## Dig-CoLa: Directed graph layout through constrained energy minimization

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Venue: | IEEE Symposium on Information Visualization, Infovis’05, 2005 |

Citations: | 12 - 4 self |

### BibTeX

@INPROCEEDINGS{Dwyer_dig-cola:directed,

author = {Tim Dwyer and Yehuda Koren},

title = {Dig-CoLa: Directed graph layout through constrained energy minimization},

booktitle = {IEEE Symposium on Information Visualization, Infovis’05, 2005},

year = {},

pages = {65--72}

}

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### Abstract

We describe a new method for visualization of directed graphs. The method combines constraint programming techniques with a high performance force-directed placement (FDP) algorithm so that the directed nature of the graph is highlighted while useful properties of FDP — such as emphasis of symmetries and preservation of proximity relations — are retained. Our algorithm automatically identifies those parts of the digraph that contain hierarchical information and draws them accordingly. Additionally, those parts that do not contain hierarchy are drawn at the same quality expected from a non-hierarchical, undirected layout algorithm. An interesting application of our algorithm is directional multidimensional scaling (DMDS). DMDS deals with low-dimensional embedding of multivariate data where we want to emphasize the overall flow in the data (e.g. chronological progress) along one of the axes.

### Citations

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(Show Context)
Citation Context ...xes of the layout are computed simultaneously as in standard undirected approaches. This process is made feasible by combining two optimization techniques: majorization [14] and quadratic programming =-=[8]-=-. 2 PREVIOUS WORK The predominant approach to drawing digraphs is based on Sugiyama et al. [21], which has evolved into many successful algorithms. Fig. 1 gives an example of a graph drawn with such a... |

459 | An Algorithm for Drawing General Undirected Graphs
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(Show Context)
Citation Context ... Euclidean distances will approach the corresponding target distances (e.g., graph-theoretic distances). Drawing undirected graphs by minimizing a stress function was made popular by Kamada and Kawai =-=[17]-=-. Specifically, we have an ideal distance dij for every pair of nodes i and j, modeled as a spring. Given a d-D layout, where node i is placed at point Xi, the energy of the system is ( ) 2 stress(X)=... |

388 |
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(Show Context)
Citation Context ...s majorization A recent report by Gansner et al. [14] suggests computing a graph layout by minimizing the stress function through majorization,following works in the field of multidimensional scaling =-=[4]-=-. Majorization is a rather global optimization process offering some distinct advantages over localized processes like gradient descent — especially guaranteed monotonic decrease of stress, improved r... |

345 |
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(Show Context)
Citation Context ...tions (edges). The latter requirement renders algorithms designed for undirected graph drawing inappropriate for digraphs. The dominant digraph-drawing strategy, rooted in the work of Sugiyama et al. =-=[21]-=- involves assigning x and y coordinates in separate stages with different objectives. Thus, the y-axis represents the directional information, or hierarchy, and the x-axis placement is adjusted for ad... |

220 | A technique for drawing directed graphs
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(Show Context)
Citation Context ... edge bends. Each of the optimization problems outlined abovehas been shown to be NP-hard so heuristic approaches have been designed that offer reasonable results in most cases. For more details see =-=[2, 11, 12, 13]-=-. Carmel et al. [5] use an alternative approach to drawing digraphs. The nodes are associated with continuous y-coordinates, in a way that can be applied to any kind of digraph, whether cyclic or acyc... |

76 | The Matrix Market: A web resource for test matrix collections,” Quality of Numerical Software: Assessment and Enhancement
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Citation Context ...mes possible by the clear separation between constraints and cost function. We provide here two examples where negative gap was advantageous. Both examples are taken from the Matrix Market collection =-=[3]-=-. The first graph, Nos4, in Fig. 7, is based on a finite element approximation to a beam structure. Here, setting G = −1 improves layout quality while still being very similar to the layout that was a... |

67 | S.: Graph drawing by stress majorization
- GANSNER, KOREN, et al.
(Show Context)
Citation Context ...digraphs (DIG-COLA) where all axes of the layout are computed simultaneously as in standard undirected approaches. This process is made feasible by combining two optimization techniques: majorization =-=[14]-=- and quadratic programming [8]. 2 PREVIOUS WORK The predominant approach to drawing digraphs is based on Sugiyama et al. [21], which has evolved into many successful algorithms. Fig. 1 gives an exampl... |

59 | Object-oriented software for quadratic programming
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(Show Context)
Citation Context ...mming solvers Many quadratic solvers are available employing a number of different optimization methods. All of them can serve for solving problem (10). They include, MOSEK [22], CPLEX [24], and OOQP =-=[9]-=-. The results reported in this paper were carried out using the MOSEK package which is based on the general purpose interior-point non-linear optimization method. However, several properties of the DI... |

