## Energy Models for Drawing Clustered Small-World Graphs (2003)

Citations: | 7 - 3 self |

### BibTeX

@TECHREPORT{Noack03energymodels,

author = {Andreas Noack},

title = {Energy Models for Drawing Clustered Small-World Graphs},

institution = {},

year = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

We introduce energy models for drawing clustered small-world graphs.

### Citations

636 |
From Data Mining to Knowledge Discovery: An Overview
- Fayyad, Piatetsky-Shapiro, et al.
(Show Context)
Citation Context ...to N P-hard problems like Minimum Ratio-Cut and Minimum Clique Partition, information about hidden order, knowledge (“valid, novel, potentially useful, and ultimately understandable patterns in data=-=” [18]-=-). The Fruchterman-Reingold model and other classical models take desired distances between adjacent nodes as input, and the output hopefully preserves these distances. Clustering energy models output... |

517 | A Heuristic for Graph Drawing
- Eades
- 1984
(Show Context)
Citation Context ...mum of U(x). 0 = U ′ (x1) 0 = nef ′ (x1) + n1n2g ′ (x1) n1n2 ne g′ (x1) = −f ′ (x1) n1n2 ne = − f ′ (x1) g ′ (x1) (because of condition 4.) ✷ 9sTable 3. Theorem 1 applied to four e=-=nergy models. Eades [15] U(x) = nex(log x ��-=-� log e) + n1n2(1 − ne 1 ) n1n2 x U ′ (x) = ne log x + (−n1n2(1 − ne n1n2 U ′′ (x) = (log e)ne 1 x + 2n1n2(1 − ne n1n2 ne ≈ x20 log x0 if ne ≪ n1n2 Fruchterman and Reingold [19] U(x)... |

465 | An Algorithm For Drawing General Undirected Graphs
- Kamada, Kawai
- 1989
(Show Context)
Citation Context ...bject to � x v∈V 2 v = � y v∈V 2 v =1, � xv = v∈V � yv =0, v∈V � Multidimensional Scaling [29] � (||pu − pv|| − δ(u, v)) u,v∈V 2 Eades [15] � ||pu − pv||(log ||pu − pv=-=|| − log e) + � Kamada and Kawai [27] Fruchterman and Reingold [19] Davidso-=-n and Harel [13] {u,v}∈E � u,v∈V � {u,v}∈E � {u,v}∈E (||pu−pv||−δ(u,v)) 2 δ(u,v) 2 1 3 ||pu − pv||3 − ||pu − pv|| 2 + � {u,v}∈V (2) � {u,v}∈V (2) {u,v}�∈E ln ||pu... |

434 | Graph Drawing by Force-directed Placement
- Fruchterman, Reingold
(Show Context)
Citation Context ...ades [15] U(x) = nex(log x − log e) + n1n2(1 − ne 1 ) n1n2 x U ′ (x) = ne log x + (−n1n2(1 − ne n1n2 U ′′ (x) = (log e)ne 1 x + 2n1n2(1 − ne n1n2 ne ≈ x20 log x0 if ne ≪ n1n2 Fruch=-=terman and Reingold [19] U(x) = ne 1 3 x3 -=-+ (−n1n2 ln x) U ′ (x) = nex 2 + (−n1n2 1 x ) U ′′ (x) = 2nex + n1n2 1 n1n2 ne = x3 0 ⇒ x0 = 3 Davidson and Harel [13] U(x) = nex 2 x2 � n1n2 ne + n1n2 1 x2 U ′ (x) = 2nex + (−2n1n2 ... |

352 |
Exploring complex networks
- Strogatz
- 2001
(Show Context)
Citation Context ...o the number of nodes. However, there are many practically important graphs with small graph-theoretic distances, like models of software systems (see Sect. 2), the World Wide Web, or social networks =-=[37, 1]. -=-Moreover, many of these graphs have clusters, i.e. subsets of nodes ⋆ Computer Science Report 07/2003, BTU Cottbusswith many internal edges (high cohesion) and few edges to outside nodes (low coupli... |

331 | Graph Drawing: Algorithms for the Visualization of Graphs - Battista, Eades, et al. - 1998 |

