Back in July I wrote:
It's true that not playing at least one more FBS team would improve the connectivity of the field, but that's only important to those of us who study computer ratings, and few Athletic Directors have us in mind when making schedules!It occurred to me that it might be interesting to find out which teams' (non-conference) schedules do contribute the most to connecting the FBS field.
For my basic connectivity measurement the important thing is the number of opponents' opponents a team contributes. In my version of the FBS Distance Matrix I count the number of unique opponents' opponents (who are not also opponents) for each team. But there's no unambiguous way to assign an O-O to any particular opponent when multiple O's have the same opponent (not in common) so for this measurement we want the maximum contributions by any team to all teams, even if they duplicate O-Os.
That's hard to understand without an example or two.
In 2008 Notre Dame plays Michigan, Michigan St, and Purdue who all play each other, so { Michigan, Michigan St, Purdue, Notre Dame } form a set of four teams each of which plays all the others. The Irish also play USC, Stanford and Washington, each of which plays all of the others and none of them play any of Michigan, Michigan St, or Purdue. So { USC, Stanford, Washington, Notre Dame } forms another such set. All 12 of ND's opponents are FBS teams, so each team in either set gets nine opponents' opponents from Notre Dame (the nine that are not in the same set as the team and Notre Dame.)
Notre Dame is also a part of four other sets of teams who've all played each other. The full list is
The Panthers don't because they've already been counted above. However, they got too much credit above because one of the O-O's they picked up turns out to be an O, so ND's score is (94+10-1=103)
If that mud isn't clear enough, an example from the other end of the scale is UL Monroe. Except for subsets that consist only of teams from the Sun Belt, UL Monroe is a member of only two sets of teams where each plays all the others:
To find the teams that contribute the most to connectivity (as opposed to are the most connected) we can follow these steps:
Call the set of lists that remain connectors.
1 Find all subsets of the teams in the FBS for which every team in the subset plays every other team.
This isn't as hard as it may seem, since there are only 678 matchups between FBS teams, so that's how many lists of 2 teams each of which plays each other there are. The intersections of those teams' opponents lists give all the rest.2 Ignore all the subsets that are themselves a proper subsets of larger ones - each list of six teams who all play each other contains 56 such lists from sizes two through five. 3 Ignore all the lists where all teams are from the same conference.
This is a bit artificial, but then so are conference affiliations.
When you use the connector sets to calculate the "connectivity score" as described above for each team in any of the lists, you get the contribution to connectivity.
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It's no surprise that the Independents provide the greatest benefit from a connectivity standpoint, since we've excluded conference games from the definition. Likewise it's not a surprise that teams that play more non-FBS teams contribute less, since they have fewer FBS O's to offer.
