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A Space-Filling Visualization Technique for Multivariate Small-World Graphs
May 2012 (vol. 18 no. 5)
pp. 797-809
Pak Chung Wong, Pacific Northwest Nat. Lab., Richland, WA, USA
H. Foote, Pacific Northwest Nat. Lab., Richland, WA, USA
P. Mackey, Pacific Northwest Nat. Lab., Richland, WA, USA
G. Chin, Pacific Northwest Nat. Lab., Richland, WA, USA
Zhenyu Huang, Pacific Northwest Nat. Lab., Richland, WA, USA
J. J. Thomas, Pacific Northwest Nat. Lab., Richland, WA, USA
We introduce an information visualization technique, known as GreenCurve, for large multivariate sparse graphs that exhibit small-world properties. Our fractal-based design approach uses spatial cues to approximate the node connections and thus eliminates the links between the nodes in the visualization. The paper describes a robust algorithm to order the neighboring nodes of a large sparse graph by solving the Fiedler vector of its graph Laplacian, and then fold the graph nodes into a space-filling fractal curve based on the Fiedler vector. The result is a highly compact visualization that gives a succinct overview of the graph with guaranteed visibility of every graph node. GreenCurve is designed with the power grid infrastructure in mind. It is intended for use in conjunction with other visualization techniques to support electric power grid operations. The research and development of GreenCurve was conducted in collaboration with domain experts who understand the challenges and possibilities intrinsic to the power grid infrastructure. The paper reports a case study on applying GreenCurve to a power grid problem and presents a usability study to evaluate the design claims that we set forth.

[1] S.T. Barnard and H.D. Simon, "A Fast Multilevel Implementation of Recursive Spectral Bisection for Partitioning Unstructured Problems," Concurrency: Practice and Experience, vol. 6, no. 2, pp. 101-117, Apr. 1994.
[2] S.K. Card, J.D. Mackinlay, and B. Shneiderman, Readings in Information Visualization, Using Vision to Think. Morgan Kaufmann, 1999.
[3] C. Chen, Information Visualization beyond the Horizon, second ed. Springer, 2006.
[4] J.K. Cullum and R.A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations: Theory. SIAM, 2002.
[5] J. Devore, Probability and Statistics for Engineering and the Sciences, sixth ed. Brooks/Cole, 2007.
[6] G. Di Battista, P. Eades, R. Tamassia, and I.G. Tollis, Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, 1999.
[7] T. Dwyer and L. Nachmanson, "Fast Edge-Routing for Large Graphs," Proc. 17th Int'l Symp. Graph Drawing, pp. 147-158, 2009.
[8] Electricity Infrastructure Operation Centers (EIOC), http:/, 2010.
[9] M. Fiedler, "Laplacian of Graphs and Algebraic Connectivity," Combinatorics and Graph Theory, vol. 25, pp. 57-70, 1989.
[10] M. Friedman, "The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance," J. Am. Statistical Assoc., vol. 32, no. 200, pp. 675-701, 1937.
[11] J.D. Glover and M.S. Sarma, Power System Analysis and Design, third ed. Brooks/Cole, 2002.
[12] GD2011, 19th In'l Symp. Graph Drawing 2011, tue.nlGD2011/, 2011.
[13] GD96 Graph Drawing Contest, data/gd/gd96B96.gph, 2011.
[14] D. Harel and Y. Koren, "A Fast Multi-Scale Algorithm for Drawing Large Graphs," J. Graph Algorithms and Applications, vol. 6, no. 3, pp. 179-202, 2003.
[15] Herman, G. Melancon, and M.S. Marshall, "Graph Visualization and Navigation in Information Visualization: A Survey," IEEE Trans. Visualization and Computer Graphics, vol. 6, no. 1, pp. 24-43, Jan.-Mar. 2000.
[16] M. Holzrichter and S. Oliveira, "A Graph Based Method for Generating the Fiedler Vector of Irregular Problems," Proc. Int'l Parallel and Distributed Processing Symp./Int'l Parallel Processing Symp. (IPDPS/IPPS), pp. 978-985, 1999.
[17] IEEE InfoVis 2011, IEEE Conf. Information Visualization, infovis-welcomeinfovis-welcome, 2011.
