaii | } | | | | |
| 20 | 1 | 20 | 2 | { | 27-West Virginia | 109-Bowling Green | } | | | | |
| 21 | 1 | 21 | 2 | { | 50-South Florida | 108-Ball State | } | | | | |
| 21 | 1 | 21 | 2 | { | 52-Navy | 108-San Jose State | } | | | | |
| 21 | 1 | 21 | 2 | { | 3-Alabama | 111-North Texas | } | | | | |
| 21 | 1 | 21 | 2 | { | 65-Syracuse | 107-Tulane | } | | | | |
| 21 | 1 | 21 | 2 | { | 27-Michigan State | 111-Florida Atlantic | } | | | | |
O-Os
added | Games | O-Os added
per game | Order | | | | | | | | |
| 21 | 1 | 21 | 2 | { | 85-Duke | 107-Tulane | } | | | | |
| 21 | 1 | 21 | 2 | { | 107-Tulane | 86-Army | } | | | | |
| 20 | 1 | 20 | 2 | { | 96-UTEP | 104-Colorado State | } | | | | |
| 20 | 1 | 20 | 2 | { | 12-Arkansas | 114-New Mexico | } | | | | |
| 21 | 1 | 21 | 2 | { | 88-Indiana | 108-Ball State | } | | | | |
| 20 | 1 | 20 | 2 | { | 109-Bowling Green | 87-Idaho | } | | | | |
| 21 | 1 | 21 | 2 | { | 88-Indiana | 111-North Texas | } | | | | |
| 19 | 1 | 19 | 2 | { | 109-Bowling Green | 96-Wyoming | } | | | | |
| 20 | 1 | 20 | 2 | { | 49-Tennessee | 116-Buffalo | } | | | | |
| 22 | 1 | 22 | 2 | { | 33-Georgia | 117-New Mexico State | } | | | | |
| 20 | 1 | 20 | 2 | { | 98-Kent State | 112-UL Lafayette | } | | | | |
| 19 | 1 | 19 | 2 | { | 43-Penn State | 117-Eastern Michigan | } | | | | |
| 22 | 1 | 22 | 2 | { | 82-Minnesota | 117-New Mexico State | } | | | | |
| 21 | 1 | 21 | 2 | { | 7-Ohio State | 119-Akron | } | | | | |
| 22 | 1 | 22 | 2 | { | 87-Ohio | 117-New Mexico State | } | | | | |
| 22 | 1 | 22 | 2 | { | 96-UTEP | 117-New Mexico State | } | | | | |
| 21 | 1 | 21 | 2 | { | 119-Akron | 82-Florida Intl | } | | | | |
| 30 | 3 | 10 | 3 | { | 45-Washington | 67-Colorado | 50-Hawaii | } | | | |
| 30 | 3 | 10 | 3 | { | 5-Oklahoma | 10-Oklahoma State | 37-Tulsa | } | | | |
| 30 | 3 | 10 | 3 | { | 5-Stanford | 67-UCLA | 108-San Jose State | } | | | |
| 30 | 3 | 10 | 3 | { | 4-Boise State | 69-Fresno State | 26-Nevada | } | | | |
| 30 | 3 | 10 | 3 | { | 114-New Mexico | 26-Nevada | 117-New Mexico State | } | | | |
| 30 | 3 | 10 | 3 | { | 96-UAB | 111-Florida Atlantic | 79-Troy | } | | | |
| 30 | 3 | 10 | 3 | { | 21-Notre Dame | 5-Stanford | 26-Southern California | } | | | |
| 29 | 3 | 9.7 | 3 | { | 24-Florida | 96-UAB | 111-Florida Atlantic | } | | | |
O-Os
added | Games | O-Os added
per game | Order | | | | | | | | |
| 29 | 3 | 9.7 | 3 | { | 60-Baylor | 47-Texas | 99-Rice | } | | | |
| 29 | 3 | 9.7 | 3 | { | 13-Virginia Tech | 79-East Carolina | 93-Marshall | } | | | |
| 29 | 3 | 9.7 | 3 | { | 80-Purdue | 49-Michigan | 21-Notre Dame | } | | | |
| 29 | 3 | 9.7 | 3 | { | 49-Michigan | 27-Michigan State | 21-Notre Dame | } | | | |
| 29 | 3 | 9.7 | 3 | { | 70-Houston | 37-Tulsa | 111-North Texas | } | | | |
| 29 | 3 | 9.7 | 3 | { | 105-UNLV | 50-Hawaii | 26-Nevada | } | | | |
| 29 | 3 | 9.7 | 3 | { | 4-Boise State | 43-San Diego State | 69-Fresno State | } | | | |
| 28 | 3 | 9.3 | 3 | { | 23-Mississippi State | 116-Memphis | 96-UAB | } | | | |
| 28 | 3 | 9.3 | 3 | { | 24-Florida | 8-Auburn | 111-Florida Atlantic | } | | | |
| 28 | 3 | 9.3 | 3 | { | 44-Illinois | 49-Michigan | 77-Western Michigan | } | | | |
| 28 | 3 | 9.3 | 3 | { | 86-Army | 108-Ball State | 44-Northern Illinois | } | | | |
