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## GMap: Visualizing Graphs and Clusters as Maps (2009)

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Citations: | 32 - 21 self |

### Citations

4590 |
Time-delayed self-organizing maps
- Kangas
- 1990
(Show Context)
Citation Context ...s Map of Online Communities [4]. While most such maps are generated in an ad hoc manner and are not strictly based on underlying data, they are often visually appealing. In self-organizing maps (SOM) =-=[19]-=- an unsupervised learning algorithm places objects on a two-dimensional grid such that similar objects are close to each other. Similar to our GMap algorithm, SOM also uses a map metaphor to visualize... |

3316 |
Principal Component Analysis
- Jolliffe
- 2002
(Show Context)
Citation Context ... to some notion of the distance between a pair of vertices. The first step in our GMap algorithm is to embed the graph in the plane. Possible embedding algorithms include principal component analysis =-=[17]-=-, multidimensional scaling (MDS) [20], a force-directed algorithm [12], or non-linear dimensionality reduction such as LLE [29] and Isomap [31]. The second step is a cluster analysis of the underlying... |

2420 |
A Global Geometric Framework for Nonlinear Dimensionality
- Tenenbaum, Silva, et al.
- 2000
(Show Context)
Citation Context ...mbedding algorithms include principal component analysis [17], multidimensional scaling (MDS) [20], a force-directed algorithm [12], or non-linear dimensionality reduction such as LLE [29] and Isomap =-=[31]-=-. The second step is a cluster analysis of the underlying graph or the embedded pointset from step one. In this step, it is important to match the clustering algorithm to the embedding algorithm. For ... |

2382 | Nonlinear Dimensionality Reduction by Locally Linear Embedding
- Roweis
- 2000
(Show Context)
Citation Context ...lane. Possible embedding algorithms include principal component analysis [17], multidimensional scaling (MDS) [20], a force-directed algorithm [12], or non-linear dimensionality reduction such as LLE =-=[29]-=- and Isomap [31]. The second step is a cluster analysis of the underlying graph or the embedded pointset from step one. In this step, it is important to match the clustering algorithm to the embedding... |

1341 | Least squares quantization in pcm
- Lloyd
- 1982
(Show Context)
Citation Context ...graph or the embedded pointset from step one. In this step, it is important to match the clustering algorithm to the embedding algorithm. For example, a geometric clustering algorithm such as k-means =-=[24]-=- may be suitable for an embedding derived from MDS, as the latter tends to place similar vertices in the same geometric region with good separation between clusters. On the other hand, with an embeddi... |

947 |
Modularity and community structure in networks
- Newman
(Show Context)
Citation Context ...e similar vertices in the same geometric region with good separation between clusters. On the other hand, with an embedding derived from a force-directed algorithm [12], a modularity based clustering =-=[26]-=- could be a better fit. The two algorithms are strongly related [27] and therefore we can expect vertices that are in the same cluster to also be physically close to each other in the embedding. In th... |

741 | From few to many: Illumination cone models for face recognition under variable lighting and pose
- Georghiades, Kriegman
- 2001
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Citation Context ... Artificial points are inserted along the spline edges. When GMap is applied, distant items are connected by “bridges” along the edges. Figure 5 shows a group of 55 photos from the Yale Face Database =-=[7]-=-, representing portraits using different expressions and lightings. The photos are embedded in 2D using MDS. The distance between two photos is calculated using principal component analysis of a matri... |

549 |
Multidimensional Scaling
- Kruskal, Wish
- 1978
(Show Context)
Citation Context ...en a pair of vertices. The first step in our GMap algorithm is to embed the graph in the plane. Possible embedding algorithms include principal component analysis [17], multidimensional scaling (MDS) =-=[20]-=-, a force-directed algorithm [12], or non-linear dimensionality reduction such as LLE [29] and Isomap [31]. The second step is a cluster analysis of the underlying graph or the embedded pointset from ... |

527 | Tree visualization with tree-maps: 2-D space-filling approach
- Shneiderman
- 1992
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Citation Context ...erlying symmetries, or minimizing the number of edge crossings. While such point-and-line representation are most commonly used, other representations have also been considered. For example, treemaps =-=[30]-=- use a recursive space filling approach to represent trees. There is also a large body of work on representing planar graphs as contact graphs [8, 14, 22], where vertices are embodied by geometrical o... |