294 |
Multidimensional scaling
- Kruskal, Wish
- 1986
(Show Context)
Citation Context ...st approximately clustering: 10 ) 1 x 3sTable 4. Energy models. Hall [24, 8, 9, 28] � ||pu − pv|| {u,v}∈E 2 subject to � x v∈V 2 v = � y v∈V 2 v =1, � xv = v∈V � yv =0, v∈V � M=-=ultidimensional Scaling [29] � (||pu − pv|| − δ(u, v))-=- u,v∈V 2 Eades [15] � ||pu − pv||(log ||pu − pv|| − log e) + � Kamada and Kawai [27] Fruchterman and Reingold [19] Davidson and Harel [13] {u,v}∈E � u,v∈V � {u,v}∈E � {u,v}∈E... |

219 | Graph Clustering by Flow Simulation - Dongen - 2000 |

197 | Recent directions in netlist partitioning: A survey
- Alpert, Kahng
- 1995
(Show Context)
Citation Context ...aw graphs with a given hierarchical cluster structure [41, 16]. But finding clusters or appropriate edge weights is difficult, because most variants of the graph clustering problem are N P-hard. (See =-=[2, 33]-=- for surveys on graph clustering.) Our energy models do not require edge weights or clusters as input parameter. It is only the attraction and repulsion between nodes that reveals the global structure... |

174 | HAREL D.: Drawing graphs nicely using simulated annealing
- DAVIDSON
- 1996
(Show Context)
Citation Context ...− ne n1n2 ne ≈ x20 log x0 if ne ≪ n1n2 Fruchterman and Reingold [19] U(x) = ne 1 3 x3 + (−n1n2 ln x) U ′ (x) = nex 2 + (−n1n2 1 x ) U ′′ (x) = 2nex + n1n2 1 n1n2 ne = x3 0 ⇒ x0 = 3 D=-=avidson and Harel [13] U(x) = nex 2 x2 �-=- n1n2 ne + n1n2 1 x2 U ′ (x) = 2nex + (−2n1n2 1 x 3 ) U ′′ (x) = 2ne + 6n1n2 1 n1n2 ne = x4 0 ⇒ x0 = 4 � n1n2 ne LinLog (Sect. 5) U(x) = nex + (−n1n2 ln x) U ′ (x) = ne + (−n1n2 1 x ... |

163 |
An r-dimensional quadratic placement algorithm
- Hall
- 1970
(Show Context)
Citation Context ...for such simple drawings will not be clustering for more realistic drawings either. Non of the energy models from Table 4 is at least approximately clustering: 10 ) 1 x 3sTable 4. Energy models. Hall =-=[24, 8, 9, 28] � ||pu − pv|| {u,v}∈E 2 subject to-=- � x v∈V 2 v = � y v∈V 2 v =1, � xv = v∈V � yv =0, v∈V � Multidimensional Scaling [29] � (||pu − pv|| − δ(u, v)) u,v∈V 2 Eades [15] � ||pu − pv||(log ||pu − pv|| − l... |

140 |
and Albert-László Barabási. Statistical mechanics of complex networks
- Albert
(Show Context)
Citation Context ...o the number of nodes. However, there are many practically important graphs with small graph-theoretic distances, like models of software systems (see Sect. 2), the World Wide Web, or social networks =-=[37, 1]. -=-Moreover, many of these graphs have clusters, i.e. subsets of nodes ⋆ Computer Science Report 07/2003, BTU Cottbusswith many internal edges (high cohesion) and few edges to outside nodes (low coupli... |

98 | Bead: Explorations in information visualization
- Chalmers, Chiston
- 1992
(Show Context)
Citation Context ...on (right drawing). All mentioned energy models easily generalize to graphs with weighted edges. Other works that use force-directed methods or energy minimization for drawing weighted graphs include =-=[11, 17, 22, 35, 43, 12]-=-. Given appropriate edge weights, the models create drawings which reveal clusters. But this means putting clusters in (in the form of edge weights) to get clusters out. Other approaches apply force-d... |

