Currently, I have been reading Geometric Deep Learning: Going beyond Euclidean Data for a multitude of reasons. The reason I searched out this topic is that I knew one could use Convolutional Neural Networks to forecast a variety of things on graphs. For example, a Conv-Net on a graph could likely forecast node characteristics in the same way as Monte Carlo methods as shown here. The second reason is not related to economics, but knot theory. From what little I know, the central question in knot theory is to define invariants, such as knots that can be untied to another knot are the same (under some definition of an equivalence class of the invariants) and knots that are incapable are different invariants. Designing invariants that capture all these quantities is an open question, which is proved to be undecidable, but current invariants don’t even come close. I figured out later that the easier way to do this would come down to computing the similarity of two knot groups (in the algebraic sense) with the Wirtinger Presentation and wouldn’t require defining a Conv-Net on a graph. Then one could run an LSTM through each “word” (I.e word representation of that group), and use some sort of comparison tool to compare the state at the end of the first word to the state at the end of the second group.
When reading this document, I discovered that you can solve differential equations on graphs. Most notably, you can solve diffusion equations analytically on a graph, which is something I want to understand more concretely and why I am writing this blog post. A huge benefit of this is that this differential equation is obviously differentiable, unlike simulated diffusion on graphs that network econometricians use a lot. This means estimation is just an application of hill-climbing as opposed to time-consuming and inaccurate brute-force grid search. Additionally, the continuous time analogs represent an expected value of diffusion, so you don’t have to simulate diffusion 100s of times to get an expected value.
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