526 | Y.: University of Florida sparse matrix collection
- Davis, Hu
- 2009
(Show Context)
Citation Context ...algorithm.. Time in brackets are for GMap without artificial points Graph |V| ncluster npoly CPU Fig. 6 340 19 27 0.26 (0.01) Fig. 1 913 26 95 0.95 (0.01) Movie map [2] 1981 41 108 2.65 (0.04) fxm3 6 =-=[10]-=- 5026 36 196 9.20 (0.52) wb-cs-stan. [10] 8929 130 540 28.24 (1.28) mus books (The Stranger, The Plague, The Fall), all well connected with the Russian classics. Graecoromania: Another large region to... |

423 |
A sweepline algorithm for Voronoi diagrams
- Fortune
- 1986
(Show Context)
Citation Context ...t of vertices and artificial points. Each test takes O(log|V |) time, for O(|V |log|V |) time overall. We next compute a Delaunay triangulation of the points, which can be done in time O(|V |log|V |) =-=[11]-=-. Next we create the corresponding Voronoi diagram of all points and merge Voronoi cells that belong to the same cluster. This step requires O(|V |) and thus the overall complexity of GMap is O(|V |lo... |

347 | The theory underlying concept maps and how to construct them. Accessed on May 12, 2005 from http://cmap.coginst.uwf.edu/info/printer.html
- Novak
(Show Context)
Citation Context ... more recent newsmaps represent hierarchical information by means of space-filling tilings, allocating area proportional to some metric. Concept maps are diagrams showing relationships among concepts =-=[10]-=-. Somewhat similar are cognitive maps and mind-maps used to represent words or ideas linked to and arranged around a central key word. In self-organizing maps (SOM) [8], an unsupervised learning algor... |

168 | Squarified treemaps
- Bruls, Huizing, et al.
- 2000
(Show Context)
Citation Context ...ulation). The map of science [5] uses vertex coloring in a graph drawing to provide an overview of the scientific landscape, based on citations of journal articles. Treemaps [30], squarified treemaps =-=[7]-=- and the more recent newsmaps [33] represent hierarchical information by means of space-filling tilings, allocating area proportional to some metric. Representing imagined places on a map as if they w... |

126 |
Kontaktprobleme der konformen abbildung. ber. s achs. akad
- Koebe
- 1936
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Citation Context ... on representing planar graphs as contact graphs [8, 14, 22], where vertices are embodied by geometrical objects and edges are shown by two objects touching in some specified fashion. Koebe’s theorem =-=[18]-=- shows that all planar graphs can be represented by touching disks. A similar representation is possible with triangles, where two adjacent vertices correspond to a vertex-to-side touching pair of tri... |

94 | Mapping the backbone of science
- Boyack, Klavans, et al.
(Show Context)
Citation Context ... is still popular today (e.g., the New York Times red-blue maps of the US, showing the presidential election results in 2000 and 2004 with states drawn proportional to population). The map of science =-=[5]-=- uses vertex coloring in a graph drawing to provide an overview of the scientific landscape, based on citations of journal articles. Treemaps [30], squarified treemaps [7] and the more recent newsmaps... |

94 | Graph drawing by stress majorization
- Gansner, Koren, et al.
- 2004
(Show Context)
Citation Context ...aph, we modify the edge weights as follows: edge lengths of intra-cluster edges are set to 1 and intra-cluster edges to a number α > 1. The embedding is obtained using a stress majorization algorithm =-=[13]-=- to give an embedding that takes into account the clustering information. Figure 10 shows the result of this defragmentation algorithm applied to Figure 5 with α = 50. Compared with force directed emb... |

84 | S.: Bubble Sets: Revealing set relations with isocontours over existing visualizations
- Collins, Penn, et al.
- 2009
(Show Context)
Citation Context ...of interest in UML 1 All the images in this paper are available in high-resolution form at www.research.att.com/ ∼ yifanhu/MAPS.Figure 1: A map of books related to “1984” from Amazon.com diagram. In =-=[9]-=-, “bubblesets,” based on isocontours, are used to depict multiple relations among a set of objects. Simonetto et al. [35] automatically generate Euler diagrams which provide one of the standard ways, ... |

57 | GraphAEL: Graph animations with evolving layouts
- Erten, Harding, et al.
- 2004
(Show Context)
Citation Context ...s and 1517 edges. The largest component has 275 vertices and 784 edges, and thus contains about 54% of all authors. The data comes from the first 10 years of the Symposium on Graph Drawing, 1994-2004 =-=[10]-=-. We look at the first eight largest connected components. This graph is cumulative, in the sense that two authors are connected with an edge if they have written at least one joint paper in the first... |

