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... Issues with regard to computational complexity and layout stability have also been associated with force-directed methods [15]. However, these issues have recently been treated by various approaches =-=[14, 17]-=-. Most of these approaches also use a deterministic model to prevent the highly unpredictable layouts that were often produced by previous force-directed methods. Another method for drawing compound g...

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...ch we based our comparisons differs slightly from the implementation originally suggested [15]. We are using the more common implementation which replaces the two nested loops with a single loop; see =-=[2, 11]-=-. As noted in BranStress 2.5 1.5 0.5 x 105 3 2 1 0 Stress 4 3.5 3 2.5 2 1.5 1 0.5 x 10 5 Majorization Kamada−Kawai Qh882 0 0.5 1 1.5 2 2.5 3 3.5 Time (sec.) Fig. 1. Stress function vs. running time f...

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... to describe the connections between connected pair of nodes ([23], [24]) or simulated annealing, which alludes to the process of heating and cooling metals ([9], [20]). Multi-scale algorithms ([18], =-=[19]-=-) are scalable versions of force-directed methods that work on a coarse representation of a large network, which refine the layout locally to achieve remarkably rapid layout for large networks (10 6 n...

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...e is still the second issue of getting out of the initial random placement. Both these issues receive adequate treatment by incorporating the multi-scale strategy as suggested by several authors; see =-=[9, 10, 11, 21]-=-. These methods considerably improve running times by rapidly constructing a simplified initial globally nice layout and then refining it locally. The fastest of them all, [21], draws a 10 5 -node gra...

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...ing operator to construct the hierarchy. The maximal matching problem is to select the largest possible set of edges in the graph such that no two edges are incident to the same node. Harel and Koren =-=[19]-=- recursively apply an approximate solution to the k-centres problem, using graph theoretic distance as the ideal distance between two nodes. The k-centres problem groups a set of points into k cluster...

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... 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...

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...of the corresponding multi-level drawings depends heavily on the initial embedding of the graph on the plane. Recently, force-directed methods have been considered for the drawing of large graphs. In =-=[37]-=-, a hierarchy of subgraphs is associated with a large graph. One fundamental step of the algorithm is the use of the all-pairs shortest paths making its applicability to very large graphs limited sinc...