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ACE: A Fast Multiscale Eigenvector Computation for Drawing Huge Graphs
, 2002
"... We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE finds an optimal drawing by minimizing a quadratic energy function due to Hall, using a novel algebraic multigrid technique. The algorithm exhibits ..."
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Cited by 64 (13 self)
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We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE finds an optimal drawing by minimizing a quadratic energy function due to Hall, using a novel algebraic multigrid technique. The algorithm exhibits an improvement of something like two orders of magnitude over the fastest algorithms we are aware of; it draws graphs of a million nodes in less than a minute. Moreover, the algorithm can deal with more general entities, such as graphs with masses and negative weights (to be defined in the text), and it appears to be applicable outside of graph drawing too.
On Spectral Graph Drawing
 Proc. 9th Inter. Computing and Combinatorics Conference (COCOON’03), LNCS 2697
, 2002
"... The spectral approach for graph visualization computes the layout of a graph using certain eigenvectors of related matrices. Some important advantages of this approach are an ability to compute optimal layouts (according to specific requirements) and a very rapid computation time. In this paper we e ..."
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Cited by 44 (10 self)
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The spectral approach for graph visualization computes the layout of a graph using certain eigenvectors of related matrices. Some important advantages of this approach are an ability to compute optimal layouts (according to specific requirements) and a very rapid computation time. In this paper we explore spectral visualization techniques and study their properties. We present a novel view of the spectral approach, which provides a direct link between eigenvectors and the aesthetic properties of the layout. In addition, we present a new formulation of the spectral drawing method with some aesthetic advantages. This formulation is accompanied by an aestheticallymotivated algorithm, which is much easier to understand and to implement than the standard numerical algorithms for computing eigenvectors.
Spectral Methods for Mesh Processing and Analysis
 EUROGRAPHICS 2007
, 2007
"... Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projections derived from appropriately defined mesh operators to carry out desired tasks. Early works in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis ..."
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Cited by 18 (0 self)
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Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projections derived from appropriately defined mesh operators to carry out desired tasks. Early works in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis of mesh geometry based on a combinatorial Laplacian aids our understanding of the lowpass filtering approach to mesh smoothing. Over the past ten years or so, the list of applications in the area of geometry processing which utilize the eigenstructures of a variety of mesh operators in different manners have been growing steadily. Many works presented so far draw parallels from developments in fields such as graph theory, computer vision, machine learning, graph drawing, numerical linear algebra, and highperformance computing. This stateoftheart report aims to provide a comprehensive survey on the spectral approach, focusing on its power and versatility in solving geometry processing problems and attempting to bridge the gap between relevant research in computer graphics and other fields. Necessary theoretical background will be provided and existing works will be classified according to different criteria — the operators or eigenstructures employed, application domains, or the dimensionality of the spectral embeddings used — and described in adequate length. Finally, despite much empirical success, there still remain many open questions pertaining to the spectral approach, which we will discuss in the report as well.
Drawing graphs by eigenvectors: Theory and practice
 Computers and Mathematics with Applications
, 2005
"... Abstract. The spectral approach for graph visualization computes the layout of a graph using certain eigenvectors of related matrices. Some important advantages of this approach are an ability to compute optimal layouts (according to specific requirements) and a very rapid computation time. In this ..."
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Cited by 15 (1 self)
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Abstract. The spectral approach for graph visualization computes the layout of a graph using certain eigenvectors of related matrices. Some important advantages of this approach are an ability to compute optimal layouts (according to specific requirements) and a very rapid computation time. In this paper we explore spectral visualization techniques and study their properties from different points of view. We also suggest a novel algorithm for calculating spectral layouts resulting in an extremely fast computation by optimizing the layout within a small vector space.
Spectral Mesh Processing
"... Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projections derived from appropriately defined mesh operators to carry out desired tasks. Early work in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis ..."
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Cited by 10 (1 self)
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Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projections derived from appropriately defined mesh operators to carry out desired tasks. Early work in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis of mesh geometry based on a combinatorial Laplacian aids our understanding of the lowpass filtering approach to mesh smoothing. Over the past fifteen years, the list of applications in the area of geometry processing which utilize the eigenstructures of a variety of mesh operators in different manners have been growing steadily. Many works presented so far draw parallels from developments in fields such as graph theory, computer vision, machine learning, graph drawing, numerical linear algebra, and highperformance computing. This paper aims to provide a comprehensive survey on the spectral approach, focusing on its power and versatility in solving geometry processing problems and attempting to bridge the gap between relevant research in computer graphics and other fields. Necessary theoretical background is provided. Existing works covered are classified according to different criteria: the operators or eigenstructures employed, application domains, or the dimensionality of the spectral embeddings used. Despite much empirical success, there still remain many open questions pertaining to the spectral approach. These are discussed as we conclude the survey and provide our perspective on possible future research.
