A Sparse Stress Model
AA Sparse Stress Model (cid:63)
Mark Ortmann, Mirza Klimenta and Ulrik Brandes
Computer & Information Science, University of Konstanz
Abstract.
Force-directed layout methods constitute the most commonapproach to draw general graphs. Among them, stress minimization pro-duces layouts of comparatively high quality but also imposes compar-atively high computational demands. We propose a speed-up methodbased on the aggregation of terms in the objective function. It is akin toaggregate repulsion from far-away nodes during spring embedding buttransfers the idea from the layout space into a preprocessing phase. Aninitial experimental study informs a method to select representatives,and subsequent more extensive experiments indicate that our methodyields better approximations of minimum-stress layouts in less time thanrelated methods.
There are two main variants of force-directed layout methods, expressed eitherin terms of forces to balance or an energy function to minimize [3, 25]. Forconvenience, we refer to the former as spring embedders and to the latter asmultidimensional scaling (MDS) methods.Force-directed layout methods are in wide-spread use and of high practi-cal significance, but their scalability is a recurring issue. Besides investigationsinto adaptation, robustness, and flexibility, much research has therefore beendevoted to speed-up methods [20]. These efforts address, e.g., the speed of con-vergence [10, 11] or the time per iteration [1, 17]. Generally speaking, the mostscalable methods are based on multi-level techniques [13, 18, 21, 34].Experiments [5] suggest that minimization of the stress function [27] s ( x ) = (cid:88) i Algorithm 1: Sparse Stress Input : Graph G = ( V, E ) with w : E → R > , and k number of pivots. Output : α − dimensional layout x ∈ ( R α ) V sample P with |P| = k calculate R , all adapted weights w (cid:48) ip , and all d ip via weighted MSSP x ← PivotMDS(G) [4] rescale x such that (cid:80) { i,j }∈ E || x i − x j || = (cid:80) { i,j }∈ E w ij while relative positional change > − do foreach i ∈ V do foreach dimension α do t α ← (cid:80) j ∈ N ( i ) w ij (cid:18) x αj + dij ( xαi − xαj ) || xi − xj || (cid:19) + (cid:80) p ∈P\ N ( i ) w (cid:48) ip (cid:18) x αp + dip ( xαi − xαp ) || xi − xp || (cid:19)(cid:80) j ∈ N ( i ) w ij + (cid:80) p ∈P\ N ( i ) w (cid:48) ij x i ← t This implies that in order to find the globally optimal position of i we further-more have to find weights w (cid:48) ip , such that w (cid:48) ip (cid:80) j ∈ N ( i ) w ij + (cid:80) p ∈P\ N ( i ) w (cid:48) ip ≈ (cid:80) j ∈R ( p ) w ij (cid:80) i (cid:54) = j w ij .Since our goal is only to reconstruct the proportions, and our model only knowsthe shortest-path distance between all nodes i ∈ V and p ∈ P , we set w (cid:48) ip = s/d ip where s ≥ 1. At the first glance setting s = |R ( p ) | seems appropriate, since p represents |R ( p ) | addends of the stress model. Nevertheless, this strongly over-estimates the weight of close partitions. Therefore, we propose to set s = |{ j ∈R ( p ) : d jp ≤ d ip / }| . This follows the idea that p is only a good representativefor the nodes in R ( p ) that are at least as close to p as to i . Since the graph-theoretic distance between i and j ∈ R ( p ) is unknown, our best guess is that j lies on the shortest path from p to i . Consequently, if d jp ≤ d ip / j mustbe at least as close to p as to i . Note that w (cid:48) pp (cid:48) does not necessarily equal w (cid:48) p (cid:48) p for p, p (cid:48) ∈ P , and if k = n our model reduces to the full stress model. Asymptotic running time: To minimize Eq. (3) in each iteration we displace allnodes i ∈ V according to Eq. (4). Since this requires | N ( i ) | + k constant timeoperations, given that all graph-theoretic distances are known, the total timeper iteration is in O ( kn + m ). Furthermore, only the distances between all i ∈ V and p ∈ P have to be known, which can be done in O ( k ( m + n log n )) time andrequires O ( kn ) additional space. If the graph-theoretic distances for all p ∈ P are