## More Flexible Radial Layout

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Citations: | 4 - 2 self |

### BibTeX

@MISC{Brandes_moreflexible,

author = {Ulrik Brandes and Christian Pich},

title = {More Flexible Radial Layout},

year = {}

}

### OpenURL

### Abstract

We describe an algorithm for radial layout of undirected graphs, in which nodes are constrained to concentric circles centered at the origin. Such constraints are typical, e.g., in the layout of social networks, when structural centrality is mapped to geometric centrality or when the primary intention of the layout is the display of the vicinity of a distinguished node. Our approach is based on an extension of stress minimization with a weighting scheme that gradually imposes radial constraints on the intermediate layout during the majorization process, and thus is an attempt to preserve as much information about the graph structure as possible. 1

### Citations

738 |
Finding and evaluating community structure in networks
- Newman, Girvan
(Show Context)
Citation Context ...he network breaks apart into two clubs because of disagreements between the administrator and the instructor, with the latter leaving the club and taking about half of the members with him. Following =-=[22]-=-, this data set has been used frequently as a benchmark for the performance of various clustering approaches. Fig. 2 shows how the same initial layout, which is computed by minimizing stress without c... |

459 | An Algorithm for Drawing General Undirected Graphs
- Kamada, Kawai
- 1989
(Show Context)
Citation Context ...is multidimensional scaling (MDS) [3, 9]. Originating in psychometrics and the social sciences, MDS has been established and widely used for graph drawing since its popularization by Kamada and Kawai =-=[19]-=-. While there is a wide range of variants and extensions, we here concentrate on the stress minimization approach [16]. Given a set of target distances among a set of n objects, the overall goal is to... |

263 |
A set of measures of centrality based on betweenness
- Freeman
- 1977
(Show Context)
Citation Context ...n is based on closeness centrality [24] 1 cv = ∑ dvt t∈V , which is simply the inverse average distance from a vertex to all others. The right column contains drawings based on betweenness centrality =-=[14]-=- cv = ∑ δ(s, t|v) , s̸=v̸=t∈V 8closeness centrality betweenness centrality emphasizing periphery emphasizing center uniform Figure 3: Centrality layouts of the karate club social network, using two c... |

170 | Information flow model for conflict and fission
- Zachary
- 1977
(Show Context)
Citation Context ...ortest-path distances to remove artifacts potentially introduced after focusing on the first node. As an example, we consider a famous social network studied by Zachary and, subsequently, many others =-=[27]-=-. It describes friendship relations among 34 members in the karate club of a U.S. university in the 1970s. Over the course of a two-year study, the network breaks apart into two clubs because of disag... |

116 |
Modern Multi-Dimensional Scaling
- Borg, Groenen
- 1997
(Show Context)
Citation Context ...n two nodes in a layout p is defined as ‖p(u) − p(v)‖ = ( (xu − xv) 2 + (yu − yv) 2) 1/2 . 3 Stress, Weights, and Constraints 3.1 Stress The foundation of our method is multidimensional scaling (MDS) =-=[3, 9]-=-. Originating in psychometrics and the social sciences, MDS has been established and widely used for graph drawing since its popularization by Kamada and Kawai [19]. While there is a wide range of var... |

84 |
The centrality index of a graph
- Sabidussi
- 1966
(Show Context)
Citation Context ...ion magnifying regions of smaller (larger) centrality scores. Examples of centrality drawings for Zachary’s karate club network are shown in Figure 3. The left column is based on closeness centrality =-=[24]-=- 1 cv = ∑ dvt t∈V , which is simply the inverse average distance from a vertex to all others. The right column contains drawings based on betweenness centrality [14] cv = ∑ δ(s, t|v) , s̸=v̸=t∈V 8clo... |

77 | Animated Exploration of Dynamic Graphs with Radial Layout. InfoVis'01
- Yee, Fisher, et al.
- 2001
(Show Context)
Citation Context ...s the radii used in the constrained layout. Note that distances are represented more clearly, while the two clusters are apparent, still. for dynamic real-time exploration of a filesharing network in =-=[26]-=-, where users interactively select a node to be moved into the center, triggering an update of the immediate surrounding of that node. A different approach is to adapt the Sugiyama framework, original... |

