Here’s the beatpath graph for after the divisional round. Looks like we’ve got a couple of exciting games coming up.

Beatpaths is an automatic graphical powerranking system based off of only wins, losses, and who beat who.
 How does it work?
 How are beatloops resolved?
 How do the power rankings work?
 What's a beatfluke?
 Reviews and Quotes
 Contact me (through my music site) 
Pages
Categories
 2008 NFL Picks (40)
 Beatpath Tech (28)
 MLB (7)
 NBA (5)
 NCAA Football (7)
 News and Notices (18)
 NFL (384)
 Uncategorized (18)
Calender
Recent Comments
What does it take to get the beatgraphs charts also? I know it isn’t under this sites control, but I’m sure the author reads here. It would be real interesting to see how the other ways of breaking beatloops rank the teams.
Chris,
Moose explained here: http://beatpaths.com/?p=335#comment96585
I currently don’t have the time to post my versions of the graphs. Hopefully I’ll get a chance to later this week to catch up. I had been working on NBA and NHL versions leading into the holidays, but coming out and into the playoffs, my work has kept me too busy. I’ll drop a note here when playoff graphs are posted.
Thanks. I hadn’t read the comments in the other thread. I understand about being busy. I hadn’t even had the time to read all the threads. I just wanted to be sure you weren’t doing it because there wasn’t more data. I saw the comment about week 17 being final and thought maybe you had moved on to other things. No pressure intended.
@chris
You might find this PDF interesting:
http://www.ncsu.edu/crsc/reports/ftp/pdf/crsctr0619.pdf
It uses the Google Pagerank algorithm to create rankings from cyclic graphs without discarding data (e.g. loops).
Given some of your previous comments about changing cyclic graphs into directed acyclic graphs…
http://beatpaths.com/?p=317#comment94219
http://beatpaths.com/?p=317#comment94481
http://beatpaths.com/?p=330#comment96126
…I thought you might be able to make better sense of particulars of the math (stochastic irreducible matricies, etc).
It seems like you can plus in any statistic you like when weighting the links between nodes: winslosses (standard), final score difference (weighted), defensive or offensive performance, and so on.
I’d be interested to know your thoughts on it.
Also, those two scholars have another PDF on the topic: http://meyer.math.ncsu.edu/Meyer/PS_Files/SASGF08RankingPaper.pdf
@tom
Absolutely cool! It does not require breaking any cycles.
But talk about life coming full circle, when I was a HS student looking for my first “job”, I interviewed with a local college professor (of Chemistry if I recall right) and he wanted to know if I knew something of eigenvectors, which I didn’t other than knowing that the computer system at the HS had a library which calculated them, so I bluffed it. I got an offer but was too scared to take it, as I was certain to be over my head. Well, I did learn a little about them in college, but have forgotten most of that, except that they are intimately tied to matrix multiplication, and since the page rank is the limit of repeated multiplication of the matrix with itself, the eigenvector must capture that essence–and that all vaguely rings a bell. The other thing I recall is that the eignenvalue is related to treating the matrix as a series of simultaneous equations and sovling them for the relvant unknowns. And, clearly from the paper, the eigenvector is the ranking of each team, i.e. we treat the games matrix as a series of simultaneous equations telling us which team is probably the better than which other and the eigenvector is the relative ranking of the teams according to those probabilities.
The rest of the paper was an easier read and the definitions of stochastic and irreducible were easy enough to follow, but I’ll summarize them for those who don’t want to read the paper.
Stochastic means each row is like probability numbers, you want the total to sum to 1. For the page rank case that is easy to image, you have a 1/n chance of leaving a page by each of the n links leaving a page (assuming that you pick one at random). Making that make sense for football games is a little less obvious. However, it does map to the way the iterative version values duplicate paths in a beatloop.
The irreducible concept is stranger yet. It basically says, one artificially introduces a beatwin between every team, making the whole graph into 1 big beatloop. However, I’m certain that the idea behind it is to give values ways to flow all through the matrix and not get stuck at winless teams.
In any case, I don’t think eigenvalues are that hard to calculate especially only for a 32×32 matrix. There 2nd paper even looks like it has code one can use to get started, presuming that it isn’t Maple or some other specialized math package. It definitely looks like something worth playing with. Like Moose, my bosses will kill me if I play with it too much now, but it is something that I can do over time.
I’m very familiar with eigenvectors; I’ve used them in my day job. I don’t have time to read those papers thoroughly right now, but the approaches look very interesting.
I know they’re not explicitly breaking cycles there, but any ranking is implicitly breaking cycles at the part of the loop that goes up the ranking list.
@chris & doktarr
Awesome! Go team.
Can this type of thing be set up in Excel, or does it require MATLAB/Maple/Mathematica type programs?
What, you don’t have Matlab on both your home and work computers?
Excel doesn’t have a function to calculate eigenvectors, as far as I know. You could set something up by hand, but it would be a real bear.
Actually, I’m pretty sure we have MATLAB on all the computer lab workstations at my university. Needless to say, I have no idea how to use it.