computed with a multi-source shortest path algorithm (MSSP), it is possibleto construct R as well as calculate all w (cid:48) ip during its execution without increasingits asymptotic running time. The full algorithm to minimize our sparse stressmodel is presented in Alg. 1. We report on two sets of experiments. The first is concerned with the evaluationof the impact of different pivot sampling strategies. The second set is designed M. Ortmann, M. Klimenta and U. Brandes Table 1. Dataset: n , m , δ ( G ), ∆ ( G ), and D ( G ) denote the number of nodes, edges,the min. and max. degree, and the diameter, respectively. Column { deg ( i ) } shows thedegree and { d ij } the distance distribution. Bipartite graphs are marked with ∗ andweighted graphs with ∗∗ graph n m δ ( G ) ∆ ( G ) D ( G ) { deg ( i ) } { d ij } graph n m δ ( G ) ∆ ( G ) D ( G ) { deg ( i ) } { d ij } dwt1005 1005 3808 3 26 34 pesa 11738 33914 2 9 2081138bus 1138 1458 1 17 31 bodyy5 18589 55346 2 8 132plat1919 1919 15240 2 18 43 finance256 20657 71866 1 54 553elt 4740 13722 3 9 65 btree (binary tree) 1023 ∗ ∗ ∗∗ ∗ ∗∗ to assess how well the different sparse stress models approximate the full stressmodel, in both absolute terms and in relation to the speed-up achieved.For the experiments we implemented the sparse stress model, Alg. 1, as wellas different sampling techniques in Java using Oracle SDK 1.8 and the yFiles2.9 graph library ( . yworks . com ). The tests were carried out on a single 64-bitmachine with a 3.60GHz quad-core Intel Core i7-4790 CPU, 32GB RAM, runningUbuntu 14.10. Times were measured using the System.currentTimeMillis() command. The reported running times were averaged over 25 iterations. We notehere that all drawing algorithms, except stated otherwise, were initialized witha 200 PivotMDS layout [4]. Furthermore, the maximum number of iterations forthe full stress algorithm was set to 500. As stress is not resilient against scaling,see Eq. (1), we optimally rescaled each drawing such that it creates the lowestpossible stress value [2]. Data: We conducted our experiments on a series of different graphs, see Tab. 1,most of them taken from the sparse matrix collection [8]. We selected thesegraphs as they differ in their structure and size, and are large enough to comparethe results of different techniques. Two of the graphs, LeHavre and commanche ,have predefined edge lengths that were derived from the node coordinates. Wedid not modify the graphs in any way, except for those that were disconnected.In this case we only kept the largest component. In Section 3 we discussed how vital the proper selection of the pivots is for ourmodel. In the optimal case we would sample pivots that are well distributed overthe graph, creating regions of equal complexity, and are central in the drawingof their regions. In order to evaluate the impact of different sampling strategieson the quality of our sparse stress model and recommend a proper samplingscheme, we compared a set of different strategies: – random: nodes are selected uniformly at random – MIS filtration: nodes are sampled according to the maximal independent setfiltration algorithm by Gajer et al. [13]. Once n ≤ k the coarsening stops. If n < k , unsampled nodes from the previous level are randomly added Sparse Stress Model 7 – max/min euclidean: starting with a uniformly randomly chosen node, P isextended by adding arg max i ∈ V \P min p ∈P || x i − x p || – max/min sp: similar to max/min euclidean except that P is extended ac-cording arg max i ∈ V \P min p ∈P d ip [4]Pretests showed that the max/min sp strategy initially favors sampling leaves,but nevertheless produces good results for large k . Thus, we also evaluated strate-gies building on this idea, yet try to overcome the problem of leaf node sampling. – max/min random sp: similar to max/min sp, yet each node i is sampled witha probability proportional to min p ∈P d ip – k-means layout: the nodes are selected via a k-means algorithm, running atmost 50 iterations, on the initial layout – k-means sp: initially k nodes with max/min