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Laplacian matrix

About: Laplacian matrix is a(n) research topic. Over the lifetime, 5716 publication(s) have been published within this topic receiving 167307 citation(s). The topic is also known as: Kirchhoff matrix & admittance matrix.

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Papers
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Open accessBook
Fan Chung1Institutions (1)
03 Dec 1996-
Abstract: Eigenvalues and the Laplacian of a graph Isoperimetric problems Diameters and eigenvalues Paths, flows, and routing Eigenvalues and quasi-randomness Expanders and explicit constructions Eigenvalues of symmetrical graphs Eigenvalues of subgraphs with boundary conditions Harnack inequalities Heat kernels Sobolev inequalities Advanced techniques for random walks on graphs Bibliography Index.

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Topics: Integral graph (63%), Spectral graph theory (61%), Laplacian matrix (60%) ...read more

6,908 Citations


Open accessJournal ArticleDOI: 10.1162/089976603321780317
Mikhail Belkin1, Partha Niyogi1Institutions (1)
01 Jun 2003-Neural Computation
Abstract: One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low-dimensional manifold embedded in a high-dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the high-dimensional data. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality-preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed.

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  • Figure 1: 2000 Random data points on the swiss roll.
    Figure 1: 2000 Random data points on the swiss roll.
  • Figure 2: Two-dimensional representations of the swiss roll data, for different values of the number of nearest neighbors N and the heat kernel parameter t. t = ∞ corresponds to the discrete weights.
    Figure 2: Two-dimensional representations of the swiss roll data, for different values of the number of nearest neighbors N and the heat kernel parameter t. t = ∞ corresponds to the discrete weights.
  • Figure 3: (Left) A horizontal and a vertical bar. (Middle) A two-dimensional representation of the set of all images using the Laplacian eigenmaps. (Right) The result of PCA using the first two principal directions to represent the data. Blue dots correspond to images of vertical bars, and plus signs correspond to images of horizontal bars.
    Figure 3: (Left) A horizontal and a vertical bar. (Middle) A two-dimensional representation of the set of all images using the Laplacian eigenmaps. (Right) The result of PCA using the first two principal directions to represent the data. Blue dots correspond to images of vertical bars, and plus signs correspond to images of horizontal bars.
  • Figure 4: The 300 most frequent words of the Brown corpus represented in the spectral domain.
    Figure 4: The 300 most frequent words of the Brown corpus represented in the spectral domain.
  • Figure 5: Fragments labeled by arrows: (left) infinitives of verbs, (middle) prepositions, and (right) mostly modal and auxiliary verbs. We see that syntactic structure is well preserved.
    Figure 5: Fragments labeled by arrows: (left) infinitives of verbs, (middle) prepositions, and (right) mostly modal and auxiliary verbs. We see that syntactic structure is well preserved.
  • + 2

Topics: Manifold alignment (63%), Diffusion map (62%), Spectral clustering (62%) ...read more

6,475 Citations


Open accessProceedings Article
Mikhail Belkin1, Partha Niyogi1Institutions (1)
03 Jan 2001-
Abstract: Drawing on the correspondence between the graph Laplacian, the Laplace-Beltrami operator 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 a higher dimensional space. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering. Several applications are considered.

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Topics: Spectral clustering (66%), Manifold alignment (62%), Laplacian matrix (62%) ...read more

4,224 Citations


Open accessJournal ArticleDOI: 10.1109/TAC.2004.834433
J.A. Fax1, Richard M. Murray2Institutions (2)
Abstract: We consider the problem of cooperation among a collection of vehicles performing a shared task using intervehicle communication to coordinate their actions. Tools from algebraic graph theory prove useful in modeling the communication network and relating its topology to formation stability. We prove a Nyquist criterion that uses the eigenvalues of the graph Laplacian matrix to determine the effect of the communication topology on formation stability. We also propose a method for decentralized information exchange between vehicles. This approach realizes a dynamical system that supplies each vehicle with a common reference to be used for cooperative motion. We prove a separation principle that decomposes formation stability into two components: Stability of this is achieved information flow for the given graph and stability of an individual vehicle for the given controller. The information flow can thus be rendered highly robust to changes in the graph, enabling tight formation control despite limitations in intervehicle communication capability.

