## An Energy Model for Visual Graph Clustering (2003)

### Cached

### Download Links

- [www-sst.informatik.tu-cottbus.de]
- [www-sst.informatik.tu-cottbus.de]
- DBLP

### Other Repositories/Bibliography

Venue: | Proceedings of the 11th International Symposium on Graph Drawing (GD 2003), LNCS 2912 |

Citations: | 41 - 4 self |

### BibTeX

@INPROCEEDINGS{Noack03anenergy,

author = {Andreas Noack},

title = {An Energy Model for Visual Graph Clustering},

booktitle = {Proceedings of the 11th International Symposium on Graph Drawing (GD 2003), LNCS 2912},

year = {2003},

pages = {425--436},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

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.

### Citations

1198 | Statistical Mechanics of Complex Networks
- Albert, Barabasi
(Show Context)
Citation Context ...he small worlds described in the next subsection to the graphs addressed by classical force-directed graph drawing. Application: Drawing Clustered Small-World Graphs. According to Albert and Barabási=-= [1]-=-, three concepts are prominent in contemporary thinking about complex networks: small worlds, clustering, and degree distributions. In small-world graphs, the nodes have small graph-theoretic distance... |

515 | A heuristic for graph drawing
- Eades
- 1984
(Show Context)
Citation Context ...els there is only a weak dependency of distance and coupling, and thus these energy models do not isolate clusters well. (See [24] for a more detailed discussion of these models.) The models of Eades =-=[12]-=-, Fruchterman and Reingold [14], and Davidson and Harel [10] strongly enforce uniform edge lengths. However, separating clusters requires some long edges. Section 3 contains example drawings of the Fr... |

457 | An algorithm for drawing general undirected graphs
- Kamada, Kawai
- 1989
(Show Context)
Citation Context ...ngths. However, separating clusters requires some long edges. Section 3 contains example drawings of the Fruchterman-Reingold model. In multidimensional scaling [20] and the model of Kamada and Kawai =-=[18]-=-, the distances between nodes in the drawing are determined by their graph-theoretic distances. But the graph-theoretic distances and the coupling are only weakly related. The energy model of Hall ([1... |

451 | The Geometry of Graphs and Some of its Algorithmic Applications
- Linial, London, et al.
- 1995
(Show Context)
Citation Context ... the third subsection. Related Work: Clustering by Minimizing Distance Ratios. For a given graph, the cut with the minimum cut ratio is called the sparsest cut. Computing sparsest cuts is N Phard. In =-=[22]-=- and [3] approximation algorithms are given that compute sparse cuts from minima of (with respect to appropriate metrics). So it is not surprising arithmean(E,p) arithmean(V (2) ,p) that minimizing th... |

428 | Graph drawing by force-directed placement
- Fruchterman, Reingold
- 1991
(Show Context)
Citation Context ... this paper.sThe new energy model and its formal analysis are presented in Sect. 2. Section 3 shows example drawings and compares them with drawings of the well-known Fruchterman-Reingold force model =-=[14]-=-. Section 4 compares our energy model with existing force and energy models, and proposes a generalization that includes among others our energy model and the Fruchterman-Reingold model. It discusses ... |

349 |
Exploring complex networks
- Strogatz
- 2001
(Show Context)
Citation Context ... drawing call graphs and similar models of object-oriented programs [21]. Further examples of clustered small-world graphs include models of computer networks, the World Wide Web, and social networks =-=[27,1]-=-. Interpretability of Drawings. Drawings of graphs are useful because we can infer properties of the graph from properties of the drawing. For valid inferences, we need precise statements which proper... |

288 |
Multidimensional scaling
- Kruskal, Wish
- 1978
(Show Context)
Citation Context ...l [10] strongly enforce uniform edge lengths. However, separating clusters requires some long edges. Section 3 contains example drawings of the Fruchterman-Reingold model. In multidimensional scaling =-=[20]-=- and the model of Kamada and Kawai [18], the distances between nodes in the drawing are determined by their graph-theoretic distances. But the graph-theoretic distances and the coupling are only weakl... |

198 | Recent directions in netlist partitioning: A survey
- Alpert, Kahng
- 1995
(Show Context)
Citation Context ...the perspective of energy minimization in this paper. Finding clusters, i.e. subsets of nodes with many internal edges and few edges to outside nodes, in graphs is an important problem in VLSI design =-=[2]-=-, parallel computing [25], software engineering [23], and graph drawing [9]. The most popular force and energy models do not clearly isolate clusters, especially in graphs with small diameter. The fir... |

