Subscribe
Issue No.11 - November (2011 vol.17)
pp: 1587-1598
Vladimir Batagelj , University of Ljubljana, Ljubljana
Franz J. Brandenburg , University of Passau, Passau
Walter Didimo , Università degli Studi di Perugia, Perugia
Giuseppe Liotta , Università degli Studi di Perugia, Perugia
Pietro Palladino , Università degli Studi di Perugia, Perugia
Maurizio Patrignani , Roma Tre University, Roma
ABSTRACT
Many different approaches have been proposed for the challenging problem of visually analyzing large networks. Clustering is one of the most promising. In this paper, we propose a new clustering technique whose goal is that of producing both intracluster graphs and intercluster graph with desired topological properties. We formalize this concept in the (X,Y) -clustering framework, where Y is the class that defines the desired topological properties of intracluster graphs and X is the class that defines the desired topological properties of the intercluster graph. By exploiting this approach, hybrid visualization tools can effectively combine different node-link and matrix-based representations, allowing users to interactively explore the graph by xpansion/contraction of clusters without loosing their mental map. As a proof of concept, we describe the system Visual Hybrid (X,Y)-clustering (VHYXY) that implements our approach and we present the results of case studies to the visual analysis of social networks.
INDEX TERMS
Large graphs, graph clustering, hybrid visualization, visual analytics.
CITATION
Vladimir Batagelj, Franz J. Brandenburg, Walter Didimo, Giuseppe Liotta, Pietro Palladino, Maurizio Patrignani, "Visual Analysis of Large Graphs Using (X,Y)-Clustering and Hybrid Visualizations", IEEE Transactions on Visualization & Computer Graphics, vol.17, no. 11, pp. 1587-1598, November 2011, doi:10.1109/TVCG.2010.265
REFERENCES
 [1] J. Abello , S. Kobourov , and R. Yusufov , “Visualizing Large Graphs with Compound-Fisheye Views and Treemaps,” Proc. Graph Drawing (GD '04), pp. 431-441, 2004. [2] J. Abello , F. van Ham , and N. Krishnan , “Ask-Graphview: A Large Scale Graph Visualization System,” IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 5, pp. 669-676, Sept./Oct. 2006. [3] D. Archambault , T. Munzner , and D. Auber , “Grouse: Feature-Based, Steerable Graph Hierarchy Exploration,” Proc. Eurographics, pp. 67-74, 2007. [4] D. Archambault , T. Munzner , and D. Auber , “Topolayout: Multilevel Graph Layout by Topological Features,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 2, pp. 305-317, Mar./Apr. 2007. [5] D. Archambault , T. Munzner , and D. Auber , “Grouseflocks: Steerable Exploration of Graph Hierarchy Space,” IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 4, pp. 900-913, July/Aug. 2008. [6] D. Auber , Y. Chiricota , F. Jourdan , and G. Melançon , “Multiscale Visualization of Small World Networks,” Proc. IEEE Symp. Information Visualization (INFOVIS '03), pp. 75-81, 2003. [7] V. Batagelj , F.J. Brandenburg , W. Didimo , G. Liotta , P. Palladino , and M. Patrignani , “Collapsing $k$ -Cliques to Get a Planar Graph Is NP-Complete,” Technical Report RT-DIA-174-2010, Dept. Computer Science and Automation, Roma Tre Univ., 2010. [8] V. Batagelj , A. Mrvar , and M. Zaveršnik , “Partitioning Approach to Visualization of Large Networks,” Proc. Graph Drawing (GD '99), pp. 90-97, 1999. [9] V. Batagelj and M. Zaveršnik , “An O(m) Algorithm for Cores Decomposition of Networks,” CoRR, cs.DS/0310049, 2003. [10] M. Baur and U. Brandes , “Crossing Reduction in Circular Layouts,” Proc. WG '04, pp. 332-343, 2004. [11] F. Bertault , “A Force-Directed Algorithm that Preserves Edge Crossing Properties,” Proc. Graph Drawing, pp. 351-358, 1999. [12] B. Bollobás , “The Evolution of Sparse Graphs,” Graph Theory and Combinatorics, pp. 35-57, Academic Press, 1984. [13] F.