{ Arizona, Arizona St, California, Oregon, Oregon St, Southern California, Stanford, UCLA, Washington, Washington St }
{ Air Force, BYU, Colorado St, New Mexico, San Diego St, TCU, UNLV, Utah, Wyoming }
{ Boise St, Fresno St, Hawaii, Idaho, Louisiana Tech, Nevada, New Mexico St, San Jose St, Utah St }
{ Arkansas St, Florida Atlantic, Florida Intl, Middle Tenn St, North Texas, Troy, UL Lafayette, UL Monroe }
{ Cincinnati, Connecticut, Louisville, Pittsburgh, Rutgers, South Florida, Syracuse, West Virginia }
†{ Western Kentucky, Florida Atlantic, Florida Intl, Middle Tenn St, North Texas, Troy }
{ Houston, Rice, SMU, Tulane, Tulsa, UTEP }
{ Florida, Georgia, Kentucky, South Carolina, Tennessee, Vanderbilt }
{ Ball State, Central Michigan, Eastern Michigan, Northern Illinois, Toledo, Western Michigan }
{ Boston College, Clemson, Florida St, Maryland, North Carolina St, Wake Forest }
{ East Carolina, Marshall, Memphis, Southern Miss, UAB, UCF }
{ Alabama, Arkansas, Auburn, LSU, Mississippi, Mississippi St }
{ Duke, Georgia Tech, Miami-Florida, North Carolina, Virginia, Virginia Tech }
{ Baylor, Oklahoma, Oklahoma St, Texas, Texas A&M, Texas Tech }
{ Colorado, Iowa State, Kansas, Kansas St, Missouri, Nebraska }
{ Akron, Buffalo, Kent St, Ohio, Temple }
{ Buffalo, Central Michigan, Ohio, Temple, Western Michigan }
{ Buffalo, Kent St, Miami-Ohio, Ohio, Temple }
{ Arkansas, Florida, LSU, Mississippi, South Carolina }
{ Indiana, Iowa, Michigan St, Northwestern, Purdue }
{ Indiana, Iowa, Michigan St, Penn State, Purdue }
{ Indiana, Iowa, Michigan St, Penn State, Wisconsin }
{ Illinois, Indiana, Iowa, Minnesota, Northwestern }
{ Indiana, Iowa, Minnesota, Northwestern, Purdue }
{ Illinois, Michigan, Minnesota, Ohio State, Wisconsin }
{ Illinois, Michigan, Ohio State, Penn State, Wisconsin }
{ Michigan, Michigan St, Ohio State, Penn State, Wisconsin }
{ Illinois, Michigan, Minnesota, Northwestern, Ohio State }
{ Michigan, Minnesota, Northwestern, Ohio State, Purdue }
{ Illinois, Indiana, Iowa, Penn State, Wisconsin }
{ Kansas, Kansas St, Nebraska, Oklahoma, Texas Tech }
{ Akron, Bowling Green, Buffalo, Kent St, Ohio }
{ Illinois, Indiana, Iowa, Minnesota, Wisconsin }
{ Michigan, Michigan St, Ohio State, Penn State, Purdue }
{ Bowling Green, Buffalo, Kent St, Miami-Ohio, Ohio }
{ Michigan, Michigan St, Northwestern, Ohio State, Purdue }
†{ Army, Akron, Buffalo, Temple }
†{ Army, Akron, Eastern Michigan, Temple }
{ Memphis, Rice, SMU, Southern Miss }
{ Memphis, Rice, SMU, Tulane }
{ Rice, SMU, Southern Miss, UTEP }
{ Houston, Marshall, Tulsa, UAB }
{ Houston, Tulane, Tulsa, UAB }
{ Marshall, Tulsa, UAB, UCF }
†{ Arkansas, Auburn, Mississippi, UL Monroe }
†{ BYU, UNLV, Utah, Utah St }
{ Florida, Mississippi, South Carolina, Vanderbilt }
{ Boston College, Clemson, Florida St, Georgia Tech }
{ Boston College, Florida St, Georgia Tech, Virginia Tech }
{ Boston College, Georgia Tech, North Carolina, Virginia Tech }
{ Marshall, Memphis, Rice, Southern Miss }
{ Akron, Ball State, Eastern Michigan, Toledo }
{ Clemson, Duke, North Carolina St, Wake Forest }
{ Akron, Bowling Green, Eastern Michigan, Toledo }
{ Bowling Green, Eastern Michigan, Northern Illinois, Toledo }
{ Alabama, Auburn, Mississippi St, Tennessee }
{ Auburn, Mississippi, Mississippi St, Vanderbilt }
{ Auburn, Mississippi St, Tennessee, Vanderbilt }
†{ Michigan, Michigan St, Purdue, Notre Dame }
{ Florida, Georgia, LSU, South Carolina }
{ East Carolina, Houston, Tulane, UAB }
