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Spectral Sparsification of Graphs: Theory and Algorithms
, 2013
"... Graph sparsification is the approximation of an arbitrary graph by a sparse graph. We explain what it means for one graph to be a spectral approximation of another and review the development of algorithms for spectral sparsification. In addition to being an interesting concept, spectral sparsificati ..."
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Cited by 7 (0 self)
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Graph sparsification is the approximation of an arbitrary graph by a sparse graph. We explain what it means for one graph to be a spectral approximation of another and review the development of algorithms for spectral sparsification. In addition to being an interesting concept, spectral
Consistency of spectral clustering
, 2004
"... Consistency is a key property of statistical algorithms, when the data is drawn from some underlying probability distribution. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of a popular family of spe ..."
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Cited by 572 (15 self)
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of spectral clustering algorithms, which cluster the data with the help of eigenvectors of graph Laplacian matrices. We show that one of the two of major classes of spectral clustering (normalized clustering) converges under some very general conditions, while the other (unnormalized), is only consistent
Laplacian eigenmaps and spectral techniques for embedding and clustering.
 Proceeding of Neural Information Processing Systems,
, 2001
"... Abstract Drawing on the correspondence between the graph Laplacian, the LaplaceBeltrami op erator on a manifold , and the connections to the heat equation , we propose a geometrically motivated algorithm for constructing a representation for data sampled from a low dimensional manifold embedded in ..."
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Cited by 668 (7 self)
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Abstract Drawing on the correspondence between the graph Laplacian, the LaplaceBeltrami op erator on a manifold , and the connections to the heat equation , we propose a geometrically motivated algorithm for constructing a representation for data sampled from a low dimensional manifold embedded
Spectral Sparsification in Dynamic Graph Streams
"... Abstract. We present a new bound relating edge connectivity in a simple, unweighted graph with effective resistance in the corresponding electrical network. The bound is tight. While we believe the bound is of independent interest, our work is motivated by the problem of constructing combinatorial a ..."
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Cited by 4 (2 self)
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and spectral sparsifiers of a graph, i.e., sparse, weighted subgraphs that preserve cut information (in the case of combinatorial sparsifiers) and additional spectral information (in the case of spectral sparsifiers). Recent results by Fung et al. (STOC 2011) and Spielman and Srivastava (SICOMP 2011) show
Finding community structure in networks using the eigenvectors of matrices
, 2006
"... We consider the problem of detecting communities or modules in networks, groups of vertices with a higherthanaverage density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as “modularity ” over possible div ..."
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Cited by 502 (0 self)
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divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a
Constructing LinearSized Spectral Sparsification in AlmostLinear Time
"... We present the first almostlinear time algorithm for constructing linearsized spectral sparsification for graphs. This improves all previous constructions of linearsized spectral sparsification, which requires Ω(n2) time [1], [2], [3]. A key ingredient in our algorithm is a novel combination of t ..."
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Cited by 1 (0 self)
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We present the first almostlinear time algorithm for constructing linearsized spectral sparsification for graphs. This improves all previous constructions of linearsized spectral sparsification, which requires Ω(n2) time [1], [2], [3]. A key ingredient in our algorithm is a novel combination
Spectral Sparsification and Restricted Invertibility
, 2010
"... In this thesis we prove the following two basic statements in linear algebra. Let B be an arbitrary n × m matrix where m ≥ n and suppose 0 < ε < 1 is given. 1. Spectral Sparsification. There is a nonnegative diagonal matrix Sm×m with at most ⌈n/ε2 ⌉ nonzero entries for which (1 − ε) 2BBT ≼ BSB ..."
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Cited by 11 (1 self)
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In this thesis we prove the following two basic statements in linear algebra. Let B be an arbitrary n × m matrix where m ≥ n and suppose 0 < ε < 1 is given. 1. Spectral Sparsification. There is a nonnegative diagonal matrix Sm×m with at most ⌈n/ε2 ⌉ nonzero entries for which (1 − ε) 2BBT
Consensus and cooperation in networked multiagent systems
 Proceedings of the IEEE
, 2007
"... Summary. This paper provides a theoretical framework for analysis of consensus algorithms for multiagent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, timedelays, and performance guarantees. An ove ..."
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Cited by 807 (4 self)
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. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis framework is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems
Results 1  10
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2,401