67 | S.: Graph drawing by stress majorization
- GANSNER, KOREN, et al.
(Show Context)
Citation Context ...y by formulating both of the above criteria as objective functions measuring how far a layout is from meeting them. While the first objective is captured by a common function known as weighted stress =-=[16]-=-, we try to accomplish the second goal using stress-like terms measuring the representation error with respect to the radial constraints, and attempt to minimize a linear combination of the two object... |

61 |
Nicheworks – interactive visualization of very large graphs
- Wills
(Show Context)
Citation Context ... in undirected graph layout, the constraints in the second criterion are specific to the application at hand. An example is the exploration of hierarchies with discrete (nominal-scale) layers [8]; in =-=[25]-=- large such hierarchies are laid out radially as a tree, followed by an incremental force-based placement. This approach was later modified 0(a) unconstrained (b) with radial constraints Figure 1: A ... |

51 | Applications of convex analysis to multidimensional scaling - LEEUW - 1977 |

46 |
Automatic display of hierarchized graphs for computer aided decision analysis
- Carpano
- 1980
(Show Context)
Citation Context ...bjective in undirected graph layout, the constraints in the second criterion are specific to the application at hand. An example is the exploration of hierarchies with discrete (nominal-scale) layers =-=[8]-=-; in [25] large such hierarchies are laid out radially as a tree, followed by an incremental force-based placement. This approach was later modified 0(a) unconstrained (b) with radial constraints Fig... |

33 | Communicating centrality in policy network drawings
- Brandes, Kenis, et al.
(Show Context)
Citation Context ...s, to radial layers [1]. In the case of fixed radii defined to represent some continuous (interval-scale) node valuation, unary constraints are imposed on the drawing. This scenarion is introduced in =-=[5]-=- to map any (structural) centrality index to visual centrality. Layouts are determined from a combination of simulated annealing, which is very flexible and allows for penalty costs, e.g., for edge cr... |

31 | Eigensolver methods for progressive multidimensional scaling of large data
- Brandes, Pich
- 2007
(Show Context)
Citation Context ...coordinates are at hand, the iterative process may be initialized at random, but more favorable and robust strategies are available. The experiments of [7] indicate that approximate classical scaling =-=[6]-=- is the method of choice. During stress majorization, new positions ˆp(u) = (ˆxu, ˆyu) for every node u ∈ V can be computed from the current positions using the update rules ∑ v̸=u ˆxu ← wuv (xv + duv... |

23 |
Convergence of the majorization method for multidimensional scaling
- Leeuw
- 1988
(Show Context)
Citation Context ...hreshold value, a predefined number of steps, or some other criterion. The sequence of layouts generated in this way can be shown to have non-increasing stress and to converge towards a local minimum =-=[11]-=-. 3.2 Weights for Constraints In early applications of MDS, each pair u, v of objects was assigned the same unit weight corresponding to wuv = 1 in (1). When a target distance is unknown for some pair... |

22 | Crossing reduction in circular layouts
- Baur, Brandes
- 2004
(Show Context)
Citation Context ...l cost, this method does not scale even to moderately sized graphs, though. For applications in social network analysis, it was therefore replaced by a combinatorial approach based on circular layout =-=[2]-=-. Our present approach addresses the task more uniformly by formulating both of the above criteria as objective functions measuring how far a layout is from meeting them. While the first objective is ... |

20 | Radial level planarity testing and embedding in linear time - Bachmaier, Brandenburg, et al. - 2003 |

16 | A radial adaption of the Sugiyama framework for visualizing hierarchical information
- Bachmaier
- 2007
(Show Context)
Citation Context ...he center, triggering an update of the immediate surrounding of that node. A different approach is to adapt the Sugiyama framework, originally designed for layout in parallel layers, to radial layers =-=[1]-=-. In the case of fixed radii defined to represent some continuous (interval-scale) node valuation, unary constraints are imposed on the drawing. This scenarion is introduced in [5] to map any (structu... |

11 | An experimental study on distancebased graph drawing
- Brandes, Pich
- 2009
(Show Context)
Citation Context ...rates an improving sequence of layouts. When no coordinates are at hand, the iterative process may be initialized at random, but more favorable and robust strategies are available. The experiments of =-=[7]-=- indicate that approximate classical scaling [6] is the method of choice. During stress majorization, new positions ˆp(u) = (ˆxu, ˆyu) for every node u ∈ V can be computed from the current positions u... |

