Statistical Foundation of Spectral Graph Theory

Spectral graph theory is undoubtedly the most favored graph data analysis technique, both in theory and practice. It has emerged as a versatile tool for a wide variety of applications including data mining, web search, quantum computing, computer vision, image segmentation, and among others. However, the way in which spectral graph theory is currently taught and practiced is rather mechanical, consisting of a series of matrix calculations that at first glance seem to have very little to do with statistics, thus posing a serious limitation to our understanding of graph problems from a statistical perspective. Our work is motivated by the following question: How can we develop a general statistical foundation of “spectral heuristics” that avoids the cookbook mechanical approach? A unified method is proposed that permits frequency analysis of graphs from a nonparametric perspective by viewing it as function estimation problem. We show that the proposed formalism incorporates seemingly unrelated spectral modeling tools (e.g., Laplacian, modularity, regularized Laplacian, diffusion map etc.) under a single general method, thus providing better fundamental understanding. It is the purpose of this paper to bridge the gap between two spectral graph modeling cultures: Statistical theory (based on nonparametric function approximation and smoothing methods) and Algorithmic computing (based on matrix theory and numerical linear algebra based techniques) to provide transparent and complementary insight into graph problems.