Speaker: Lek-Heng Lim, University of Chicago

Title: Incompressive fluid flows, time-dependent electric fields, and the top 100 Netflix movies

Abstract:

In physics, one learns early on that incompressible fluid flow has a potential flow component and a vorticity component. Also, essentially the same principle applies to an electrodynamic field, which decomposes into a space derivative of the electrostatic scalar potential and a time derivative of the magnetic vector potential. These are special cases of the Helmholtz decomposition of a vector field. For domains that are not simply connected, one introduces an additional harmonic component to ensure uniqueness. Somewhat unexpectedly, a discrete version of this applies to rank aggregation.

The basic idea is that when a large number of viewers rate a (small fraction of a) large number of movies, the problem behaves like water flowing in a pipe. Like fluid flow, the average rating can be broken up into three components: a part that flows from a high-pressured region to low-pressured one, a part that swirls around, and a part that loops around the entire pipe (asumming that we join the two ends of the pipe into a donut-shaped tube).

What does this tell you about ranking movies? The first part allows you to rank all the movies in the Netflix inventory according to preferences of all viewers. The second and third parts tell you how much disagreement there is among the viewers and therefore tell you how reliable the ranking given by the first part is. If there's a lot of disagreement, then it says that there is essentially no consensus among the viewers; but otherwise there is.

There are interesting things that one can say about the second and third parts too. For example, the second component would tell you whether there's disagreement in the ranking of three very bad movies (or three very good movies) whereas the third component would tell you whether there's disagreement in the ranking of a good, an average, and a bad movie.

Of course movies and viewers are just placeholder here. This technique may be applied whenever we have a problem requiring some ranking of alternatives by a collection of voters. Alternatives could be books, colleges, restaurants, websites, politicians, TV show contestants, genes, events, etc.

This is joint work with Xiaoye Jiang, Yuan Yao, and Yinyu Ye.