Week 3 of 2019 FBS rankings (09/17/2019) demonstrates one situation with a need for this:
Wisconsin has not given up a single point--therefore as the Markov Chain proceeds, the probability of ending up at Wisconsin approaches 1, and the probability of all other teams approaches 0. In essence, Wisconsin should be #1, and all other teams should be tied for #2. However, the algorithm stops before this because of the tolerance. This might be unavoidable, as we have to stop calculating somewhere.
Another interesting point about this ranking calculation is App St, Charlotte, Rutgers, and UMass. Together, these 4 teams have not scored a single point against any other team in FBS. Rutgers was the only to play a team outside of the 4 (Iowa), in which they got shut out. These 4 teams were given a rating of 0, but only because we read the ratings from row 1 of the matrix (in this case Air Force). It is true that these ratings will also approach zero as the Markov Chain proceeds, but for a different reason than the rest of the teams in FBS.
Both of these situations arise because the graph is not strongly connected yet. It is not guaranteed to be connected, as a team could theoretically go the entire season without scoring a point or without giving up a point. This should be considered when deciding on an end condition.
jbadlato
commented
5 years ago
Week 3 of 2019 FBS rankings (09/17/2019) demonstrates one situation with a need for this:
Wisconsin has not given up a single point--therefore as the Markov Chain proceeds, the probability of ending up at Wisconsin approaches 1, and the probability of all other teams approaches 0. In essence, Wisconsin should be #1, and all other teams should be tied for #2. However, the algorithm stops before this because of the tolerance. This might be unavoidable, as we have to stop calculating somewhere.
Another interesting point about this ranking calculation is App St, Charlotte, Rutgers, and UMass. Together, these 4 teams have not scored a single point against any other team in FBS. Rutgers was the only to play a team outside of the 4 (Iowa), in which they got shut out. These 4 teams were given a rating of 0, but only because we read the ratings from row 1 of the matrix (in this case Air Force). It is true that these ratings will also approach zero as the Markov Chain proceeds, but for a different reason than the rest of the teams in FBS.
Both of these situations arise because the graph is not strongly connected yet. It is not guaranteed to be connected, as a team could theoretically go the entire season without scoring a point or without giving up a point. This should be considered when deciding on an end condition.