sp are sampled succeeded byk-means sampling using the shortest path entries of these pivots – k-means + max/min sp: P is initialized with k/ k ∈ { , , . . . , } pivots. Forall tests the sparse stress algorithm terminated after 200 iterations. Since alltechniques at some point rely on a random decision, we repeated each execution20 times in order to ensure we do not rest our results upon outliers. To distinguishthe applicability of the different techniques to our model, we used two measures.The first measure is the normalized stress, which is the stress value divided by (cid:0) n (cid:1) . While the normalized stress measures the quality of our drawing, we alsocalculated the Procrustes statistic, which measures how well the layout matchesthe full stress drawing [30]. The range of the Procrustes statistic is [0 , k by max/min sp, is pesa . The reason for this result is that k-means spmainly samples pivots in the center of the left arm, see Tab. 4, creating twists.Max/min sp for small k in contrast mostly samples nodes on the contour ofthe arm, yet once k reaches a certain threshold the resulting distribution of thepivots prevents twists, yielding a lower normalized stress value.The explanation of the poor behavior for lpship04l is strongly related to itsstructure. The low diameter of 13 causes, after a few iterations, the max/minsp strategy to repeatedly sample nodes that are part of the same cluster, seeTab. 4, and consequently are structurally very similar. As k-means sp builds on M. Ortmann, M. Klimenta and U. Brandes ll llllll ll llllll ll llllll ll llllll ll llllll ll llllll ll llllll ll llllll ll llllll ll llllll ll llllll ll llllll ll llllll ll llllll c o mm an c he pe s a number of pivots no r m a li z ed s t r e ss sampling strategy llllllll max/min euclideanrandomMIS filtrationmax/min spk−means layoutk−means + max/min spmax/min random spk−means sp Fig. 1. Comparison of different sampling strategies and number of pivots w.r.t. theresulting normalized stress value ll llll ll ll llll ll ll llll ll ll llll ll ll llll ll ll llll ll ll llll ll ll llll ll ll llll ll ll llll ll ll llll ll ll llll ll ll llll ll ll llll ll b t r ee U S po w e r G r i d number of pivots P r o c r u s t e s s t a t i s t i c sampling strategy llllllll max/min euclideanMIS filtrationrandomk−means layoutk−means + max/min spmax/min random spmax/min spk−means sp Fig. 2. Comparison of different sampling strategies and number of pivots w.r.t. theresulting Procrustes statistic max/min sp, it can only slightly improve the pivot distribution. The argumentthat the problem is related to the structure is reinforced by the outcome ofthe random strategy. Still, except for these two graphs k-means sp generatesthe best outcomes, and since this strategy is also strongly favorable over theothers subject to the Procrustes statistics, see Fig. 2, our following evaluationalways relies on this sampling strategy. However, we note that the Procrustesstatistic for btree and lpship04l are by magnitudes larger than for any othertested graph. While for lpship04l this is mostly caused by the quality of thedrawings, this is only partly true for btree . The other factor contributing to thehigh Procrustes statistic for btree is caused by the restricted set of operationsprovided by the Procrustes analysis. As dilation, translation, and rotation areused to find the best match between two layouts, the Procrustes analysis cannotresolve reflections. Therefore, if in the one layout of btree , the subtree T of v is drawn to the right of subtree T of v and vice versa in the second drawing,although the two layouts are identical, the statistic will be high. This symmetryproblem mainly explains the low performance w.r.t. btree . The next set of experiments is designed to assess how well our sparse stress modelusing k-means sp sampling, as well as related sparse stress techniques, resemblesthe full stress model. For this we compared the median stress layout over 25repetitions on the same graph of our sparse stress model with k ∈ { , , } , Sparse Stress Model 9 Table 