Minicourse: Introduction to Spectral Graph Theory and Applications

We will discuss the basics of spectral graph theory, which studies random walks on graphs, and related objects such as the Laplacian and its eigenfunctions, on a weighted graph. This can be thought as a discrete analogue to spectral geometry, albeit the geometry of graphs and their discrete nature gives rise to issues not generally considered in the continuous, smooth case of Riemannian manifolds. We will present some classical connections between properties of the random walks and the geometry of the graph. We will then discuss disparate applications: the solution of sparse linear systems by multiscale methods based on random walks; analysis of large data sets (images, web pages, etc...), in particular how to find systems of coordinates on them, performing dimensionality reduction, and performing multiscale analysis on them; tasks in learning, such as spectral clustering, classification and regression on data sets.

References

• F. Chung's book "Spectral Graph Theory"
• D. Spielman notes for his course at Yale
• several papers on specific applications, dependent on the attendant's interests.

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