37 |
Drawing Graphs
- Kaufmann, Wagner
- 2001
(Show Context)
Citation Context ... algorithms convert the relational structure of a graph (or network) to a diagram. Many approaches to graph drawing have been developed for different types of graphs and different application domains =-=[2, 12]-=-. In this paper, we consider the problem of drawing directed graphs (digraphs). Drawing digraphs is a challenging task, requiring algorithms that faithfully represent the relative connectivity of the ... |

35 |
Drawing graphs to convey proximity: an incremental arrangement method
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Citation Context ... ability to preserve proximity relations (as reflected by conservation of default edge length), clear decomposition to clusters when they naturally exist (especially when using linearnetwork distance =-=[6]-=-), reliable display of graph symmetries (see [7]) and a natural capability to handle multidimensional layouts. However, FDP does possess a few shortcomings, and our method is not an exception. A notab... |

19 |
A simple and unified method for drawing graphs: Magnetic-spring algorithm
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- 1995
(Show Context)
Citation Context ...r a force model) whose minimization generally produces an acceptable node placement. In particular, we are aware of two other occasions where FDP was suggested for digraph drawing: Sugiyama and Misue =-=[20]-=- and Kamps et al. [18]. In both cases the force model was extended by ‘angle forces’ that encourage all directed edges to point in the same direction. However, this adds additional complexity to the f... |

17 | Exact and heuristic algorithms for 2layer straightline crossing minimization
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(Show Context)
Citation Context ... edge bends. Each of the optimization problems outlined abovehas been shown to be NP-hard so heuristic approaches have been designed that offer reasonable results in most cases. For more details see =-=[2, 11, 12, 13]-=-. Carmel et al. [5] use an alternative approach to drawing digraphs. The nodes are associated with continuous y-coordinates, in a way that can be applied to any kind of digraph, whether cyclic or acyc... |

15 |
Constraint-Based SpringModel Algorithm for Graph Layout
- Kamps, Kleinz, et al.
- 1995
(Show Context)
Citation Context ... minimization generally produces an acceptable node placement. In particular, we are aware of two other occasions where FDP was suggested for digraph drawing: Sugiyama and Misue [20] and Kamps et al. =-=[18]-=-. In both cases the force model was extended by ‘angle forces’ that encourage all directed edges to point in the same direction. However, this adds additional complexity to the force model and we are ... |

13 |
An open graph visualization system and its applications
- Gansner, North
- 1999
(Show Context)
Citation Context ...Hierarchical layout (all directed edges point downwards) Undirected layout (red edges point upwards) Figure 1: Two layouts of a gene network using standard layout algorithms from the GRAPHVIZ toolkit =-=[15]-=-, blue edges are undirected In this work we present a new approach to drawing digraphs. Following the common convention, we also dedicate one of the axes to conveying hierarchical information. However... |

9 | Combining hierarchy and energy for drawing directed graphs
- Carmel, Harel, et al.
(Show Context)
Citation Context ...zation problems outlined abovehas been shown to be NP-hard so heuristic approaches have been designed that offer reasonable results in most cases. For more details see [2, 11, 12, 13]. Carmel et al. =-=[5]-=- use an alternative approach to drawing digraphs. The nodes are associated with continuous y-coordinates, in a way that can be applied to any kind of digraph, whether cyclic or acyclic, and which requ... |

8 | Stress majorization with orthogonal ordering constraints
- Dwyer, Koren, et al.
- 2006
(Show Context)
Citation Context ...change between iterations. Also, the solution to one iteration is a good approximation to the solution of the next. Full details of a solver which takes advantage of these attributes are to appear in =-=[1]-=-. Fixing a single node The convexity of the quadratic program is thanks to the fact that the matrix Lw is positive semi-definite. This means that every x ∈ Rn satisfies xT Lwx � 0. This inequality can... |

2 |
Spring algorithms and symmetry, Theoret
- Eades, Lin
- 2000
(Show Context)
Citation Context ...lected by conservation of default edge length), clear decomposition to clusters when they naturally exist (especially when using linearnetwork distance [6]), reliable display of graph symmetries (see =-=[7]-=-) and a natural capability to handle multidimensional layouts. However, FDP does possess a few shortcomings, and our method is not an exception. A notable shortcoming is handling of edge crossings. Wh... |

2 | Layout of Bayesian Networks - Marriott, Moulder, et al. - 2005 |