Spectral Analysis of Laplacians of an Unweighted and Weighted Multidimensional Grid Graph -- Combinatorial versus Normalized and Random Walk Laplacians

Abstract

In this paper we generalise the results on eigenvalues and eigenvectors of unnormalized (combinatorial) Laplacian of two-dimensional grid presented by Edwards:2013 first to a grid graph of any dimension, and second also to other types of Laplacians, that is unoriented Laplacians, normalized Laplacians, and random walk Laplacians. While the closed-form or nearly closed form solutions to the eigenproblem of multidimensional grid graphs constitute a good test suit for spectral clustering algorithms for the case of no structure in the data, the multidimensional weighted grid graphs, presented also in this paper can serve as testbeds for these algorithms as graphs with some predefined cluster structure. The weights permit to simulate node clusters not perfectly separated from each other. This fact opens new possibilities for exploitation of closed-form or nearly closed form solutions eigenvectors and eigenvalues of graphs while testing and/or developing such algorithms and exploring their theoretical properties. Besides, the differences between the weighted and unweighted case allow for new insights into the nature of normalized and unnormalized Laplacians.