2. Stress and Procrustes statistics: sparse model values are highlighted when nolarger than minimum over previous methods graph full stress sparse 200 sparse 100 sparse 50 maxent MARS 200 MARS 100 GRIP 1-stress PivotMDS stress dwt1005 10729 Procrustes statistic dwt1005 Table 3. Runtime in seconds: fastest sparse model yielding lower stress than bestprevious method, c.f. Table 2, is highlighted. Marked implementations written in C/C++ with time measured via clock() command graph full stress sparse 200 sparse 100 sparse 50 maxent ∗ MARS 200 ∗ MARS 100 ∗ GRIP ∗ with MARS, maxent, PivotMDS, 1-stress, and the weighted version of GRIP. The number of iterations of our model as well as for MARS and 1-stress havebeen limited to 200. Furthermore, we tested MARS with 100 and 200 pivotsand report the layout with the smallest stress from the drawings obtained byrunning mars with argument -p ∈ { , } combined with a PivotMDS or randomlyinitialized layout.Besides comparing the resulting stress values and Procrustes statistics, wecompared the distribution of pairwise euclidean distances subject to their graph-theoretic distances. Since the Procrustes statistic has problems with symmetries,as we pointed out in the previous subsection, we propose to evaluate the similar-ity of the sparse stress layouts with the full stress layout via Gabriel graphs [12].The Gabriel graph of a given layout x contains an edge between a pair of points https://github . com/marckhoury/mars We are grateful to Yifan Hu for providing us with the code. . cs . arizona . edu/~kobourov/GRIP/ ll llllll l ll llllll l ll llllll l ll llllll l ll llllll lll llllll l ll llllll l ll llllll l ll llllll l ll llllll l d w t H a v r e k−neighborhood J a cc a r d algorithm lllllllll GRIPMARS 1001−stressMARS 200PivMDSmaxentsparse 50sparse 100sparse 200 Fig. 3. The similarity of the Gabriel Graph of the full stress layout and the GabrielGraph of the layout algorithms under consideration as a function of k . For each nodeof the graph the k-neighborhood in the Gabriel Graph of the full stress layout and thelayout algorithm are compared by calculating the Jaccard coefficient. A higher valueindicates that the nodes share a high percentage of common neighbors in the differentGabriel Graphs. if and only if the disc associated with the diameter of the endpoints does notcontain any other point. Since the treatment of identical positions is not definedfor Gabriel Graphs, we resolve this by adding edges between each pair of iden-tical positions. We assess the similarity between the Gabriel Graph of the fullstress layout and the sparse stress layouts by comparing the k-neighborhoods ofa node in the graphs using the Jaccard coefficient.A further measure we introduce evaluates the visual error. More preciselywe measure for a given node v the percentage of nodes that lie in the drawingarea of the k-neighborhood, but are not part of it. We calculate this value bycomputing the convex hull induced by the k-neighborhood and then test foreach other node if it belongs to the hull or not. This number is then divided by n − |{ w ∈ V | d vw ≤ k }| . Therefore, a low value implies that there are only a fewnodes lying in the region, while high values indicate we cannot distinguish nonk-neighborhood and k-neighborhood nodes in the drawing. This measure is to acertain extend similar to the precision of neighborhood preservation [15].The results of all these experiments, see Tabs. 2 and 4, Figs. 3 and 4, andthe Appendix, reveal that our model is more adequate in resembling the fullstress drawing than any other of the tested algorithm, while showing comparablerunning times that scale nicely with k , cf. Tab. 3. The error plots in Tab. 4 exposethe strength of our approximation scheme. We can see that, while all approacheswork very well in representing short distances, our approach is more precise inapproximating middle and especially long distances, explaining our good results.As the evaluation clearly shows that our approach yields better approximationsof the full stress model, we rather want to