{ East Carolina, Houston, Tulane, UTEP }
{ East Carolina, Memphis, Tulane, UAB }
{ Florida St, Georgia Tech, Miami-Florida, Virginia Tech }
{ Alabama, Arkansas, Kentucky, Mississippi St }
{ Alabama, Auburn, Georgia, LSU }
{ Florida St, Miami-Florida, North Carolina St, Wake Forest }
{ Boston College, Maryland, North Carolina, Virginia Tech }
{ Maryland, North Carolina, Virginia, Virginia Tech }
{ Alabama, Auburn, Georgia, Tennessee }
{ Auburn, Georgia, Tennessee, Vanderbilt }
{ Ball State, Kent St, Miami-Ohio, Northern Illinois }
{ Clemson, Maryland, Virginia, Wake Forest }
{ Clemson, Duke, Georgia Tech, Virginia }
{ Clemson, Duke, Virginia, Wake Forest }
{ Duke, Miami-Florida, Virginia, Wake Forest }
{ East Carolina, Houston, Marshall, UAB }
{ Alabama, Georgia, Kentucky, Tennessee }
{ Duke, Miami-Florida, North Carolina, North Carolina St }
{ Central Michigan, Eastern Michigan, Temple, Western Michigan }
{ East Carolina, Southern Miss, UCF, UTEP }
{ SMU, Southern Miss, UCF, UTEP }
{ SMU, Tulsa, UCF, UTEP }
†{ Notre Dame, Southern California, Stanford, Washington }
{ Alabama, Kentucky, Mississippi St, Tennessee }
{ Arkansas, Florida, Kentucky, South Carolina }
{ Baylor, Nebraska, Oklahoma, Texas Tech }
{ Memphis, SMU, Southern Miss, UCF }
{ Colorado, Kansas, Missouri, Texas }
{ Colorado, Missouri, Oklahoma St, Texas }
{ Colorado, Oklahoma St, Texas, Texas A&M }
{ Kansas St, Oklahoma, Texas A&M, Texas Tech }
{ Baylor, Iowa State, Missouri, Nebraska }
{ Boston College, Maryland, North Carolina, North Carolina St }
{ Baylor, Iowa State, Oklahoma St, Texas A&M }
{ Colorado, Iowa State, Oklahoma St, Texas A&M }
{ Ball State, Miami-Ohio, Northern Illinois, Toledo }
{ Houston, Marshall, Rice, Tulsa }
{ Baylor, Iowa State, Missouri, Oklahoma St }
{ Kentucky, Mississippi St, Tennessee, Vanderbilt }
{ Baylor, Missouri, Oklahoma St, Texas }
{ Colorado, Iowa State, Missouri, Oklahoma St }
{ Bowling Green, Kent St, Miami-Ohio, Northern Illinois }
{ Bowling Green, Miami-Ohio, Northern Illinois, Toledo }
{ Kansas, Oklahoma, Texas, Texas Tech }
{ Colorado, Iowa State, Kansas St, Texas A&M }
{ Boston College, Florida St, Maryland, Virginia Tech }
{ Duke, Miami-Florida, North Carolina St, Wake Forest }
†{ Clemson, North Carolina St, South Carolina }
†{ Army, Navy, Temple }
†{ Indiana, Western Kentucky, Ball State }
†{ South Florida, Kansas, Florida Intl }
†{ Kansas St, North Texas, UL Lafayette }
†{ Minnesota, Bowling Green, Northern Illinois }
†{ Pittsburgh, Bowling Green, Buffalo }
†{ Southern Miss, UTEP, UL Lafayette }
†{ Southern Miss, Arkansas St, UL Lafayette }
†{ UNLV, Nevada, Utah St }
†{ Virginia, Virginia Tech, East Carolina }
†{ Baylor, Oklahoma St, Washington St }
†{ Bowling Green, Ohio, Wyoming }
†{ Louisville, Kentucky, Middle Tenn St }
†{ Navy, Ball State, Northern Illinois }
†{ Alabama, LSU, Tulane }
†{ West Virginia, East Carolina, Marshall }
†{ Florida St, Miami-Florida, Florida }
†{ LSU, North Texas, Troy }
†{ Michigan, Miami-Ohio, Toledo }
†{ Nebraska, New Mexico St, San Jose St }
†{ Rice, Tulsa, North Texas }
†{ Northwestern, Ohio State, Ohio }
†{ Oregon, Boise St, Utah St }
†{ Cincinnati, Syracuse, Akron }
†{ Texas, Rice, UTEP }
†{ Wake Forest, Mississippi, Vanderbilt }
†{ UCLA, Washington, BYU }
†{ North Carolina, Connecticut, Rutgers }
{ Akron, Ball State, Kent St }
†{ Alabama, Kentucky, Western Kentucky }
†{ Memphis, Southern Miss, Arkansas St }
†{ North Carolina, Virginia, Connecticut }
†{ Duke, Wake Forest, Navy }
