Sharp Bounds on Random Walk Eigenvalues via Spectral Embedding

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, NJW01]. 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 connected graph with n vertices and all k ≥ 2, the kth largest eigenvalue is at most 1 − Ω(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 − Ω(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.

The exclusion process mixes (almost) faster than independent particles

“…We verify now the claimed bound on r * ( ) of (5). For (n-vertex) regular graphs, P t (v, v) − 1 n (t + 1) −1/2 (e.g., [4,35]) for all t. Hence r * ( ) ≤ C( )(log n) 4 for some constant C depending only on . As…”

“…See also [36], Lemma 10.46, where G is not assumed to be infinite.) mix in [35] via the spectral measure is often better than the one obtained via t evolving−sets . (iv) For all r ∈ I , conditioned on Y 0 = r and S = 0 we have that c −i 5 (1 − Y i n−k+1 ) is a supermartingale (c 5 = c 5 (α, p), where α is as in the definition of (r)).…”

“…in the right-hand side of (64) is at most n −10 × n × n 2 log n (using max i β i (α) ≤ n −10 and t sp (ε) t rel log(n/ε) n 2 log(n/ε) for ε ≤ 1/2, e.g., [4,35]).…”

Estimating graph parameters with random walks

“…In particular, the bound of Lemma 3 is valid up to t rel . The following powerful result on the sum of return probabilities was established by Lyons and Oveis Gharan [17].…”

Three conjectures of Ostrander on digraph Laplacian eigenvectors

“…It is natural to expect that the shape of an eigenfunction of a Laplacian operator should somehow follow the shape of the underlying space; That is, you may be able to tell/predict the shape of space from some time-invariant data. Classical Fourier analysis provides deep understanding of signals over regular domains, to process graph-supported data we should accordingly develop spectral graph theory or the theory of graph Fourier transforms [3,9,12,21,27,39,41,46,50]. There are already quite some results and questions on the relationship between the oscillations of Laplacian eigenvector and the landscape of the underlying digraph, say those related to Courant's nodal line The Laplacian L M of a digraph M is a linear operator from a suitable linear subspace U of K V(M ) to some other subspace of K V(M ) such that…”