174 | HAREL D.: Drawing graphs nicely using simulated annealing
- DAVIDSON
- 1996
(Show Context)
Citation Context ..., and thus these energy models do not isolate clusters well. (See [24] for a more detailed discussion of these models.) The models of Eades [12], Fruchterman and Reingold [14], and Davidson and Harel =-=[10]-=- strongly enforce uniform edge lengths. However, separating clusters requires some long edges. Section 3 contains example drawings of the Fruchterman-Reingold model. In multidimensional scaling [20] a... |

159 |
An r-dimensional quadratic placement algorithm
- Hall
- 1970
(Show Context)
Citation Context ...8], the distances between nodes in the drawing are determined by their graph-theoretic distances. But the graph-theoretic distances and the coupling are only weakly related. The energy model of Hall (=-=[16]-=-, recently used e.g. in [8,19]) places some non-adjacent nodes and loosely coupled subgraphs very close or at the same position. All mentioned energy models easily generalize to graphs with weighted e... |

126 | An O(log k) Approximate Min-cut Max-flow Theorem and Approximation Algorithm for Multi-Commodity Flows
- Aumann, Rabani
- 1998
(Show Context)
Citation Context ...d subsection. Related Work: Clustering by Minimizing Distance Ratios. For a given graph, the cut with the minimum cut ratio is called the sparsest cut. Computing sparsest cuts is N Phard. In [22] and =-=[3]-=- approximation algorithms are given that compute sparse cuts from minima of (with respect to appropriate metrics). So it is not surprising arithmean(E,p) arithmean(V (2) ,p) that minimizing the simila... |

124 | Using Automatic Clustering to Produce High-level System Organizations of Source Code
- Mancoridis, Mitchell, et al.
- 1998
(Show Context)
Citation Context ...r. Finding clusters, i.e. subsets of nodes with many internal edges and few edges to outside nodes, in graphs is an important problem in VLSI design [2], parallel computing [25], software engineering =-=[23]-=-, and graph drawing [9]. The most popular force and energy models do not clearly isolate clusters, especially in graphs with small diameter. The first main result of this paper is an energy model whos... |

94 | Graph Drawing: Algorithms for the Visualization of Graphs - Battista, Eades, et al. - 1999 |

88 | A multilevel algorithm for force-directed graph drawing, in
- Walshaw
- 2000
(Show Context)
Citation Context ... implement, adaptable to different drawing criteria, and give satisfactory results for many graphs ([4, Chap. 10], [7]). Since the introduction of multi-scale algorithms they are even quite efficient =-=[28,15,17,26,29]-=-. 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 | A fast multi-scale method for drawing large graphs
- Harel, Koren
- 2000
(Show Context)
Citation Context ... implement, adaptable to different drawing criteria, and give satisfactory results for many graphs ([4, Chap. 10], [7]). Since the introduction of multi-scale algorithms they are even quite efficient =-=[28,15,17,26,29]-=-. 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... |

62 | Ace: A fast multiscale eigenvectors computation for drawing huge graphs
- Koren, Harel
- 2002
(Show Context)
Citation Context ...des in the drawing are determined by their graph-theoretic distances. But the graph-theoretic distances and the coupling are only weakly related. The energy model of Hall ([16], recently used e.g. in =-=[8,19]-=-) places some non-adjacent nodes and loosely coupled subgraphs very close or at the same position. All mentioned energy models easily generalize to graphs with weighted edges. Given appropriate edge w... |

46 | Navigating clustered graphs using force-directed methods
- Eades, Huang
(Show Context)
Citation Context ...lusters. But this means putting clusters in (in the form of edge weights) to get clusters out. Other approaches apply force-directed methods to draw graphs with a given hierarchical cluster structure =-=[30,13]-=-. But finding clusters or appropriate edge weights is difficult, because most variants of the graph clustering problem are N P-hard. In the LinLog energy model, the attraction and repulsion between th... |

44 | An Experimental Comparison of ForceDirected and Randomized Graph Drawing Algorithms
- Brandenburg, Himsolt, et al.
- 1995
(Show Context)
Citation Context ...ike small and uniform edge lengths, well-distributed nodes, or well-separated clusters. Empirical studies were performed to evaluate to what degree force and energy models fulfill such criteria (e.g. =-=[6]-=-), but few analytical results exist. A formal characterization of the minimum energy drawings of our energy model is the second main result of this paper.sThe new energy model and its formal analysis ... |

43 |
FADE: graph drawing, clustering, and visual abstraction, in
- Quigley, Eades
- 2000
(Show Context)
Citation Context ... implement, adaptable to different drawing criteria, and give satisfactory results for many graphs ([4, Chap. 10], [7]). Since the introduction of multi-scale algorithms they are even quite efficient =-=[28,15,17,26,29]-=-. 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... |

40 | Graph Partitioning Algorithms with Applications to Scientific Computing
- Pothen
(Show Context)
Citation Context ...y minimization in this paper. Finding clusters, i.e. subsets of nodes with many internal edges and few edges to outside nodes, in graphs is an important problem in VLSI design [2], parallel computing =-=[25]-=-, software engineering [23], and graph drawing [9]. The most popular force and energy models do not clearly isolate clusters, especially in graphs with small diameter. The first main result of this pa... |