[18] J. Graph Algorithms and Applications, http:/, 2011.
[19] LAPACK, http://www.netlib.orglapack, 2011.
[20] M. Manguoglu, "A Highly Efficient Parallel Algorithm for Computing the Fiedler Vector," CoRR abs/1008.1700, 2010.
[21] S. Milgram, "The Small-World Problem," Psychology Today, vol. 2, pp. 60-67, 1967.
[22] C. Muelder and K.-L. Ma, "Rapid Graph Layout Using Space Filling Curves," IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 6, pp. 1301-1308, Nov./Dec. 2008.
[23] C. Mueller, B. Martin, and A. Lumsdaine, "A Comparison of Vertex Ordering Algorithms for Large Graph Visualization," Proc. Asia Pacific Symp. Information Visualisation (APVIS), 2007.
[24] C.C. Paige and M.A. Saunders, "Solution of Sparse Indefinite Systems of Linear Equations," SIAM J. Numerical Analysis, vol. 12, no. 4, pp. 617-629, 1975.
[25] B.N. Parlett, The Symmetric Eigenvalue Problem. SIAM, 1987.
[26] H.-O. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals, New Frontiers of Science. Springer, 1993.
[27] TreeMap,, 2011.
[28] UMFPACK, , 2011.
[29] D. Voorhies, "Space-Filling Curves and A Measure of Coherence," Graphics Gems, pp. 26-30, Academic Press, 1991.
[30] C. Walshaw, "A Multilevel Algorithm for Forced-Directed Graph-Drawing," J. Graph Algorithms and Applications, vol. 7, no. 3, pp. 253-285, 2003.
[31] S. Wasserman and K. Faust, Social Network Analysis-Methods and Applications. Cambridge Univ. Press, 1999.
[32] Western Electricity Coordinating Council (WECC), http:/, 2011.
[33] Why P = 0.05?, 2011.
[34] F. Wilcoxon, "Individual Comparisons by Ranking Methods," Biometrics Bull., vol. 1, no. 6, pp. 80-83, Dec. 1945.
[35] N. Wirth, Algorithms +Data Structures = Programs, Prentice-Hall, 1976.
[36] P.C. Wong, G. ChinJR, H. Foote, P. Mackey, and J. Thomas, "Have Green-A Visual Analytics Framework for Large Semantic Graphs," Proc. IEEE Symp. Visual Analytics Science and Technology, pp. 67-74, Oct. 2006.
[37] P.C. Wong, H. Foote, P. Mackey, G. ChinJR, H. Sofia, and J. Thomas, "A Dynamic Multiscale Magnifying Tool for Exploring Large Sparse Graphs," Information Visualization, vol. 7, no. 2, pp. 105-117, 2008.
[38] P.C. Wong, K. Schneider, P. Mackey, H. Foote, G. ChinJR, R. Guttromson, and J. Thomas, "A Novel Visualization Technique for Electric Power Analytics," IEEE Trans. Visualization and Computer Graphics, vol. 15, no. 3, pp. 410-423, May/June 2009.
[39] P.C. Wong, K.K. Wong, H. Foote, and J. Thomas, "Global Visualization and Alignments of Whole Bacteria Genomes," IEEE Trans. Visualization and Computer Graphics, vol. 9, no. 3, pp. 361-371, July-Sept. 2003.

Index Terms:
vectors,curve fitting,data visualisation,graph theory,power engineering computing,power grids,usability study,space-filling visualization technique,multivariate small-world graphs,GreenCurve technique,multivariate sparse graph,small-world property,fractal-based design approach,spatial cue,node connection,visualization node,Fiedler vector,Laplacian graph,space-filling fractal curve,power grid infrastructure,electric power grid operation,domain expert,power grid problem,Data visualization,Laplace equations,Power grids,Layout,Fractals,Sparse matrices,Partitioning algorithms,visualization techniques and methodologies.,Data and knowledge visualization,information visualization
Pak Chung Wong, H. Foote, P. Mackey, G. Chin, Zhenyu Huang, J. J. Thomas, "A Space-Filling Visualization Technique for Multivariate Small-World Graphs," IEEE Transactions on Visualization and Computer Graphics, vol. 18, no. 5, pp. 797-809, May 2012, doi:10.1109/TVCG.2011.99
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