88 | A multi-level algorithm for force-directed graph-drawing
- Walshaw
- 2003
(Show Context)
Citation Context ...nt, adaptable to different drawing criteria, and give satisfactory results for many graphs ([7], [4, Chap. 10]). Since the introduction of multi-scale drawing algorithms they are even quite efficient =-=[38, 20, 25, 34, 40]-=-. Energy-based methods generally have two parts: an energy model, and an algorithm that searches a state with minimum total energy. In force-directed methods, the model is a force system, and an algor... |

85 |
A linear iteration time layout algorithm for visualising high-dimensional data
- Chalmers
(Show Context)
Citation Context ...be speeded up by using forces with finite range (e.g. [19, 12]). A similar approach is to compute for each node only the repulsion to some close nodes and a small random sample of the remaining nodes =-=[10]-=-. But our goal are drawings with interpretable global structures, like separated clusters with interpretable distances. These structures emerge only when forces act globally. Technically, while in mos... |

79 | A fast multi-scale method for drawing large graphs
- Harel, Koren
(Show Context)
Citation Context ...nt, adaptable to different drawing criteria, and give satisfactory results for many graphs ([7], [4, Chap. 10]). Since the introduction of multi-scale drawing algorithms they are even quite efficient =-=[38, 20, 25, 34, 40]-=-. Energy-based methods generally have two parts: an energy model, and an algorithm that searches a state with minimum total energy. In force-directed methods, the model is a force system, and an algor... |

79 |
Ratio Cut Partitioning for Hierarchical Designs
- Wei, Cheng
- 1992
(Show Context)
Citation Context ... of nodes V . We use the density density(V1, V2) between V1 and V2 (defined in Sect. 1.1) as a measure of coupling between V1 and V2. The density between two subgraphs was first used by Wei and Cheng =-=[42]-=- as quality measure for partitions of circuits. Wei and Cheng called it ratio of the cut, but we prefer the term density to have parallel names for the parallel concepts of the density in a subgraph a... |

74 | Navigating Large Networks with Hierarchies
- Eick, Wills
(Show Context)
Citation Context ...on (right drawing). All mentioned energy models easily generalize to graphs with weighted edges. Other works that use force-directed methods or energy minimization for drawing weighted graphs include =-=[11, 17, 22, 35, 43, 12]-=-. Given appropriate edge weights, the models create drawings which reveal clusters. But this means putting clusters in (in the form of edge weights) to get clusters out. Other approaches apply force-d... |

62 | Ace: A fast multiscale eigenvectors computation for drawing huge graphs
- Koren, Carmel, et al.
(Show Context)
Citation Context ...for such simple drawings will not be clustering for more realistic drawings either. Non of the energy models from Table 4 is at least approximately clustering: 10 ) 1 x 3sTable 4. Energy models. Hall =-=[24, 8, 9, 28] � ||pu − pv|| {u,v}∈E 2 subject to-=- � x v∈V 2 v = � y v∈V 2 v =1, � xv = v∈V � yv =0, v∈V � Multidimensional Scaling [29] � (||pu − pv|| − δ(u, v)) u,v∈V 2 Eades [15] � ||pu − pv||(log ||pu − pv|| − l... |

61 |
Nicheworks – interactive visualization of very large graphs
- Wills
(Show Context)
Citation Context ...on (right drawing). All mentioned energy models easily generalize to graphs with weighted edges. Other works that use force-directed methods or energy minimization for drawing weighted graphs include =-=[11, 17, 22, 35, 43, 12]-=-. Given appropriate edge weights, the models create drawings which reveal clusters. But this means putting clusters in (in the form of edge weights) to get clusters out. Other approaches apply force-d... |

46 | Navigating clustered graphs using force-directed methods
- Eades, Huang
(Show Context)
Citation Context ...o get clusters out. Other approaches apply force-directed 11 ne forsFig. 4. Three minimum energy drawings for the Hall energy model. methods to draw graphs with a given hierarchical cluster structure =-=[41, 16]-=-. But finding clusters or appropriate edge weights is difficult, because most variants of the graph clustering problem are N P-hard. (See [2, 33] for surveys on graph clustering.) Our energy models do... |

