Week three is done (and I guess week four has started, but I’m slow…) and I worked on All Things Spectral. In short, to any districting plan, either as a collection of shapes or an abstract graph, we can associate a certain operator called the Laplacian which measures how geometrically similar the neighborhood of each point is to that of nearby points. These operators have eigenfunctions in the geometric case and eigenvectors in the graph case, and the spectrum tell you something about the shape of the original object. This week, I looked at a lot of eigenvalues. I’d like to write up a little primer on these spectral methods in more detail, but I’d like to get this post up without any further delay.
As in much of research mathematics, a lot of time was spent on things that didn’t work. In reality, it’s kind of surprising that we managed to find anything interesting at all in the brief four day work week. What ended up being kind of remarkable were the normalized laplacian eigenvalues of the graphs representing districting plans. In the short video below, there is a graph of the 200 smallest eigenvalues for various plans above an animated map of North Carolina. The black curve is the eigenvalues for the graph representing the entire state, the blue curve for the current (115th Congress) districting plan, the green curve for the districting plan which was in place in 2012 before being invalidated by the courts, and the red curve for the plan seen in the animation in the lower frame, which is generated by a random MCMC process. You can see that as the MCMC plan becomes more and more spider-y, the red curve moves downward, suggesting that this does measure something about district compactness!
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Zach is a PhD student at the University of Pennsylvania, Department of Computer and Information Science. His research interests include game theory and mechanism design, machine learning, and computing for the social sciences.