Sharp Bounds on Random Walk Eigenvalues via Spectral Embedding

TL;DR

This paper develops sharp bounds on the eigenvalues of random walk matrices using spectral embedding, leading to improved estimates on mixing times, return probabilities, and a new local algorithm for counting spanning trees.

Contribution

It introduces spectral embedding as a unifying framework for bounding all eigenvalues of graphs, extending previous results and providing new tools for analyzing random walks and graph parameters.

Findings

Eigenvalues are bounded by 1 - Omega(k^3/n^3) for all k >= 2

Bounds improve to 1 - Omega(k^2/n^2) for regular graphs

Spectral embedding yields sharper bounds on mixing times and return probabilities

Abstract

Spectral embedding of graphs uses the top k non-trivial eigenvectors of the random walk matrix to embed the graph into R^k. The primary use of this embedding has been for practical spectral clustering algorithms [SM00,NJW02]. Recently, spectral embedding was studied from a theoretical perspective to prove higher order variants of Cheeger's inequality [LOT12,LRTV12]. We use spectral embedding to provide a unifying framework for bounding all the eigenvalues of graphs. For example, we show that for any finite graph with n vertices and all k >= 2, the k-th largest eigenvalue is at most 1-Omega(k^3/n^3), which extends the only other such result known, which is for k=2 only and is due to [LO81]. This upper bound improves to 1-Omega(k^2/n^2) if the graph is regular. We generalize these results, and we provide sharp bounds on the spectral measure of various classes of graphs, including vertex-transitive graphs and infinite graphs, in terms of specific graph parameters like the volume growth. As a consequence, using the entire spectrum, we provide (improved) upper bounds on the return probabilities and mixing time of random walks with considerably shorter and more direct proofs. Our work introduces spectral embedding as a new tool in analyzing reversible Markov chains. Furthermore, building on [Lyo05], we design a local algorithm to approximate the number of spanning trees of massive graphs.