45 |
Fade: Graph drawing, clustering, and visual abstraction
- Quigley, Eades
- 2000
(Show Context)
Citation Context ...nt, adaptable to different drawing criteria, and give satisfactory results for many graphs ([7], [4, Chap. 10]). Since the introduction of multi-scale drawing algorithms they are even quite efficient =-=[38, 20, 25, 34, 40]-=-. Energy-based methods generally have two parts: an energy model, and an algorithm that searches a state with minimum total energy. In force-directed methods, the model is a force system, and an algor... |

44 | An Experimental Comparison of ForceDirected and Randomized Graph Drawing Algorithms
- Brandenburg, Himsolt, et al.
- 1995
(Show Context)
Citation Context ...perties, like small and uniform edge lengths, well-distributed nodes, or well-separated clusters. Empirical studies were performed to evaluate to what degree energy models fulfill such criteria (e.g. =-=[6]-=-), but few analytical results exist. Formal characterizations of minimum energy drawings of new and well-known energy models are the second major result of this paper. In Sect. 2, we argue that a usef... |

43 | A new approach to effective circuit clustering
- Hagen, Kahng
- 1992
(Show Context)
Citation Context ... edge {u, v} is pint if u and v belong to the same cluster and pext otherwise, and all edges are chosen independently. (This class of pseudo-random graphs with expected clusters is also used in [21], =-=[23]-=-, [30], [39, Chap. 12].) 8.1 Edge probabilities The first examples in Figs. 6 and 7 are used to introduce pseudo-random graphs and a problem with their use for evaluating clustering algorithms. Fig. 6... |

40 | Graph Partitioning Algorithms with Applications to Scientific Computing
- Pothen
(Show Context)
Citation Context ...aw graphs with a given hierarchical cluster structure [41, 16]. But finding clusters or appropriate edge weights is difficult, because most variants of the graph clustering problem are N P-hard. (See =-=[2, 33]-=- for surveys on graph clustering.) Our energy models do not require edge weights or clusters as input parameter. It is only the attraction and repulsion between nodes that reveals the global structure... |

36 | A multi-dimensional approach to force-directed layouts of large graphs - Gajer, Goodrich, et al. - 2000 |

34 |
Hut P.: A hierarchical O(N log n) force-calculation algorithm
- Barnes
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Citation Context ...is not correct for drawings of the new energy models of this paper (LinLog, QuadLin, r-PolyLog for r < 2). The same argumentation applies to the similar method of Morrison et al. [32]. Barnes and Hut =-=[3]-=- proposed another type of multi-scale algorithm for the simulation of astronomical systems. The algorithm was introduced to graph drawing by [38, 34]. It accelerates the computation of the repulsive f... |

30 | H-BLOB: A Hierarchical Visual Clustering Method Using Implicit Surfaces
- Sprenger, Brunella, et al.
- 2000
(Show Context)
Citation Context ...algorithm gets trapped in bad local minima, but a more systematic examination is necessary. Visual Abstraction. Groups of close nodes can be surrounded by semitransparent implicit surfaces, as e.g in =-=[35, 36, 31]-=-. This makes the clusters more distinct and decreases visual complexity without significant loss of information. 10 Summary 10.1 Energy Models for Clustered Small-World Graphs The practically importan... |

28 |
Finding clusters in VLSI circuits
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Citation Context ... of an edge {u, v} is pint if u and v belong to the same cluster and pext otherwise, and all edges are chosen independently. (This class of pseudo-random graphs with expected clusters is also used in =-=[21]-=-, [23], [30], [39, Chap. 12].) 8.1 Edge probabilities The first examples in Figs. 6 and 7 are used to introduce pseudo-random graphs and a problem with their use for evaluating clustering algorithms. ... |

26 | Visualization of bibliographic networks with a reshaped landscape metaphor
- Brandes, Willhalm
- 2002
(Show Context)
Citation Context ...for such simple drawings will not be clustering for more realistic drawings either. Non of the energy models from Table 4 is at least approximately clustering: 10 ) 1 x 3sTable 4. Energy models. Hall =-=[24, 8, 9, 28] � ||pu − pv|| {u,v}∈E 2 subject to-=- � x v∈V 2 v = � y v∈V 2 v =1, � xv = v∈V � yv =0, v∈V � Multidimensional Scaling [29] � (||pu − pv|| − δ(u, v)) u,v∈V 2 Eades [15] � ||pu − pv||(log ||pu − pv|| − l... |

