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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 Article
Rie K. Ando1, Tong Zhang2
04 Dec 2006
TL;DR: This work considers a general form of transductive learning on graphs with Laplacian regularization, and derive margin-based generalization bounds using appropriate geometric properties of the graph, and suggests a limitation of the standard degree-based normalization.
Abstract: We consider a general form of transductive learning on graphs with Laplacian regularization, and derive margin-based generalization bounds using appropriate geometric properties of the graph. We use this analysis to obtain a better understanding of the role of normalization of the graph Laplacian matrix as well as the effect of dimension reduction. The results suggest a limitation of the standard degree-based normalization. We propose a remedy from our analysis and demonstrate empirically that the remedy leads to improved classification performance.

140 citations

01 Mar 2009
TL;DR: This work shows how power grid vulnerability analysis can be studied as a bilevel mixed integer nonlinear programming problem, and reveals a special structure in the formulation that can be exploited to avoid nonlinearity and approximate the original problem as a pure combinatorial problem.
Abstract: Identifying small groups of lines, whose removal would cause a severe blackout, is critical for the secure operation of the electric power grid. We show how power grid vulnerability analysis can be studied as a mixed integer nonlinear programming (minlp) problem. Our analysis reveals a special structure in the formulation that can be exploited to avoid nonlinearity and approximate the original problem as a pure combinatorial problem. The key new observation behind our analysis is the correspondence between the Jacobian matrix (a representation of the feasibility boundary of the equations that describe the flow of power in the network) and the Laplacian matrix in spectral graph theory (a representation of the graph of the power grid). The reduced combinatorial problem is known as the network inhibition problem, for which we present a mixed integer linear programming formulation. Our experiments on benchmark power grids show that the reduced combinatorial model provides an accurate approximation, to enable vulnerability analyses of real-sized problems with more than 10,000 power lines.

139 citations

Journal ArticleDOI
TL;DR: In this article, power grid vulnerability analysis can be studied as a bilevel mixed integer nonlinear programming problem, which can be exploited to avoid nonlinearity and approximate the original problem as a pure combinatorial problem.
Abstract: Identifying small groups of lines, whose removal would cause a severe blackout, is critical for the secure operation of the electric power grid. We show how power grid vulnerability analysis can be studied as a bilevel mixed integer nonlinear programming problem. Our analysis reveals a special structure in the formulation that can be exploited to avoid nonlinearity and approximate the original problem as a pure combinatorial problem. The key new observation behind our analysis is the correspondence between the Jacobian matrix (a representation of the feasibility boundary of the equations that describe the flow of power in the network) and the Laplacian matrix in spectral graph theory (a representation of the graph of the power grid). The reduced combinatorial problem is known as the network inhibition problem, for which we present a mixed integer linear programming formulation. Our experiments on benchmark power grids show that the reduced combinatorial model provides an accurate approximation, to enable vulnerability analyses of real-sized problems with more than 16,520 power lines.

138 citations

Journal ArticleDOI
TL;DR: This framework generalizes the recently proposed spectral graph wavelet transform (SGWT) and proposes a design for multislice graphs that is based on the higher-order singular value decomposition (HOSVD), a powerful tool from multilinear algebra.
Abstract: We present a framework for the design of wavelet transforms tailored to data defined on multislice graphs (i.e., multiplex or dynamic graphs). Graphs with multiple types of interactions are ubiquitous in real life, motivating the extension of wavelets to these complex domains. Our framework generalizes the recently proposed spectral graph wavelet transform (SGWT) [D. Hammond, P. Vandergheynst, and R. Gribonval, “Wavelets on Graphs via Spectral Graph Theory,” Appl. Comput. Harmon. Anal., vol. 30, pp. 129-150, Mar. 2011], which is designed in the spectral (frequency) domain of an arbitrary finite weighted graph. We extend the SGWT to form a tight frame, which conserves energy in the wavelet domain, and define the relationship between conventional and spectral graph wavelets. We then propose a design for multislice graphs that is based on the higher-order singular value decomposition (HOSVD), a powerful tool from multilinear algebra. In particular, the multiple adjacency matrices are stacked to form a tensor and the HOSVD decomposition provides information about its third-order structure, analogous to that provided by matrix factorizations. We obtain a set of “eigennetworks” and from these graph wavelets, which exploit the variability across the graphs. We demonstrate the feasibility of our method 1) by capturing different orientations of a gray-scale image and 2) by decomposing brain signals from functional magnetic resonance imaging. We show its effectiveness to identify variability across graph edges and provide meaningful decompositions.

136 citations

Posted Content
TL;DR: A novel metric learning method to evaluate distance between graphs that leverages the power of convolutional neural networks, while exploiting concepts from spectral graph theory to allow these operations on irregular graphs to be exploited.
Abstract: Evaluating similarity between graphs is of major importance in several computer vision and pattern recognition problems, where graph representations are often used to model objects or interactions between elements. The choice of a distance or similarity metric is, however, not trivial and can be highly dependent on the application at hand. In this work, we propose a novel metric learning method to evaluate distance between graphs that leverages the power of convolutional neural networks, while exploiting concepts from spectral graph theory to allow these operations on irregular graphs. We demonstrate the potential of our method in the field of connectomics, where neuronal pathways or functional connections between brain regions are commonly modelled as graphs. In this problem, the definition of an appropriate graph similarity function is critical to unveil patterns of disruptions associated with certain brain disorders. Experimental results on the ABIDE dataset show that our method can learn a graph similarity metric tailored for a clinical application, improving the performance of a simple k-nn classifier by 11.9% compared to a traditional distance metric.

133 citations


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