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  • Figure 1: Hexagon stabilization: (a) initial condition and sensing topology, (b) sensed information only, (c) sensed and communicated information flow.
    Figure 1: Hexagon stabilization: (a) initial condition and sensing topology, (b) sensed information only, (c) sensed and communicated information flow.
  • Figure 2: Formation graph and Nyquist plot.
    Figure 2: Formation graph and Nyquist plot.
  • Figure 3 shows various eigenvalue regions for −L and the corresponding regions for −L−1. The region bounded by the solid line is the Perron disk in which all eigenvalues must lie. Its inverse is the LHP shifted by -0.5. The dashed region is a bound in the magnitude of the nonzero eigenvalues of L. It corresponds to a shifted circle on the right-hand side of Figure 3. Finally, the dash-dot line corresponds to a bound on the real component of the eigenvalues. The inverse of this bound corresponds to a circle which touches the origin. The shaded region represents the “desirable” region, in which the eigenvalues’ locations do not differ substantially from −1.
    Figure 3 shows various eigenvalue regions for −L and the corresponding regions for −L−1. The region bounded by the solid line is the Perron disk in which all eigenvalues must lie. Its inverse is the LHP shifted by -0.5. The dashed region is a bound in the magnitude of the nonzero eigenvalues of L. It corresponds to a shifted circle on the right-hand side of Figure 3. Finally, the dash-dot line corresponds to a bound on the real component of the eigenvalues. The inverse of this bound corresponds to a circle which touches the origin. The shaded region represents the “desirable” region, in which the eigenvalues’ locations do not differ substantially from −1.
  • Figure 3 shows various eigenvalue regions for −L and the corresponding regions for −L−1. The region bounded by the solid line is the Perron disk in which all eigenvalues must lie. Its inverse is the LHP shifted by -0.5. The dashed region is a bound in the magnitude of the nonzero eigenvalues of L. It corresponds to a shifted circle on the right-hand side of Figure 3. Finally, the dash-dot line corresponds to a bound on the real component of the eigenvalues. The inverse of this bound corresponds to a circle which touches the origin. The shaded region represents the “desirable” region, in which the eigenvalues’ locations do not differ substantially from −1.
    Figure 3 shows various eigenvalue regions for −L and the corresponding regions for −L−1. The region bounded by the solid line is the Perron disk in which all eigenvalues must lie. Its inverse is the LHP shifted by -0.5. The dashed region is a bound in the magnitude of the nonzero eigenvalues of L. It corresponds to a shifted circle on the right-hand side of Figure 3. Finally, the dash-dot line corresponds to a bound on the real component of the eigenvalues. The inverse of this bound corresponds to a circle which touches the origin. The shaded region represents the “desirable” region, in which the eigenvalues’ locations do not differ substantially from −1.
  • Table 1: Sample Graphs, Spectra, and Nyquist Locations.
    Table 1: Sample Graphs, Spectra, and Nyquist Locations.
  • + 10

Topics: Graph (abstract data type) (59%), Laplacian matrix (58%), Graph theory (57%) ...read more

4,093 Citations


Open accessJournal ArticleDOI: 10.1103/PHYSREVE.74.036104
Mark Newman1Institutions (1)
11 Sep 2006-Physical Review E
Abstract: We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average 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 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 number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.

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Topics: Modularity (networks) (61%), Adjacency matrix (60%), Graph partition (60%) ...read more

4,062 Citations


Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202210
2021406
2020543
2019448
2018439
2017496

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Topic's top 5 most impactful authors

Edwin R. Hancock

55 papers, 1.4K citations

Gene Cheung

33 papers, 581 citations

Pascal Frossard

16 papers, 532 citations

Ravindra B. Bapat

15 papers, 607 citations

Feiping Nie

13 papers, 1K citations

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