discuss the low performance of ourmodel for lpship04l and thereby expose one weakness of our approach.Looking at the sparse 50 drawing of lpship04l in Tab. 4, we can see that alarge portion of nodes share a similar or even the same position. This is because Sparse Stress Model 11 lll llllll l lll llllll l lll llllll l lll llllll l lll llllll llll llllll l lll llllll l lll llllll l lll llllll l lll llllll l bod yy f i nan c e256 k−neighborhood E rr o r algorithm llllllllll MARS 1001−stressMARS 200maxentGRIPPivMDSsparse 50sparse 100sparse 200full stress Fig. 4. Error charts as a function of k . For each node of the graph the convex hullw.r.t. the coordinates of the nodes in the k-neighborhood is computed. For each of theconvex hulls the error is calculated by counting the number of non k-neighborhoodnodes that lie inside or on the contour of this hull divided by their total number. lpship04l has a lot of nodes that share very similar graph-theoretic distance vec-tors, exhibit highly overlapping neighborhoods, and are drawn in close proximityin the initial PivotMDS layout. While our model would rely on small variationsof the graph-theoretic distances to create a good drawing we diminish thesedifferences even further by restricting our model to P . Consequently, the posi-tional vote for two similar non-pivot nodes i and j that lie in the same partitionwill only slightly differ, mainly caused by their distinct neighbors. However, asthese neighbors are also in close proximity in the initial drawing of lpship04l the distance between i and j will not increase. Therefore, if the graph has a lotof structurally very similar nodes and the initial layout has poor quality, ourapproach will inevitably create drawings where nodes are placed very close toone another. In this paper we proposed a sparse stress model that requires O ( kn + m ) spaceand time per iteration, and a preprocessing time of O ( k ( m + n log n )). WhileBarnes & Hut derive their representatives from a given partitioning, we arguedthat for our model it is more appropriate to first select the pivots and then topartition the graph only relying on its structure. Since the approximation qualityheavily depends on the proper selection of these pivots, we evaluated differentsampling techniques, showing that k-means sp works very well in practice.Furthermore, we compared a variety of sparse stress models w.r.t. their per-formance in approximating the full stress model. We therefore proposed two newmeasures to assemble the similarity between two layouts of the same graph. Forthe tested graphs, all our experiments clearly showed that our proposed sparsestress model exceeds related approaches in approximating the full stress layoutwithout compromising the computation time. Table 4. Layouts and error charts of the algorithms. Each chart shows the zero ycoordinate (black horizontal line), the median (red line), the 25 and 75 percentiles(black/gray ribbon) and the min/max error (outer black dashed line). The error (y-axis) is the difference between the euclidean distance and the graph-theoretic distance(x-axis). 1000 bins have been used for weighted graphs graph full stress sparse 200 sparse 100 sparse 50 maxent MARS 200 MARS 100 GRIP 1-stress PivotMDS d w t bu s p l a t e l t c o mm a n c h e p e s a fin a n c e q h l p s h i p l ibliography [1] Barnes, J., Hut, P.: A hierarchical O( n log n ) force-calculation algorithm.Nature 324(6096), 446–449 (1986), http://dx . doi . org/10 . [2] Borg, I., Groenen, P.J.: Modern Multidimensional Scaling: Theory and Ap-plications. Springer (2005)[3] Brandes, U.: Drawing on physical analogies. In: Kaufmann, M., Wagner, D.(eds.) Drawing Graphs: Methods and Models. LNCS, vol. 2025, pp. 71–86.Springer (2001), http://dx . doi . org/10 . [4] Brandes, U., Pich, C.: Eigensolver methods for progressive multidimensionalscaling of large data. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS,vol. 4372, pp. 42–53. Springer (2007), http://dx . doi . org/10 . [5] Brandes, U., Pich, C.: An experimental study on distance-based graphdrawing. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417,pp. 218–229. Springer (2009), http://dx . doi . org/10 . [6] Brandes, U., Schulz, F., Wagner, D., Willhalm, T.: Travel planning withself-made maps. In: Buchsbaum, A.L., Snoeyink, J. (eds.) ALENEX2001. LNCS, vol. 2153, pp. 132–144. Springer (2001), http://dx . doi . org/10 . [7] Cohen, J.D.: Drawing graphs to convey proximity: An incremental arrange-ment method. ACM Transactions on Computer-Human Interaction 4(3),197–229 (1997), http://doi . acm . org/10 . . [8] Davis, T.A., Hu, Y.: The University of Florida sparse matrix collection.ACM Transactions on Mathematical Software 38(1), 1:1–1:25 (2011), . cise . ufl . edu/research/sparse/matrices [9] Drineas, P., Frieze, A.M., Kannan, R., Vempala, S., Vinay, V.:Clustering large graphs via the singular value decomposition. Ma-chine Learning 56(1-3), 9–33 (2004), http://dx . doi . org/10 . . . . [10] Frick, A., Ludwig, A., Mehldau, H.: A fast adaptive layout algorithm forundirected graphs. In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol.894, pp. 388–403. Springer (1995), http://dx . doi . org/10 . [11] Fruchterman, T.M.J., Reingold, E.M.: Graph drawing by force-directedplacement. Software: Practice and Experience 21(11), 1129–1164 (1991), http://dx . doi . org/10 . . [12] Gabriel, R.K., Sokal, R.R.: A new statistical approach to geographic varia-tion analysis. Systematic Zoology 18(3), 259–278 (1969)[13] Gajer, P., Goodrich, M.T., Kobourov, S.G.: A multi-dimensional approachto force-directed layouts of large graphs. In: Marks, J. (ed.) GD 2000. LNCS,vol. 1984, pp. 211–221. Springer (2001), http://dx . doi . org/10 . [14] Gansner, E.R., Hu, Y., Krishnan, S.: COAST: A convex optimization ap-proach to stress-based embedding. In: Wismath, S.K., Wolff, A. (eds.) GD2013. LNCS, vol. 8242, pp. 268–279. Springer (2013), http://dx . doi . org/10 . [15] Gansner, E.R., Hu, Y., North, S.C.: A maxent-stress model for graph layout.IEEE Transactions on Visualization and Computer Graphics 19(6), 927–940(2013), http://dx . doi . org/10 . . . [16] Gansner, E.R., Koren, Y., North, S.: Graph drawing by stress majorization.In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 239–250. Springer (2004), http://dx . doi . org/10 . [17] Greengard, L.: The Rapid evaluation of potential fields in particle systems.ACM distinguished dissertations, Cambridge, Mass. MIT Press (1988), http://opac . inria . fr/record=b1086802 [18] Hachul, S., J¨unger, M.: Drawing large graphs with a potential-field-basedmultilevel algorithm. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 285–295. Springer (2004), http://dx . doi . org/10 . [19] Hall, K.M.: An r -dimensional quadratic placement algorithm. ManagmentScience 17(3), 219–229 (1970), http://dx . doi . org/10 . . . . [20] Hu, Y., Shi, L.: Visualizing large graphs. Wiley Interdisciplinary Re-views: Computational Statistics 7(2), 115–136 (2015), http://dx . doi . org/10 . . [21] Hu, Y.F.: Efficient and high quality force-directed graph drawing.The Mathematica Journal 10, 37–71 (2005), . mathematica-journal . com/issue/v10i1/contents/graph draw/graph draw . pdf [22] Ingram, S., Munzner, T.: Glint: An MDS framework for costly distance func-tions. In: Kerren, A., Seipel, S. (eds.) Proceedings of SIGRAD 2012, Interac-tive Visual Analysis of Data. Link¨oping Electronic Conference Proceedings,vol. 81, pp. 29–38. Link¨oping University Electronic Press (2012), . ep . liu . se/ecp article/index . en . aspx?issue=081;article=005 [23] Khoury, M., Hu, Y., Krishnan, S., Scheidegger, C.E.: Drawing large graphsby low-rank stress majorization. Computer Graphics Forum 31(3), 975–984(2012), http://dx . doi . org/10 . . . . . x [24] Klimenta, M., Brandes, U.: Graph drawing by classical multidimensionalscaling: New perspectives. In: Didimo, W., Patrignani, M. (eds.) GD 2012.LNCS, vol. 7704, pp. 55–66. Springer (2013), http://dx . doi . org/10 . [25] Kobourov, S.G.: Force-directed drawing algorithms. In: Tamassia, R. (ed.)Handbook of Graph Drawing and Visualization, pp. 383–408. CRC