26 | M (2002) A Hybrid Layout Algorithm for Sub–Quadratic Multidimensional Scaling
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Citation Context ...nce. This assumption is not correct for drawings of the new energy models of this paper (LinLog, QuadLin, r-PolyLog for r < 2). The same argumentation applies to the similar method of Morrison et al. =-=[32]-=-. Barnes and Hut [3] proposed another type of multi-scale algorithm for the simulation of astronomical systems. The algorithm was introduced to graph drawing by [38, 34]. It accelerates the computatio... |

24 | Drawing on physical analogies
- Brandes
- 2001
(Show Context)
Citation Context ...lar for creating straight-line drawings of general undirected graphs. They are comparatively easy to implement, adaptable to different drawing criteria, and give satisfactory results for many graphs (=-=[7]-=-, [4, Chap. 10]). Since the introduction of multi-scale drawing algorithms they are even quite efficient [38, 20, 25, 34, 40]. Energy-based methods generally have two parts: an energy model, and an al... |

24 | Cluster stability and the use of noise in interpretation of clustering
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24 | Visualizing Information on a Sphere
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21 | Narcissus: Visualising information
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Citation Context ...or clusters as input parameter. It is only the attraction and repulsion between nodes that reveals the global structure of the graph. In this respect our approach is similar to that of Hendley et al. =-=[26]-=- who, however, did not publish their force model and an analysis of its properties. The next subsection discusses some candidates for clustering energy models. Two of them, LinLog and QuadLin, are exa... |

20 | Visual ranking of link structures
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(Show Context)
Citation Context |

19 | A Numerical Optimization Approach to General Graph Drawing
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19 |
Generating customized layouts
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Citation Context ...o get clusters out. Other approaches apply force-directed 11 ne forsFig. 4. Three minimum energy drawings for the Hall energy model. methods to draw graphs with a given hierarchical cluster structure =-=[41, 16]-=-. But finding clusters or appropriate edge weights is difficult, because most variants of the graph clustering problem are N P-hard. (See [2, 33] for surveys on graph clustering.) Our energy models do... |

18 |
The Effect of Graph Layout on Inference from Social Network Data.'' Symposium on Graph Drawing GD 95
- Blythe, McGrath, et al.
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(Show Context)
Citation Context ...is that observers view graph drawings hierarchically, that they separate them into interconnected subgraphs [14]. McGrath et al. conducted an empirical study in the context of social network analysis =-=[5], -=-and conclude that the Euclidean distance between nodes has a significant effect on the viewers’ assignment of nodes to groups. 3 Making the Clustering Property Precise This section gives precise def... |

14 | The shape of shakespeare: Visualizing text using implicit surfaces
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Citation Context |

12 |
Human Perception of Laid-out Graphs
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Citation Context ... nodes: In an empirical study of Dengler and Cowan, the most important semantic attribution is that observers view graph drawings hierarchically, that they separate them into interconnected subgraphs =-=[14]. -=-McGrath et al. conducted an empirical study in the context of social network analysis [5], and conclude that the Euclidean distance between nodes has a significant effect on the viewers’ assignment ... |

11 |
A stochastic heuristic for visualising graph clusters in a bi-dimensional space prior to partitioning
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Citation Context ...{u, v} is pint if u and v belong to the same cluster and pext otherwise, and all edges are chosen independently. (This class of pseudo-random graphs with expected clusters is also used in [21], [23], =-=[30]-=-, [39, Chap. 12].) 8.1 Edge probabilities The first examples in Figs. 6 and 7 are used to introduce pseudo-random graphs and a problem with their use for evaluating clustering algorithms. Fig. 6 shows... |

3 |
Density surfaces for immersive explorative data analyses
- Maˇzeika, Böhlen, et al.
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Citation Context ...algorithm gets trapped in bad local minima, but a more systematic examination is necessary. Visual Abstraction. Groups of close nodes can be surrounded by semitransparent implicit surfaces, as e.g in =-=[35, 36, 31]-=-. This makes the clusters more distinct and decreases visual complexity without significant loss of information. 10 Summary 10.1 Energy Models for Clustered Small-World Graphs The practically importan... |