Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Accelerated graph-based spectral polynomial filters

Abstract: Graph-based spectral denoising is a low-pass filtering using the eigendecomposition of the graph Laplacian matrix of a noisy signal. Polynomial filtering avoids costly computation of the eigendecomposition by projections onto suitable Krylov subspaces. Polynomial filters can be based, e.g., on the bilateral and guided filters. We propose constructing accelerated polynomial filters by running flexible Krylov subspace based linear and eigenvalue solvers such as the Block Locally Optimal Preconditioned Conjugate Gradient (LOBPCG) method.

Bounding robustness in complex networks under topological changes through majorization techniques

Abstract: Measuring robustness is a fundamental task for analyzing the structure of complex networks. Indeed, several approaches to capture the robustness properties of a network have been proposed. In this paper we focus on spectral graph theory where robustness is measured by means of a graph invariant called Kirchhoff index, expressed in terms of eigenvalues of the Laplacian matrix associated to a graph. This graph metric is highly informative as a robustness indicator for several realworld networks that can be modeled as graphs. We discuss a methodology aimed at obtaining some new and tighter bounds of this graph invariant when links are added or removed. We take advantage of real analysis techniques, based on majorization theory and optimization of functions which preserve the majorization order (Schurconvex functions). Applications to simulated graphs show the effectiveness of our bounds, also in providing meaningful insights with respect to the results obtained in the literature.

A spectral graph theoretic study of predator-prey networks

Abstract: Predator-prey networks originating from different aqueous and terrestrial environments are compared to assess if the difference in environments of these networks produce any significant difference in the structure of such predator-prey networks. Spectral graph theory is used firstly to discriminate between the structure of such predator-prey networks originating from aqueous and terrestrial environments and secondly to establish that the difference observed in the structure of networks originating from these two environments are precisely due to the way edges are oriented in these networks and are not a property of random networks.We use random projections in $\mathbb{R^2}$ and $\mathbb{R^3}$ of weighted spectral distribution (WSD) of the networks belonging to the two classes viz. aqueous and terrestrial to differentiate between the structure of these networks. The spectral theory of graph non-randomness and relative non-randomness is used to establish the deviation of structure of these networks from having a topology similar to random networks.We thus establish the absence of a universal structural pattern across predator-prey networks originating from different environments.

Geary’s c and Spectral Graph Theory

Abstract: Spatial autocorrelation, of which Geary’s c has traditionally been a popular measure, is fundamental to spatial science. This paper provides a new perspective on Geary’s c. We discuss this using concepts from spectral graph theory/linear algebraic graph theory. More precisely, we provide three types of representations for it: (a) graph Laplacian representation, (b) graph Fourier transform representation, and (c) Pearson’s correlation coefficient representation. Subsequently, we illustrate that the spatial autocorrelation measured by Geary’s c is positive (resp. negative) if spatially smoother (resp. less smooth) graph Laplacian eigenvectors are dominant. Finally, based on our analysis, we provide a recommendation for applied studies.