## On Spectral Graph Drawing (2002)

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Venue: | Proc. 9th Inter. Computing and Combinatorics Conference (COCOON’03), LNCS 2697 |

Citations: | 44 - 10 self |

### BibTeX

@INPROCEEDINGS{Koren02onspectral,

author = {Yehuda Koren},

title = {On Spectral Graph Drawing},

booktitle = {Proc. 9th Inter. Computing and Combinatorics Conference (COCOON’03), LNCS 2697},

year = {2002},

pages = {496--508},

publisher = {Springer-Verlag}

}

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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. 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 aesthetically-motivated algorithm, which is much easier to understand and to implement than the standard numerical algorithms for computing eigenvectors.

### Citations

2756 | Normalized cuts and image segmentation
- Shi, Malik
- 2000
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Citation Context ...lacian. In fact, Mohar [11] claims that the Laplacian spectrum is more fundamental than this of the adjacency matrix. Related areas where the spectral approach has been popularized include clustering =-=[13]-=-, partitioning [12], and ordering [7]. However, these areas use discrete quantizations of the eigenvectors, unlike graph drawing, which employs the eigenvectors without any modification. Regarding thi... |

517 |
Partitioning sparse matrices with eigenvectors of graphs
- Pothen, Simon, et al.
- 1990
(Show Context)
Citation Context ...har [11] claims that the Laplacian spectrum is more fundamental than this of the adjacency matrix. Related areas where the spectral approach has been popularized include clustering [13], partitioning =-=[12]-=-, and ordering [7]. However, these areas use discrete quantizations of the eigenvectors, unlike graph drawing, which employs the eigenvectors without any modification. Regarding this aspect, it is mor... |

391 | How to draw a graph
- Tutte
- 1963
(Show Context)
Citation Context ...same location. Hence, to incorporate this aesthetic criterion into a graph drawing algorithm, it should be modified appropriately. Presumably the earliest graph drawing algorithm, formulated by Tutte =-=[15]-=-, is based on placing each node on the weighted centroid of its neighbors. To avoid the degenerate solution, Tutte arbitrarily chose a certain number of nodes to be anchors, i.e. he fixed their coordi... |

169 |
An r-dimensional quadratic placement algorithm
- Hall
- 1970
(Show Context)
Citation Context ... eigenvectors of certain matrices associated with the graph. To get some feeling, we provide results for three graphs in Fig. 1. This spectral approach is quite old, originating with the work of Hall =-=[6]-=- in 1970. However, since then it has not been used much. In fact, spectral graph drawing algorithms are almost absent in the graph-drawing literature (e.g., they are not mentioned in the two books [4,... |

155 | The Laplacian spectrum of graphs
- Mohar
- 1991
(Show Context)
Citation Context ...techniques utilize eigenvalues and eigenvectors of the adjacency matrix of the graph. More recently, the interest has shifted somewhat to the spectrum of the closely related Laplacian. In fact, Mohar =-=[11]-=- claims that the Laplacian spectrum is more fundamental than this of the adjacency matrix. Related areas where the spectral approach has been popularized include clustering [13], partitioning [12], an... |

64 | Ace: A fast multiscale eigenvectors computation for drawing huge graphs
- Koren, Carmel, et al.
(Show Context)
Citation Context ...st all other formulations result in an NP-hard problem, which can only be approximated. The second advantage is computation speed. Spectral drawings can be computed extremely fast as we have shown in =-=[9]-=-. This is very important because the amount of information to be visualized is constantly growing exponentially.(a) (b) (c) Fig. 1. Drawings obtained from the Laplacian eigenvectors. (a) The 4970 gra... |

57 |
Optimal linear labelings and eigenvalues of graphs
- Juvan, Mohar
- 1992
(Show Context)
Citation Context ...t the Laplacian spectrum is more fundamental than this of the adjacency matrix. Related areas where the spectral approach has been popularized include clustering [14], partitioning [12], and ordering =-=[7]-=-. However, these areas use discrete quantizations of the eigenvectors, unlike graph drawing, which employs the eigenvectors without any modification. Regarding this aspect, it is more fundamental to e... |