42 |
On triangle contact graphs
- Fraysseix, Mendez, et al.
- 1994
(Show Context)
Citation Context ...ons have also been considered. For example, treemaps [30] use a recursive space filling approach to represent trees. There is also a large body of work on representing planar graphs as contact graphs =-=[8, 14, 22]-=-, where vertices are embodied by geometrical objects and edges are shown by two objects touching in some specified fashion. Koebe’s theorem [18] shows that all planar graphs can be represented by touc... |

40 |
Graph drawing by force directed placement
- Fruchterman, Reingold
- 1991
(Show Context)
Citation Context ...step in our GMap algorithm is to embed the graph in the plane. Possible embedding algorithms include principal component analysis [17], multidimensional scaling (MDS) [20], a force-directed algorithm =-=[12]-=-, or non-linear dimensionality reduction such as LLE [29] and Isomap [31]. The second step is a cluster analysis of the underlying graph or the embedded pointset from step one. In this step, it is imp... |

39 | D.: Fully automatic visualisation of overlapping sets
- Simonetto, Auber, et al.
- 2009
(Show Context)
Citation Context ...hu/MAPS.Figure 1: A map of books related to “1984” from Amazon.com diagram. In [9], “bubblesets,” based on isocontours, are used to depict multiple relations among a set of objects. Simonetto et al. =-=[35]-=- automatically generate Euler diagrams which provide one of the standard ways, along with Venn diagrams, for visualizing subset relationships. Apart from differences in the algorithms used to generate... |

36 | Two improved algorithms for envelope and wavefront reduction
- Kumfert, Pothen
- 1997
(Show Context)
Citation Context ...ong the edges is wellstudied. For example, in the context of minimum bandwidth or wavefront reduction ordering for sparse matrices, it is known that the problem is NP-hard, and a number of heuristics =-=[16, 21]-=- were proposed. One such heuristic is to order vertices using the Fiedler vector. Motivated by this approach, we approximate (1) by max ∑ wi, j(ci − c j) 2 , subject to ∑ c 2 k {i, j}∈Ec k∈Vc = 1 (2) ... |

32 |
Modularity clustering is force-directed layout
- Noack
(Show Context)
Citation Context ... between clusters. On the other hand, with an embedding derived from a force-directed algorithm [12], a modularity based clustering [26] could be a better fit. The two algorithms are strongly related =-=[27]-=- and therefore we can expect vertices that are in the same cluster to also be physically close to each other in the embedding. In the third step the two-dimensional embedding together with the cluster... |

31 |
The rectangular statistical cartogram
- Raisz
- 1934
(Show Context)
Citation Context ...an space. Examples of the second kind of problem are found in cartograms, where the goal is to redraw a map so that the country areas are proportional to some metric, an idea which dates back to 1934 =-=[28]-=- and is still popular today (e.g., the New York Times red-blue maps of the US, showing the presidential election results in 2000 and 2004 with states drawn proportional to population). The map of scie... |

26 | Visualization of areas of interest in software architecture diagrams
- Byelas, Telea
- 2006
(Show Context)
Citation Context ...so that the overall outline of the point set often follows that shape. Also related is work on visualizing subsets of a set of items using geometric regions to indicate the grouping. Byelas and Telea =-=[8]-=- use deformed convex hulls to highlight areas of interest in UML 1 All the images in this paper are available in high-resolution form at www.research.att.com/ ∼ yifanhu/MAPS.Figure 1: A map of books ... |

25 |
Efficient and high quality force-directed graph drawing
- HU
(Show Context)
Citation Context ...that two authors are connected with an edge if they have written at least one joint paper in the first ten years of the symposium. Even when drawn with a high-quality scalable forcedirected algorithm =-=[15]-=- and after applying a node-overlap removal step, the resulting graph looks more like a hairball than anything else; see Fig. 5. On the other hand, the corresponding map, as shown in Fig. 6, seems much... |

23 | Methods for Realistic Landscape Imaging
- Musgrave
- 1993
(Show Context)
Citation Context ... therefore overall outline of the point set often follows that shape. Generating synthetic geography has a large literature, connected to its use in computer games and movies. Most of the work (e.g., =-=[23, 25]-=-) relies on variations of a fractal model. These techniques could provide additional photorealism, and may be used in future extention of our work. 3 THE GMAP ALGORITHM The input to our algorithm is a... |

20 | A multilevel algorithm for wavefront reduction
- Hu, Scott
(Show Context)
Citation Context ...ong the edges is wellstudied. For example, in the context of minimum bandwidth or wavefront reduction ordering for sparse matrices, it is known that the problem is NP-hard, and a number of heuristics =-=[16, 21]-=- were proposed. One such heuristic is to order vertices using the Fiedler vector. Motivated by this approach, we approximate (1) by max ∑ wi, j(ci − c j) 2 , subject to ∑ c 2 k {i, j}∈Ec k∈Vc = 1 (2) ... |

