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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.


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Proceedings ArticleDOI
01 Dec 2006
TL;DR: It is shown that effective resistances provide bounds on the spectrum of the graph Laplacian matrix and the Dirichlet graph La Placian, which can be used to characterize the stability and convergence rate of several distributed algorithms that appeared in the literature.
Abstract: We introduce the concept of matrix-valued effective resistance for undirected matrix-weighted graphs. Effective resistances are defined to be the square blocks that appear in the diagonal of the inverse of the matrix-weighted Dirichlet graph Laplacian matrix. However, they can also be obtained from a "generalized" electrical network that is constructed from the graph, and for which currents, voltages and resistances take matrix values. Effective resistances play an important role in several problems related to distributed control and estimation. They appear in least-squares estimation problems in which one attempts to reconstruct global information from relative noisy measurements; as well as in motion control problems in which agents attempt to control their positions towards a desired formation, based on noisy local measurements. We show that in either of these problems, the effective resistances have a direct physical interpretation. We also show that effective resistances provide bounds on the spectrum of the graph Laplacian matrix and the Dirichlet graph Laplacian. These bounds can be used to characterize the stability and convergence rate of several distributed algorithms that appeared in the literature.

190 citations

Journal ArticleDOI
TL;DR: In this paper, three mathematical notions, namely nodal regions for eigenfunctions of the Laplacian, covering theory, and fiber products, were studied in the context of graph theory and spectral theory for graphs.
Abstract: We study three mathematical notions, that of nodal regions for eigenfunctions of the Laplacian, that of covering theory, and that of fiber products, in the context of graph theory and spectral theory for graphs. We formulate analogous notions and theorems for graphs and their eigenpairs. These techniques suggest new ways of studying problems related to spectral theory of graphs. We also perform some numerical experiments suggesting that the fiber product can yield graphs with small second eigenvalue.

188 citations

Proceedings ArticleDOI
14 Mar 2010
TL;DR: This paper proposes an online algorithm that uses concepts from unsupervised learning and spectral graph theory to infer this 'correct' graph structure, and allows each node to locally identify and adjust to the optimal operating point, and achieves good performance in all scenarios considered.
Abstract: Delay Tolerant Networks (DTN) are networks of self-organizing wireless nodes, where end-to-end connectivity is intermittent. In these networks, forwarding decisions are generally made using locally collected knowledge about node behavior (e.g., past contacts between nodes) to predict future contact opportunities. The use of complex network analysis has been recently suggested to perform this prediction task and improve the performance of DTN routing. Contacts seen in the past are aggregated to a social graph, and a variety of metrics (e.g., centrality and similarity) or algorithms (e.g., community detection) have been proposed to assess the utility of a node to deliver a content or bring it closer to the destination. In this paper, we argue that it is not so much the choice or sophistication of social metrics and algorithms that bears the most weight on performance, but rather the mapping from the mobility process generating contacts to the aggregated social graph. We first study two well-known DTN routing algorithms - SimBet and BubbleRap - that rely on such complex network analysis, and show that their performance heavily depends on how the mapping (contact aggregation) is performed. What is more, for a range of synthetic mobility models and real traces, we show that improved performances (up to a factor of 4 in terms of delivery ratio) are consistently achieved for a relatively narrow range of aggregation levels only, where the aggregated graph most closely reflects the underlying mobility structure. To this end, we propose an online algorithm that uses concepts from unsupervised learning and spectral graph theory to infer this 'correct' graph structure; this algorithm allows each node to locally identify and adjust to the optimal operating point, and achieves good performance in all scenarios considered.

187 citations

Journal ArticleDOI
TL;DR: A spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed, which provides a fundamental tradeoff between a signal's localization on a graph and in its spectral domain.
Abstract: The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed. Just as the classical result provides a tradeoff between signal localization in time and frequency, this result provides a fundamental tradeoff between a signal's localization on a graph and in its spectral domain. Using the eigenvectors of the graph Laplacian as a surrogate Fourier basis, quantitative definitions of graph and spectral “spreads” are given, and a complete characterization of the feasibility region of these two quantities is developed. In particular, the lower boundary of the region, referred to as the uncertainty curve, is shown to be achieved by eigenvectors associated with the smallest eigenvalues of an affine family of matrices. The convexity of the uncertainty curve allows it to be found to within e by a fast approximation algorithm requiring O(e-1/2) typically sparse eigenvalue evaluations. Closed-form expressions for the uncertainty curves for some special classes of graphs are derived, and an accurate analytical approximation for the expected uncertainty curve of Erd-s-Renyi random graphs is developed. These theoretical results are validated by numerical experiments, which also reveal an intriguing connection between diffusion processes on graphs and the uncertainty bounds.

185 citations

Journal ArticleDOI
TL;DR: In this article, a graph Laplacian regularizer is proposed for image denoising in the continuous domain, and the convergence of the regularizer to a continuous domain functional is analyzed.
Abstract: Inverse imaging problems are inherently underdetermined, and hence, it is important to employ appropriate image priors for regularization. One recent popular prior—the graph Laplacian regularizer—assumes that the target pixel patch is smooth with respect to an appropriately chosen graph. However, the mechanisms and implications of imposing the graph Laplacian regularizer on the original inverse problem are not well understood. To address this problem, in this paper, we interpret neighborhood graphs of pixel patches as discrete counterparts of Riemannian manifolds and perform analysis in the continuous domain, providing insights into several fundamental aspects of graph Laplacian regularization for image denoising. Specifically, we first show the convergence of the graph Laplacian regularizer to a continuous-domain functional, integrating a norm measured in a locally adaptive metric space. Focusing on image denoising, we derive an optimal metric space assuming non-local self-similarity of pixel patches, leading to an optimal graph Laplacian regularizer for denoising in the discrete domain. We then interpret graph Laplacian regularization as an anisotropic diffusion scheme to explain its behavior during iterations, e.g., its tendency to promote piecewise smooth signals under certain settings. To verify our analysis, an iterative image denoising algorithm is developed. Experimental results show that our algorithm performs competitively with state-of-the-art denoising methods, such as BM3D for natural images, and outperforms them significantly for piecewise smooth images.

180 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
202316
202236
202153
202086
201981