Press(2013)[26] Koren, Y., Carmel, L., Harel, D.: ACE: A fast multiscale eigenvectors com-putation for drawing huge graphs. In: Wong, P.C., Andrews, K. (eds.) Info-Vis 2002. pp. 137–144. IEEE Computer Society (2002), http://dx . doi . org/10 . . . [27] McGee, V.E.: The multidimensional analysis of ”elastic” distances. BritishJournal of Mathematical and Statistical Psychology 19(2), 181–196 (1966), http://dx . doi . org/10 . . . . tb00367 . x Sparse Stress Model 15 [28] Meyerhenke, H., N¨ollenburg, M., Schulz, C.: Drawing large graphs by mul-tilevel maxent-stress optimization. In: Giacomo, E.D., Lubiw, A. (eds.) GD2015. LNCS, vol. 9411, pp. 30–43. Springer (2015), http://dx . doi . org/10 . [29] Quigley, A.J.: Large Scale Relational Information Visualization, Clustering,and Abstraction. Ph.D. thesis, University of Newcastle (2000)[30] Sibson, R.: Studies in the robustness of multidimensional scaling: Procrustesstatistics. Journal of the Royal Statistical Society. Series B (Methodological)40(2), 234–238 (1978), . jstor . org/stable/2984761 [31] de Silva, V., Tenenbaum, J.B.: Global versus local methods in non-linear dimensionality reduction. In: Becker, S., Thrun, S., Ober-mayer, K. (eds.) NIPS 2002. pp. 705–712. MIT Press (2002), http://papers . nips . cc/paper/2141-global-versus-local-methods-in-nonlinear-dimensionality-reduction [32] Tunkelang, D.: JIGGLE: Java interactive graph layout environment. In:Whitesides, S. (ed.) GD 1998. LNCS, vol. 1547, pp. 412–422. Springer(1998), http://dx . doi . org/10 . [33] Tunkelang, D.: A Numerical Optimization Approach to General GraphDrawing. Ph.D. thesis, Carnegie Mellon University (1999)[34] Walshaw, C.: A multilevel algorithm for force-directed graph-drawing.Journal of Graph Algorithms and Applications 7(3), 253–285 (2003), . cs . brown . edu/publications/jgaa/accepted/2003/Walshaw2003 . . . pdf Appendix Table 5. Layouts and error charts of the algorithms. Each chart shows the zero ycoordinate (black horizontal line), the median (red line), the 25 and 75 percentiles(black/gray ribbon) and the min/max error (outer black dashed line). The error (y-axis) is the difference between the euclidean distance and the graph-theoretic distance(x-axis). 1000 bins have been used for weighted graphs graph full stress sparse 200 sparse 100 sparse 50 maxent MARS 200 MARS 100 GRIP 1-stress PivotMDS d w t bu s p l a t e l t U Sp o w e r G r i d Sparse Stress Model 17 graph full stress sparse 200 sparse 100 sparse 50 maxent MARS 200 MARS 100 GRIP 1-stress PivotMDS c o mm a n c h e L e H a v r e p e s a b o d yy fin a n c e b t r ee q h l p s h i p l ll ll l l ll ll ll l l ll ll ll l l ll ll ll l l ll ll ll l l ll ll ll l l ll ll ll l l ll commanche0.02250.02500.0275 50 100 150 200 number of pivots no r m a li z ed s t r e ss sampling strategy llllllll max/min euclideanrandomMIS filtrationmax/min spk−means layoutk−means + max/min spmax/min random spk−means sp lll l ll lllll lll lllll llllllll llllllll llllllll lllllll l llllll l ll ll llll ll ll llll ll ll llll ll ll llll ll ll llll ll llllll ll lllll l ll ll llll ll ll llll ll ll llll ll ll llll ll ll llll ll llllll ll llll ll lll l llll lll lllll l ll lllll l ll lllll lll lllll llllllll lllllll l ll ll lll l ll ll llll ll ll llll ll ll llll ll ll llll ll llllll ll lllll ll l lllll ll l lllllll l lllllll llllllll llllllll llllllll lllll lll l lllllll l lllllll llllllll llllllll llllllll llllllll llllllll ll ll llll ll ll llll ll ll llll ll ll llll ll ll llll ll ll llll ll ll lll ll ll llll ll ll llll ll llllll ll llllll ll llllll ll llllll llllll ll lll lllll llllllll llllllll llllllll llllllll llllllll lllllllll l lllllll l lllllll lllllll l lllllll llllllll llllllll lllllllll l lllllll llllllll llllllll llllllll llllllll llllllll lllll . . . 072 0 . . . . 044 0 . . . 023 0 . . . 16 0 . . . . . . . . . . . . 034 0 . . . . 012 0 . . . . 12 0 . . . . 024 0 . . . . . . . . . 