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

28 |
The geometry of graphs and some of its algorithmic applications
- Rabinovich
- 1995
(Show Context)
Citation Context ...n the third subsection. Related Work: Clustering by Minimizing Distance Ratios. For a given graph, the cut with the minimum cut ratio is called the sparsest cut. Computing sparsest cuts is NPhard. In =-=[22]-=- and [3] approximation algorithms are given that compute sparse cuts from minima of arithmean(E,p) arithmean(V (2) ,p) (with respect to appropriate metrics). So it is not surprising that minimizing th... |

24 | Drawing on physical analogies
- Brandes
- 2001
(Show Context)
Citation Context ... creating straight-line drawings of undirected graphs. They are comparatively easy to implement, adaptable to different drawing criteria, and give satisfactory results for many graphs ([4, Chap. 10], =-=[7]-=-). Since the introduction of multi-scale algorithms they are even quite efficient [28,15,17,26,29]. Energy-based methods generally have two parts: an energy model, and an algorithm that searches a sta... |

22 | CrocoCosmos – 3D visualization of large object-oriented programs
- Lewerentz, Noack
- 2003
(Show Context)
Citation Context ... energy models are not suitable for drawing such graphs. The LinLog model can reveal the clusters in small worlds. We applied it for drawing call graphs and similar models of object-oriented programs =-=[21]-=-. Further examples of clustered small-world graphs include models of computer networks, the World Wide Web, and social networks [27,1]. Interpretability of Drawings. Drawings of graphs are useful beca... |

20 | Visual ranking of link structures
- Brandes, Cornelsen
(Show Context)
Citation Context ...des in the drawing are determined by their graph-theoretic distances. But the graph-theoretic distances and the coupling are only weakly related. The energy model of Hall ([16], recently used e.g. in =-=[8,19]-=-) places some non-adjacent nodes and loosely coupled subgraphs very close or at the same position. All mentioned energy models easily generalize to graphs with weighted edges. Given appropriate edge w... |

19 | A Numerical Optimization Approach to General Graph Drawing
- Tunkelang
- 1999
(Show Context)
Citation Context |

19 |
Generating customized layouts
- Wang, Miyamoto
- 1996
(Show Context)
Citation Context ...lusters. But this means putting clusters in (in the form of edge weights) to get clusters out. Other approaches apply force-directed methods to draw graphs with a given hierarchical cluster structure =-=[30,13]-=-. But finding clusters or appropriate edge weights is difficult, because most variants of the graph clustering problem are N P-hard. In the LinLog energy model, the attraction and repulsion between th... |

18 |
The Effect of Graph Layout on Inference from Social Network Data.'' Symposium on Graph Drawing GD 95
- Blythe, McGrath, et al.
- 1995
(Show Context)
Citation Context ...s [11]. Blythe et al. concluded from their study in the context of social network analysis, that the Euclidean distance of nodes has a significant effect on the viewers’ assignment of nodes to group=-=s [5]-=-. The Role of Theory in the Development and Evaluation of Energy Models. To date, force and energy models have been evaluated mainly empirically. Complementing this empirical evaluation with theoretic... |

11 |
Human perception of laid-out graphs
- Dengler, Cowan
- 1998
(Show Context)
Citation Context ...espondence exists. In a study of Dengler and Cowan, the most important semantic attribution is that observers view graph drawings hierarchically, that they separate them into interconnected subgraphs =-=[11]. -=-Blythe et al. concluded from their study in the context of social network analysis, that the Euclidean distance of nodes has a significant effect on the viewers’ assignment of nodes to groups [5]. T... |

10 | Drawing clusters and hierarchies
- Brockenauer, Cornelsen
(Show Context)
Citation Context .... subsets of nodes with many internal edges and few edges to outside nodes, in graphs is an important problem in VLSI design [2], parallel computing [25], software engineering [23], and graph drawing =-=[9]-=-. The most popular force and energy models do not clearly isolate clusters, especially in graphs with small diameter. The first main result of this paper is an energy model whose minimum energy drawin... |

7 | Energy models for drawing clustered small-world graphs
- Noack
- 2003
(Show Context)
Citation Context ..., where the scaling is applied to a minimum LinLog energy drawing. Then this function has a minimum at the minimum energy drawing, i.e. at a scaling factor of 1. The details of the proof are given in =-=[24]. -=-✷ Theorem 2. Let G = (V, E) be a connected graph. Then p 0 is a drawing of G with minimum LinLog energy if and only if p 0 is a drawing that minimizes arithmean(E,p) geomean(V (2) ,p) and satisfies ... |