19 | Compact floor-planning via orderly spanning trees
- Liao, Lu, et al.
(Show Context)
Citation Context ...ons have also been considered. For example, treemaps [30] use a recursive space filling approach to represent trees. There is also a large body of work on representing planar graphs as contact graphs =-=[8, 14, 22]-=-, where vertices are embodied by geometrical objects and edges are shown by two objects touching in some specified fashion. Koebe’s theorem [18] shows that all planar graphs can be represented by touc... |

18 |
On floor-plan of plane graphs
- He
- 1999
(Show Context)
Citation Context ...ons have also been considered. For example, treemaps [30] use a recursive space filling approach to represent trees. There is also a large body of work on representing planar graphs as contact graphs =-=[8, 14, 22]-=-, where vertices are embodied by geometrical objects and edges are shown by two objects touching in some specified fashion. Koebe’s theorem [18] shows that all planar graphs can be represented by touc... |

11 |
Selecting good color schemes for maps. www.colorbrewer.org
- Brewer
- 2002
(Show Context)
Citation Context ...everal disjoint regions we must use the same color for all regions to avoid ambiguity. Thus, four colors (or even five or six) are not enough. In GMap we start with a coloring scheme from ColorBrewer =-=[6]-=-, which typically has 5 easy to differentiate base colors, and generate as many colors as the number of countries by blending the base colors. As a result our color space is linear and discrete. Becau... |

11 | Putting recommendations on the map: visualizing clusters and relations
- Gansner, Hu, et al.
- 2009
(Show Context)
Citation Context ...kLand observed that the “gateway” to Fringistan is Rand’s Atlas Shrugged. We plan to perform formal user studies of the interaction with graphs and maps, in the context of visualizing recommendations =-=[17]-=-. There are practical and theoretical obstacles to obtaining “perfect” maps, that is, maps that do not omit or distort the underlying information. However, a similar drawback plagues any 2dimensional ... |

9 | Optimal polygonal representation of planar graphs, Algorithmica (2011
- Gansner, Hu, et al.
(Show Context)
Citation Context ... maps also can have lakes, islands, and peninsulas, similar to those found in real geographic maps. 1 2 RELATED WORK There is a large body of work on contact polygonal representation of planar graphs =-=[11, 16, 19, 23]-=-. While some theoretical problems from this area are related to our work, the emphasis in this paper is not on the strict preservation of adjacency information, but on a practical approach to visualiz... |

8 | Choosing colors for geometric graphs via color space embeddings
- Dillencourt, Eppstein, et al.
- 2007
(Show Context)
Citation Context ...not connected countries having very similar colors by making the country graph a complete graph, and assigning edge weights to be the inverse of the distance between two countries. Dillencourt et al. =-=[9]-=- investigated the case where all colors in the color spectrum are available. They proposed a force-directed model aimed at selecting |Vc| colors as far apart as possible in the color space. However in... |

4 |
Gmap: Drawing graphs and clusters as map
- Gansner, Hu, et al.
- 2010
(Show Context)
Citation Context ... and clustering information so that we can create the map representation and make it aesthetically appealing. Our presentation is narrative and informal. We refer the interested reader to the article =-=[5]-=- for technical details and more references. Making the map Given the placement of the points from the first step, and their clustering from the second, we want to create a map, with inner boundaries s... |

3 |
R.: A fractal-based relaxation algorithm for shape from terrain image. Computer Vision and Image Understanding 109
- Liao, Petrou, et al.
- 2008
(Show Context)
Citation Context ... therefore overall outline of the point set often follows that shape. Generating synthetic geography has a large literature, connected to its use in computer games and movies. Most of the work (e.g., =-=[23, 25]-=-) relies on variations of a fractal model. These techniques could provide additional photorealism, and may be used in future extention of our work. 3 THE GMAP ALGORITHM The input to our algorithm is a... |

2 |
The Shaping of Middle-Earth
- Tolkien
- 1986
(Show Context)
Citation Context ...ing tilings, allocating area proportional to some metric. Representing imagined places on a map as if they were real countries also has a long history, e.g., the 1930’s Map of Middle Earth by Tolkien =-=[32]-=- and the Bücherlandes map by Woelfle from the same period [1]. More recent popular maps include xkcd’s Map of Online Communities [4]. While most such maps are generated in an ad hoc manner and are not... |