†{ Houston, Air Force, Colorado St }
†{ Kentucky, Mississippi St, Middle Tenn St }
†{ Kentucky, Western Kentucky, Middle Tenn St }
†{ Oregon St, Washington St, Hawaii }
†{ San Diego St, Idaho, San Jose St }
†{ South Carolina, Tennessee, UAB }
†{ Indiana, Ball State, Central Michigan }
†{ Pittsburgh, Rutgers, Navy }
†{ Rutgers, Army, Navy }
†{ Indiana, Purdue, Central Michigan }
†{ Pittsburgh, Syracuse, Notre Dame }
†{ Cincinnati, West Virginia, Marshall }
†{ Army, Navy, Air Force }
†{ Rice, Tulane, Army }
†{ Boston College, North Carolina, Notre Dame }
†{ Pittsburgh, Notre Dame, Navy }
†{ Duke, Wake Forest, Vanderbilt }
†{ Wake Forest, Baylor }
†{ UTEP, Buffalo }
†{ Vanderbilt, Miami-Ohio }
†{ Arizona, Idaho }
†{ Michigan St, California }
†{ Clemson, Alabama }
†{ Florida, Hawaii }
†{ Boston College, Kent St }
†{ Mississippi St, Louisiana Tech }
†{ Michigan, Utah }
†{ Mississippi, Memphis }
†{ Illinois, Missouri }
†{ Nebraska, Western Michigan }
†{ Syracuse, Northwestern }
†{ Texas, Florida Atlantic }
†{ Texas A&M, Arkansas St }
†{ Virginia, Southern California }
†{ Wisconsin, Akron }
†{ Colorado, Colorado St }
†{ Rutgers, Fresno St }
†{ UCLA, Tennessee }
†{ Arizona, Toledo }
†{ Auburn, Southern Miss }
†{ South Florida, UCF }
†{ Duke, Northwestern }
†{ UAB, Florida Atlantic }
†{ Georgia, Central Michigan }
†{ Iowa, Florida Intl }
†{ Iowa State, Kent St }
†{ Kansas, Louisiana Tech }
†{ Michigan St, Eastern Michigan }
†{ Maryland, Middle Tenn St }
†{ Texas Tech, Nevada }
†{ Texas A&M, New Mexico }
†{ Notre Dame, San Diego St }
†{ Cincinnati, Oklahoma }
†{ Oklahoma St, Houston }
†{ Penn State, Oregon St }
†{ Connecticut, Temple }
†{ Wisconsin, Marshall }
†{ Arizona St, UNLV }
†{ Bowling Green, Boise St }
†{ Wisconsin, Fresno St }
†{ Western Michigan, Idaho }
†{ Illinois, UL Lafayette }
†{ Iowa, Iowa State }
†{ Maryland, California }
†{ Michigan St, Florida Atlantic }
†{ Missouri, Nevada }
†{ Arizona, New Mexico }
†{ Purdue, Oregon }
†{ Ohio State, Southern California }
†{ Syracuse, Penn State }
†{ Stanford, TCU }
†{ Texas, Arkansas }
†{ Texas Tech, SMU }
†{ Vanderbilt, Rice }
†{ Oklahoma, Washington }
†{ Louisville, Kansas St }
†{ West Virginia, Colorado }
†{ Connecticut, Baylor }
†{ Arizona St, Georgia }
†{ Boston College, UCF }
†{ Cincinnati, Miami-Ohio }
†{ Georgia Tech, Mississippi St }
†{ Kent St, UL Lafayette }
†{ Maryland, Eastern Michigan }
†{ Minnesota, Florida Atlantic }
†{ Missouri, Buffalo }
†{ North Carolina St, East Carolina }
†{ Ohio State, Troy }
†{ Penn State, Temple }
†{ Pittsburgh, Iowa }
†{ SMU, TCU }
†{ Stanford, San Jose St }
†{ Miami-Florida, Texas A&M }
†{ Toledo, Fresno St }
†{ Tulane, UL Monroe }
†{ Tulsa, New Mexico }
†{ Iowa State, UNLV }
†{ UTEP, New Mexico St }
†{ California, Colorado St }
†{ Florida St, Colorado }
†{ Virginia Tech, Nebraska }
†{ New Mexico, New Mexico St }
†{ North Carolina St, South Florida }
†{ Oklahoma, TCU }
†{ Oklahoma St, Troy }
†{ Texas A&M, Army }
†{ Toledo, Florida Intl }
†{ UCLA, Fresno St }
†{ Oregon St, Utah }
†{ Tennessee, Northern Illinois }
†{ Virginia Tech, Western Kentucky }
†{ Louisville, Memphis }
†{ Miami-Florida, UCF }
†{ Southern Miss, Boise St }
†{ West Virginia, Auburn }
†{ Army, Louisiana Tech }
†{ SMU, Navy }
†{ Alabama, Arkansas St }
†{ Arkansas, Tulsa }
†{ Tennessee, Wyoming }
†{ Illinois, Western Michigan }
†{ Georgia Tech, Georgia }
†{ Cincinnati, Hawaii }
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