38 |
Drawing Graphs
- Kaufmann, Wagner
- 2001
(Show Context)
Citation Context ...e readability of the resulting layout, that is, the drawing algorithm’s ability to convey the meaning of the diagram quickly and clearly. To date, many approaches to graph drawing have been developed =-=[4, 8]-=-. There are many kinds of graph-drawing problems, such as drawing di-graphs, drawing planar graphs and others. Here we investigate the problem of drawing undirected graphs with straight-line edges. In... |

27 | Visualization of bibliographic networks with a reshaped landscape metaphor
- BRANDES, WILLHALM
- 2002
(Show Context)
Citation Context ...cy matrix instead of those of the Laplacian. E.g., the work of [10], which uses the adjacency matrix eigenvectors to draw molecular graphs. Recently, eigenvectors of a modified Laplacian were used in =-=[1]-=- for the visualization of bibliographic networks. In fact, for regular graphs of uniform degree deg, the eigenvectors of the Laplacian equal those of the adjacency matrix, but in a reversed order, bec... |

11 | Visualizing and Classifying Odors Using a Similarity Matrix
- Carmel, Koren, et al.
- 2003
(Show Context)
Citation Context ...tive importance in the related local scale. For example, consider Fig. 2, where we visualize 300 odors as measured by an electronic nose. Computation of the similarities between the odors is given in =-=[2]-=-. The odors are known to be classified into 30 groups, which determine the color of each odor in the figure. Figure 2(a) shows the visualization of the odors by the eigenvectors of the Laplacian. As c... |

11 |
Characterizing graph drawing with eigenvectors
- Pisanski, Taylor
(Show Context)
Citation Context ...h-drawing algorithm was that of Hall [6]; it uses the low eigenvectors of the Laplacian. Henceforth, we will refer to this method as the eigenprojection method. Later, a similar idea was suggested in =-=[13]-=-, where the results are shown to satisfy several desired aesthetic properties. A few other researchers utilize the top eigenvectors of the adjacency matrix instead of those of the Laplacian. For examp... |

6 |
Molecular Graphs, Point Groups and Fullerenes
- Manolopoulos, Fowler
- 1992
(Show Context)
Citation Context ...nceforth, we will refer to this method as the eigenprojection method. A few other researchers utilize the top eigenvectors of the adjacency matrix instead of those of the Laplacian. E.g., the work of =-=[10]-=-, which uses the adjacency matrix eigenvectors to draw molecular graphs. Recently, eigenvectors of a modified Laplacian were used in [1] for the visualization of bibliographic networks. In fact, for r... |

4 | A new feature extraction technique for electronic noses - Carmel, Levy, et al. - 2002 |

4 |
Spectral Graph Theory, CBMS Reg
- Chung
- 1997
(Show Context)
Citation Context ...4970, |E| = 7400. (b) The 4elt graph. |V | = 15606, |E| = 45878. (c) The Crack graph. |V | = 10240, |E| = 30380. Spectral methods have become standard techniques in algebraic graph theory; see, e.g., =-=[3]-=-. The most widely used techniques utilize eigenvalues and eigenvectors of the adjacency matrix of the graph. More recently, the interest has shifted somewhat to the spectrum of the closely related Lap... |

1 |
Solution of Constrained Quadratic Optimization Problems In this appendix we study a certain kind of constrained optimization problem, whose solution is a generalized eigenvector. We use two matrices: (1) A —ann×n real symmetric positive-semidefinite matri
- Webb, Recognition, et al.
- 1999
(Show Context)
Citation Context ...ructure would be better visualized with non-balanced aspect 1 The strategy to require no correlation between the axes is used in many other visualization techniques like Principal Components Analysis =-=[16]-=- and Classical Multidimensional Scaling [16].sratio. This can be achieved by replacing the minimization problem (1) by the following maximization problem max x Var(x) (4) given: x T Lx =1. Here we max... |