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73
A Fast MultiScale Method for Drawing Large Graphs
 JOURNAL OF GRAPH ALGORITHMS AND APPLICATIONS
, 2002
"... We present a multiscale layout algorithm for the aesthetic drawing of undirected graphs with straightline edges. The algorithm is extremely fast, and is capable of drawing graphs that are substantially larger than those we have encountered in prior work. For example, the paper contains a drawi ..."
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Cited by 92 (10 self)
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We present a multiscale layout algorithm for the aesthetic drawing of undirected graphs with straightline edges. The algorithm is extremely fast, and is capable of drawing graphs that are substantially larger than those we have encountered in prior work. For example, the paper contains a drawing of a graph with over 15,000 vertices. Also we achieve "nice" drawings of 1000 vertex graphs in about 1 second. The proposed algorithm embodies a new multiscale scheme for drawing graphs, which was motivated by the earlier multiscale algorithm of Hadany and Harel [HH99]. In principle, it could significantly improve the speed of essentially any forcedirected method (regardless of that method's ability of drawing weighted graphs or the continuity of its costfunction).
Streaming Meshes
, 2005
"... Recent years have seen an immense increase in the complexity of geometric data sets. Today's gigabytesized polygon models can no longer be completely loaded into the main memory of common desktop PCs. Unfortunately, current mesh formats do not account for this. They were designed years ago whe ..."
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Cited by 86 (18 self)
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Recent years have seen an immense increase in the complexity of geometric data sets. Today's gigabytesized polygon models can no longer be completely loaded into the main memory of common desktop PCs. Unfortunately, current mesh formats do not account for this. They were designed years ago when meshes were orders of magnitudes smaller. Using such formats to store large meshes is inefficient and unduly complicates all subsequent processing.
Graph Drawing by HighDimensional Embedding
 In GD02, LNCS
, 2002
"... We present a novel approach to the aesthetic drawing of undirected graphs. The method has two phases: first embed the graph in a very high dimension and then project it into the 2D plane using PCA. Experiments we have carried out show the ability of the method to draw graphs of 10 nodes in few seco ..."
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Cited by 73 (10 self)
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We present a novel approach to the aesthetic drawing of undirected graphs. The method has two phases: first embed the graph in a very high dimension and then project it into the 2D plane using PCA. Experiments we have carried out show the ability of the method to draw graphs of 10 nodes in few seconds. The new method appears to have several advantages over classical methods, including a significantly better running time, a useful inherent capability to exhibit the graph in various dimensions, and an effective means for interactive exploration of large graphs.
An Energy Model for Visual Graph Clustering
 Proceedings of the 11th International Symposium on Graph Drawing (GD 2003), LNCS 2912
, 2003
"... We introduce an energy model whose minimum energy drawings reveal the clusters of the drawn graph. Here a cluster is a set of nodes with many internal edges and few edges to nodes outside the set. The drawings of the bestknown force and energy models do not clearly show clusters for graphs whose ..."
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Cited by 60 (4 self)
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We introduce an energy model whose minimum energy drawings reveal the clusters of the drawn graph. Here a cluster is a set of nodes with many internal edges and few edges to nodes outside the set. The drawings of the bestknown force and energy models do not clearly show clusters for graphs whose diameter is small relative to the number of nodes. We formally characterize the minimum energy drawings of our energy model. This characterization shows in what sense the drawings separate clusters, and how the distance of separated clusters to the other nodes can be interpreted.
Growing wellconnected graphs
, 2006
"... The algebraic connectivity of a graph is the second smallest eigenvalue of the graph Laplacian, and is a measure of how wellconnected the graph is. We study the problem of adding edges (from a set of candidate edges) to a graph so as to maximize its algebraic connectivity. This is a difficult comb ..."
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Cited by 56 (1 self)
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The algebraic connectivity of a graph is the second smallest eigenvalue of the graph Laplacian, and is a measure of how wellconnected the graph is. We study the problem of adding edges (from a set of candidate edges) to a graph so as to maximize its algebraic connectivity. This is a difficult combinatorial optimization, so we seek a heuristic for approximately solving the problem. The standard convex relaxation of the problem can be expressed as a semidefinite program (SDP); for modest sized problems, this yields a cheaply computable upper bound on the optimal value, as well as a heuristic for choosing the edges to be added. We describe a new greedy heuristic for the problem. The heuristic is based on the Fiedler vector, and therefore can be applied to very large graphs.
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 51 (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.
Eigensolver Methods for Progressive Multidimensional Scaling of Large Data
, 2007
"... We present a novel samplingbased approximation technique for classical multidimensional scaling that yields an extremely fast layout algorithm suitable even for very large graphs. It produces layouts that compare favorably with other methods for drawing large graphs, and it is among the fastest me ..."
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Cited by 50 (9 self)
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We present a novel samplingbased approximation technique for classical multidimensional scaling that yields an extremely fast layout algorithm suitable even for very large graphs. It produces layouts that compare favorably with other methods for drawing large graphs, and it is among the fastest methods available. In addition, our approach allows for progressive computation, i.e. a rough approximation of the layout can be produced even faster, and then be refined until satisfaction.
A MultiScale Algorithm for the Linear Arrangement Problem
 Proc. 28th Inter. Workshop on GraphTheoretic Concepts in Computer Science (WG’02), LNCS 2573
, 2002
"... Finding a linear ordering of the vertices of a graph is a common problem arising in diverse applications. In this paper we present a lineartime algorithm for this problem, based on the multiscale paradigm. Experimental results are similar to those of the best known approaches, while the running ti ..."
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Cited by 31 (4 self)
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Finding a linear ordering of the vertices of a graph is a common problem arising in diverse applications. In this paper we present a lineartime algorithm for this problem, based on the multiscale paradigm. Experimental results are similar to those of the best known approaches, while the running time is significantly better, enabling it to deal with much larger graphs. The paper contains a general multiscale construction, which may be used for a broader range of ordering problems.
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 31 (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.