Lecture : Local Spectral Methods ( 2 of 4 )

Last time, we talked about spectral ranking methods, and we observed that they can be computed as eigenvectors of certain matrices or as the solution to systems of linear equations of certain constraint matrices. These computations can be performed with a black box solver, but they can also be done with specialized solvers that take advantage of the special structure of these matrices. (For example, a vanilla spectral ranking method, e.g., one with a preference vector ~v that is an all-ones vector ~1, has a large eigenvalue gap, and this means that one can obtain a good solution with just a few steps of a simple iterative method.) As you can imagine, this is a large topic. Here, we will focus in particular on how to solve for these spectral rankings in the particular case when the preference vector ~v is has small support, i.e., when it has its mass localized on a small seed set of nodes. This is a particularly important use case, and the methods developed for it are useful much more generally.