062 0 . . . . Fig. 5. Comparison of different sampling strategies and number of pivots w.r.t. theresulting normalized stress value Sparse Stress Model 19 ll ll l l ll ll ll l l ll ll ll l l ll ll ll l l ll ll ll l l ll ll ll l l ll ll ll l l ll USpowerGrid0.010.020.030.04 50 100 150 200 number of pivots P r o c r u s t e s s t a t i s t i c sampling strategy llllllll max/min euclideanMIS filtrationrandomk−means layoutk−means + max/min spmax/min random spmax/min spk−means sp ll ll ll llll ll ll llll llllllll llllllll llllllll llllllll lllllll l ll ll lll l ll ll llll ll ll llll ll ll llll ll llllll ll llllll lllllll l ll ll lll l ll ll llll ll ll llll ll llllll ll llllll ll llllll llllll ll lll l llll lll l llll llll llll ll lll ll lll lll ll llllll ll llllll lll ll l l llll ll l l llll ll ll llll ll l l llll ll ll llll ll ll llll llll lll l ll ll lll l ll ll lll l ll ll lll l ll ll lll l ll ll llll ll llllll ll llll l ll l l llllll ll llllll ll l lllll ll ll llll ll ll lll lll ll lll lll ll l lll l ll lllll l ll lllll l lllllll lllllllllllllllllllllllllllllll lllllllll llll llll llll llllllll llllllll llllllll llllllll l l l ll lllll l ll lll lll lllll lll lllll lll lllll lll lll ll llllll ll l ll llllllll llllllll llllllll llllllll llllllll lllllllllllllll ll l ll llllll llll ll ll llllll ll llllll llllllll llllllll llllll . . . 06 0 . . . . . 020 0 . . . . 006 0 . . . . . . . . 015 0 . . . 009 0 . . . . . . . . 005 0 . . . . . . 015 0 . . . . 03 0 . . . . . Fig. 6. Comparison of different sampling strategies and number of pivots w.r.t. theProcrustes statistic0 M. Ortmann, M. Klimenta and U. Brandes ll llllll l ll llllll l ll llllll l ll llllll l ll llllll l dwt10050.000.250.500.751.00 1 2 3 4 5 k−neighborhood J a cc a r d algorithm lllllllll GRIPMARS 1001−stressMARS 200PivMDSmaxentsparse 50sparse 100sparse 200 lllll ll lllllll l l lllllll l l lllllll l l ll lllll l l ll lllll ll lllllll ll lllllll l l lllllll l l lllllll l l lllllll l ll llllll l ll llllll l ll llllll l ll llllll l ll l lllll l llll llll l llll llll l llll llll l llll llll l lll lllll l llllllll l llllllll l llllllll l lll lllll l lll l lllll llll llll l llll llll lllll llll lllll llll llll lllll l llllllll l llllllll l llllllll l llllllll l lll llllll llll lllll llll lllll llll lllll llll lllll llll llll ll lllllll ll lllllll ll lllllll ll lllllll ll ll l llll l l ll lllll l l ll lllll l lll lllll l lll lllll l lll llllll l lll llll l l lllllll l l lllllll l l ll lllll l l ll l llll l llllllll l llllllll l lll lllll l lll lllll l lll . . . . . . . . . 00 0 . . . . . 00 0 . . . . . . . . . . . . . . . . . . . . . . 00 0 . . . . . . . . . . 00 0 . . . . . 00 0 . . . . . . . . Fig. 7. The similarity of the Gabriel Graph of the full stress layout and the GabrielGraph of the layout algorithms under consideration as a function of k . For each nodeof the graph the k-neighborhood in the Gabriel Graph of the full stress layout and thelayout algorithm are compared by calculating the Jaccard coefficient. A higher valueindicates that the nodes share a high percentage of common neighbors in the differentGabriel Graphs. Sparse Stress Model 21 lll llllll l lll llllll l lll llllll l lll llllll l lll llllll l finance2560.000.010.020.030.04 1 2 3 4 5 k−neighborhood E rr o r algorithm llllllllll MARS 1001−stressMARS 200maxentGRIPPivMDSsparse 50sparse 100sparse 200full stress llllllllllllllllllllllllllllllllllllllllll l lllllll lllllllllllllllllll llllllllll llll llllll llll llllll l llllllllllllllllllll llllllllll lll lllllll lll lllll ll lllllllllllll llllllllll lllllll lll lllllll lll lllll ll lllllllllllll llllllllll llllllllll llllllllll lllllll llllllllllllllllllllllllllll llllllllll llllllllll ll llllllllllllllllllllllllllll llllllllll llllllllll ll ll lllllllllll llllll l lll lllll l l lll llllll llll llllll l lllllllllllllllllll llllllllll l lllllllll l lllllllll l lll llllllllll llllllllll llllll llll llllll llll llllll llllllllllllllllllllllllllllll llllllllll llllllllll l llllllllllllllllllllllllllllllllllllll llllllllll ll . . . . . 20 0 . . . . 06 0 . . . . 015 0 . . . . . . . . . . . 025 0 . . . . 15 0 . . . . . . 025 0 . . . . . . . . . . 100 0 . . . . 09 0 . . . . 15 0 . . . . . Fig. 8. Error charts as a function of kk