An Improved Springbased Graph Embedding Algorithm and LayoutShow: a Java Environment for Graph Drawing
, 1999
"... Algorithms based on forcedirected placement and virtual physical models have become one of the most effective techniques for drawing undirected graphs. Springbased algorithms that are the subject of this thesis are one type of forcedirected algorithms. Spring algorithms are simple. They produce g ..."
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Cited by 4 (0 self)
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Algorithms based on forcedirected placement and virtual physical models have become one of the most effective techniques for drawing undirected graphs. Springbased algorithms that are the subject of this thesis are one type of forcedirected algorithms. Spring algorithms are simple. They produce graphs with approximately uniform edge lengths, distribute nodes reasonably well, and preserve graph symmetries. A problem with these algorithms is that depending on their initial layout, it is possible that they find undesirable drawings associated with some local minimum criteria. In addition, it has always been a challenge to determine when a layout is stable in order to stop the algorithm. In this thesis, we develop a simple but effective cost function that can determine a node layout quality as well as the quality of the entire graph layout during the execution of a Spring algorithm. We use this cost function for producing the initial layout of the algorithm, for helping nodes move out ...
GraphDrawing Algorithms Geometries Versus Molecular Mechanics in Fullerenes
, 1996
"... The algorithms of KamadaKawai (KK) and FruchtermanReingold (FR) have been recently generalized (Pisanski et al., Croat. Chem. Acta 68 (1995) 283.) in order to draw molecular graphs in threedimensional space. The quality of KK and FR geometries is studied here by comparing them with the molecular ..."
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Cited by 3 (2 self)
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The algorithms of KamadaKawai (KK) and FruchtermanReingold (FR) have been recently generalized (Pisanski et al., Croat. Chem. Acta 68 (1995) 283.) in order to draw molecular graphs in threedimensional space. The quality of KK and FR geometries is studied here by comparing them with the molecular mechanics (MM) and the adjacency matrix eigenvectors (AME) algorithm geometries.
LayoutShow: a Signed Applet/Application for Graph Drawing and Experimentation (System Demonstration)
 Graph Drawing: Proc 7th International Symposium GD’99
, 1999
"... . LayoutShow is a Javabased multithreaded applet/application for experimentation with graph drawing algorithms, particularly, forcedirected algorithms. The motivation behind the development of this software is the lack of features that would help to experiment, and as a result, understand the ..."
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. LayoutShow is a Javabased multithreaded applet/application for experimentation with graph drawing algorithms, particularly, forcedirected algorithms. The motivation behind the development of this software is the lack of features that would help to experiment, and as a result, understand the behavior of forcedirected algorithms in the existing graph drawing software. Some of these features include smooth nodebased and iterationbased animations, display of runningtime and iteration counts, and variety of initial layout algorithms. LayoutShow supports a number of forcedirected graph drawing algorithms as well as layouts based on eigenvectors. Nodebased and iterationbased animations have been implemented. In addition, the software provides some algorithms for producing nonrandom initial layouts for forcedirected algorithms. File I/O using GML le format has been implemented. Furthermore, users of LayoutShow applet can choose to perform local le I/O since LayoutS...
Spectral Graph Drawing: A Survey
, 2004
"... We give an overview of graph drawing techniques as discussed by Godsil/Royle, Koren and Pisanski/ShaweTaylor. These techniques employ the eigenvectors of the Laplacian matrix to create an ndimensional representation of an undirected graph. ..."
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We give an overview of graph drawing techniques as discussed by Godsil/Royle, Koren and Pisanski/ShaweTaylor. These techniques employ the eigenvectors of the Laplacian matrix to create an ndimensional representation of an undirected graph.
ABSTRACT A HighDimensional Approach to
"... illustrating relations between objects. Interactive graph layout is often important since it is difficult to statically lay out complex graphs such as general undirected graphs. In this paper, we propose a novel approach to interactive layout of general undirected graphs. The basic idea of our appro ..."
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illustrating relations between objects. Interactive graph layout is often important since it is difficult to statically lay out complex graphs such as general undirected graphs. In this paper, we propose a novel approach to interactive layout of general undirected graphs. The basic idea of our approach is to use static graph layouts in highdimensional spaces to dynamically find twodimensional layouts according to user interaction. The resulting method that we present exhibits the following two characteristics: (1) it efficiently updates twodimensional graph layouts during user interaction; (2) it follows users ’ node dragging operations by actively moving other closely related nodes. Our method adopts eigenvectorbased multidimensional scaling to compute highdimensional graph layouts, and performs constraint satisfaction to determine appropriate twodimensional planes onto which the highdimensional layouts will be projected. Keywords information visualization, interactive graph layout, general undirected graphs, multidimensional scaling 1.