10 |
Multidimensional Scaling. CRC/Chapman and
- Cox, Cox
- 2001
(Show Context)
Citation Context ...n two nodes in a layout p is defined as ‖p(u) − p(v)‖ = ( (xu − xv) 2 + (yu − yv) 2) 1/2 . 3 Stress, Weights, and Constraints 3.1 Stress The foundation of our method is multidimensional scaling (MDS) =-=[3, 9]-=-. Originating in psychometrics and the social sciences, MDS has been established and widely used for graph drawing since its popularization by Kamada and Kawai [19]. While there is a wide range of var... |

10 | Efficient node overlap removal using a proximity stress model
- Gansner, Hu
- 2009
(Show Context)
Citation Context ...the stress term have been used for drawing graphs with explicitly formulated aesthetic criteria, such as the uniform scattering of the nodes in a graph over a unit disk [20], penalizing node overlaps =-=[15]-=-, or preserving a given topology [13]. 1All these approaches modify the target distances themselves in one form or another, while the approach presented here is based on engineering the weights used ... |

10 | A method for depicting social relationships obtained by sociometric testing - Northway - 1940 |

7 | Topology Preserving Constrained Graph Layout
- Dwyer, Marriott, et al.
- 2008
(Show Context)
Citation Context ...awing graphs with explicitly formulated aesthetic criteria, such as the uniform scattering of the nodes in a graph over a unit disk [20], penalizing node overlaps [15], or preserving a given topology =-=[13]-=-. 1All these approaches modify the target distances themselves in one form or another, while the approach presented here is based on engineering the weights used in the stress minimization model. The... |

5 | The binary stress model for graph drawing
- Koren, Civril
- 2008
(Show Context)
Citation Context ... recently, other extensions of the stress term have been used for drawing graphs with explicitly formulated aesthetic criteria, such as the uniform scattering of the nodes in a graph over a unit disk =-=[20]-=-, penalizing node overlaps [15], or preserving a given topology [13]. 1All these approaches modify the target distances themselves in one form or another, while the approach presented here is based o... |

4 |
A model and algorithm for multidimensional scaling with external constraints on the distances
- Borg, Lingoes
- 1980
(Show Context)
Citation Context ...minimizing just σW (p). Then, constraints are granted more and more control over the layout by dynamically changing coefficients in this combination, shifting the bias from one criterion to the other =-=[4]-=-. The relative influence of distance and radial components is determined by the coefficients in the linear combination σ (1−t)·W +t·Z = (1 − t) · σW (p) + t · σZ(p) . (5) This is easily incorporated i... |

4 |
Constrained multidimensional scaling, including confirmation
- Heiser, Meulman
- 1983
(Show Context)
Citation Context ...minimized, and the current iterate preconditions the next step, thus smoothing the sequence of iterates. The multidimensional scaling literature sometimes distinguishes different forms of constraints =-=[17]-=-. With soft (weak) constraints, solutions are allowed to deviate from the given constraints, and this deviation is penalized by additional stress. With hard (strong) constraints, only solutions which ... |

4 |
The multidimensional scaling of "elastic" distances
- McGee
(Show Context)
Citation Context ...pair, it is simply ignored by using a zero weight for its contribution to the stress. The standard weighted scheme for graph drawing uses wuv = d −2 uv . It was introduced as elastic scaling by McGee =-=[21]-=-, and is equal to the one used by Kamada and Kawai [19]. Its superiority is due to an emphasis of small distances over large ones. This is because the fit of local distances is visually important, but... |

3 | Constrained graph layout by stress majorization and gradient projection
- Dwyer, Koren, et al.
(Show Context)
Citation Context ...h the greater degree of freedom during most of the process, it is possible that stress majorization with penalty functions is not only simpler, but also more effective than gradientprojection methods =-=[12]-=- which maintain a feasible solution throughout. An in-depth comparison is therefore an important direction for future research. Other possible extensions of this work include its use in approximate co... |

1 |
do Nascimiento. The metro map layout problem
- Hong, Merrick, et al.
- 2004
(Show Context)
Citation Context ...and readability are more important criteria than the accurate representation of actual geographic positions. In the graph drawing literature, this drawing style is called metro map layout (see, e.g., =-=[18]-=- for a force-directed approach). The seminal design is Harry Beck’s map of London Underground, commonly known as the Tube. It has been and still is being reworked and improved, and it has inspired sim... |

1 | Travel time tube map - Carden |

1 | do Nascimiento, H.A.D.: The metro map layout problem - Hong, Merrick |