| 28 | 3 | 9.3 | 3 | { | 29-Pittsburgh | 50-South Florida | 21-Notre Dame | } | | | |
| 28 | 3 | 9.3 | 3 | { | 10-Oklahoma State | 39-Arizona | 112-UL Lafayette | } | | | |
| 28 | 3 | 9.3 | 3 | { | 20-South Carolina | 79-East Carolina | 52-Navy | } | | | |
| 28 | 3 | 9.3 | 3 | { | 61-Louisville | 27-West Virginia | 93-Marshall | } | | | |
| 28 | 3 | 9.3 | 3 | { | 116-Memphis | 99-Arkansas State | 99-Middle Tenn State | } | | | |
| 28 | 3 | 9.3 | 3 | { | 42-North Carolina | 13-Virginia Tech | 79-East Carolina | } | | | |
| 28 | 3 | 9.3 | 3 | { | 52-Navy | 79-Troy | 112-Western Kentucky | } | | | |
| 28 | 3 | 9.3 | 3 | { | 66-Cincinnati | 119-Akron | 75-Miami-Ohio | } | | | |
| 28 | 3 | 9.3 | 3 | { | 45-Texas Tech | 114-New Mexico | 26-Nevada | } | | | |
| 28 | 3 | 9.3 | 3 | { | 86-Army | 72-Temple | 108-Ball State | } | | | |
| 28 | 3 | 9.3 | 3 | { | 104-Colorado State | 108-San Jose State | 98-Utah State | } | | | |
| 28 | 3 | 9.3 | 3 | { | 71-Mississippi | 69-Fresno State | 83-Louisiana Tech | } | | | |
| 28 | 3 | 9.3 | 3 | { | 86-Army | 75-Miami-Ohio | 72-Temple | } | | | |
| 28 | 3 | 9.3 | 3 | { | 79-East Carolina | 57-Southern Miss | 52-Navy | } | | | |
O-Os
added | Games | O-Os added
per game | Order | | | | | | | | |
| 27 | 3 | 9.0 | 3 | { | 61-Kentucky | 9-LSU | 112-Western Kentucky | } | | | |
| 27 | 3 | 9.0 | 3 | { | 49-Michigan | 117-Eastern Michigan | 77-Western Michigan | } | | | |
| 27 | 3 | 9.0 | 3 | { | 51-Connecticut | 29-Pittsburgh | 116-Buffalo | } | | | |
| 27 | 3 | 9.0 | 3 | { | 42-North Carolina | 61-Louisville | 86-Rutgers | } | | | |
| 27 | 3 | 9.0 | 3 | { | 85-Washington State | 43-San Diego State | 105-UNLV | } | | | |
| 27 | 3 | 9.0 | 3 | { | 41-Oregon State | 27-Utah | 39-BYU | } | | | |
| 27 | 3 | 9.0 | 3 | { | 40-Clemson | 20-South Carolina | 8-Auburn | } | | | |
| 27 | 3 | 9.0 | 3 | { | 71-Mississippi | 23-Mississippi State | 83-Louisiana Tech | } | | | |
| 27 | 3 | 9.0 | 3 | { | 21-Notre Dame | 52-Navy | 35-Air Force | } | | | |
| 27 | 3 | 9.0 | 3 | { | 57-Southern Miss | 71-SMU | 52-Navy | } | | | |
| 27 | 3 | 9.0 | 3 | { | 86-Rutgers | 86-Army | 52-Navy | } | | | |
| 26 | 3 | 8.7 | 3 | { | 86-Army | 35-Air Force | 43-San Diego State | } | | | |
| 26 | 3 | 8.7 | 3 | { | 104-Colorado State | 96-Wyoming | 98-Utah State | } | | | |
| 26 | 3 | 8.7 | 3 | { | 86-Army | 52-Navy | 35-Air Force | } | | | |
| 34 | 6 | 5.7 | 4 | { | 4-Boise State | 114-New Mexico | 105-UNLV | 26-Nevada | } | | |
| 33 | 6 | 5.5 | 4 | { | 52-Boston College | 38-Maryland | 85-Wake Forest | 21-Notre Dame | } | | |
| 40 | 15 | 2.7 | 6 | { | 39-BYU | 50-Hawaii | 87-Idaho | 117-New Mexico State | 108-San Jose State | 98-Utah State | } |
|
Only 26 of the games that provide information about relative team strength we wouldn't be able to otherwise infer are between teams in the pre-season top 60.
We say that the adjancency/incidence matrix is "sparse" because only 1348 of its 14400 cells (9.36 percent) have non-zero value. Consider:
There are 2120-2 =
1,329,227,995,784,915,872,903,807,060,280,344,574
proper subsets of the set of the set of 120 FBS teams. Only 3,827 of them can be defined by following connections. That's 2.879×10-31 percent. Talk about sparse!