Topic
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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TL;DR: In this article, the adjacency, incidence and Laplacian matrices of a complex unit gain graph are studied and eigenvalue bounds for the adjACency matrix are derived.
101 citations
06 Apr 2018
TL;DR: In this article, the authors proposed a distributed graph multiplier operator based on shifted Chebyshev polynomials, whose recurrence relations make them readily amenable to distributed computation.
Abstract: Unions of graph multiplier operators are an important class of linear operators for processing signals defined on graphs. We present a novel method to efficiently distribute the application of these operators. The proposed method features approximations of the graph multipliers by shifted Chebyshev polynomials, whose recurrence relations make them readily amenable to distributed computation. We demonstrate how the proposed method can be applied to distributed processing tasks such as smoothing, denoising, inverse filtering, and semi-supervised classification, and show that the communication requirements of the method scale gracefully with the size of the network.
99 citations
01 Jun 2013
TL;DR: In this paper, the spectral partitioning algorithm is shown to be a constant factor approximation algorithm for finding a sparse cut if lk is a constant for some constant k. This bound is improved to O(k) l2/√lk by Cheeger's inequality.
Abstract: Let φ(G) be the minimum conductance of an undirected graph G, and let 0=λ1 ≤ λ2 ≤ ... ≤ λn ≤ 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k ≥ 2, [φ(G) = O(k) l2/√lk,] and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any $k$. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if lk is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to spectral algorithms for other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut.
98 citations
TL;DR: A new (constructive) characterization of tight wavelet frames on non-flat domains in both continuum setting, i.e. on manifolds, and discrete setting, and how fast tight wavelets frame transforms can be computed and how they can be effectively used to process graph data is introduced.
98 citations
Posted Content•
TL;DR: A large amount of robustness measures on simple, undirected and unweighted graphs are surveyed in order to offer a tool for network administrators to evaluate and improve the robustness of their network.
Abstract: Network robustness research aims at finding a measure to quantify network robustness. Once such a measure has been established, we will be able to compare networks, to improve existing networks and to design new networks that are able to continue to perform well when it is subject to failures or attacks. In this paper we survey a large amount of robustness measures on simple, undirected and unweighted graphs, in order to offer a tool for network administrators to evaluate and improve the robustness of their network. The measures discussed in this paper are based on the concepts of connectivity (including reliability polynomials), distance, betweenness and clustering. Some other measures are notions from spectral graph theory, more precisely, they are functions of the Laplacian eigenvalues. In addition to surveying these graph measures, the paper also contains a discussion of their functionality as a measure for topological network robustness.
97 citations