-J. Brandenburg , “Graph Clustering 1: Circles of Cliques,” Proc. Int'l Symp. Graph Drawing (GD '97), pp. 158-168, 1997. [14] U. Brandes , M. Eiglsperger , I. Herman , M. Himsolt , and M. Marshall , “Graphml Progress Report: Structural Layer Proposal,” Proc. Ninth Int'l Symp. Graph Drawing, 2002. [15] A. Carmignani , G.D. Battista , W. Didimo , F. Matera , and M. Pizzonia , “Visualization of the High Level Structure of the Internet with HERMES,” J. Graph Algorithms and Applications, vol. 6, no. 3, pp. 281-311, 2002. [16] B. Craft and P. Cairns , “Beyond Guidelines: What Can We Learn from the Visual Information Seeking Mantra?,” Proc. IEEE Ninth Int'l Conf. Information Visualization, pp. 110-118, 2005. [17] G. Di Battista , W. Didimo , M. Patrignani , and M. Pizzonia , “Orthogonal and Quasi-Upward Drawings with Vertices of Prescribed Sizes,” Proc. Graph Drawing (GD '99), pp. 297-310, 1999. [18] G. Di Battista , P. Eades , R. Tamassia , and I.G. Tollis , Graph Drawing. Prentice Hall, 1999. [19] E. Di Giacomo , W. Didimo , L. Grilli , and G. Liotta , “Graph Visualization Techniques for Web Clustering Engines,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 2, pp. 294-304, Mar. 2007. [20] E. Di Giacomo , W. Didimo , G. Liotta , and P. Palladino , “Visual Analysis of One-to-Many Matched Graphs,” to be published in, J. Graph Algorithms Application—Special Issues of GD 2008, 2009. [21] P. Doreian , V. Batagelj , and A. Ferligoj , Generalized Blockmodeling. Cambridge Univ. Press, 2005. [22] S.N. Dorogovtsev , A.V. Goltsev , and J.F.F. Mendes , “k-Core Organization of Complex Networks,” CoRR, abs/cond-mat/0509102, 2005. [23] T. Dwyer , K. Marriott , and M. Wybrow , “Topology Preserving Constrained Graph Layout,” Proc. Graph Drawing, pp. 230-241, 2008. [24] P. Eades , “A Heuristic for Graph Drawing,” Proc. Congressus Numerantium, vol. 42, pp. 149-160, 1984. [25] P. Eades , R.F. Cohen , and M.L. Huang , “Online Animated Graph Drawing for Web Navigation,” Proc. Fifth Int'l Symp. Graph Drawing (GD '97), pp. 330-335, 1997. [26] P. Eades and M.L. Huang , “Navigating Clustered Graphs Using Force-Directed Methods,” J. Graph Algorithms and Applications, vol. 4, no. 3, pp. 157-181, 2000. [27] M. Eiglsperger , M. Siebenhaller , and M. Kaufmann , “An Efficient Implementation of Sugiyama's Algorithm for Layered Graph Drawing,” Proc. Graph Drawing (GD '05), pp. 155-166, 2005. [28] M. Gaertler and M. Patrignani , “Dynamic Analysis of the Autonomous System Graph,” Proc. Int'l Workshop Inter-Domain Performance and Simulation (IPS '04), pp. 13-24, 2004. [29] E.R. Gansner , Y. Koren , and S.C. North , “Topological Fisheye Views for Visualizing Large Graphs,” IEEE Trans. Visualization and Computer Graphics, vol. 11, no. 4, pp. 457-468, July/Aug. 2005. [30] M. Ghoniem , J.-D. Fekete , and P. Castagliola , “On the Readability of Graphs Using Node-Link and Matrix-Based Representations: A Controlled Experiment and Statistical Analysis,” Information Visualization, vol. 4, no. 2, pp. 114-135, 2005. [31] A.V. Goltsev , S.N. Dorogovtsev , and J.F.F. Mendes , “k-Core (Bootstrap) Percolation on Complex Networks: Critical Phenomena and Nonlocal Effects,” CoRR, abs/cond-mat/0602611, 2006. [32] S. Hachul and M. Jünger , “Large-Graph Layout Algorithms at Work: An Experimental Study,” J. Graph Algorithms and Applications, vol. 11, no. 2, pp. 345-369, 2007. [33] N. Henry , J.-D. Fekete , and M.J. McGuffin , “Nodetrix: A Hybrid Visualization of Social Networks,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 6, pp. 1302-1309, Nov./Dec. 2007. [34] I. Herman , G. Melançon , 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. [35] T. Itoh , C. Muelder , K. Ma , and J. Sese , “A Hybrid Space-Filling and Force-Directed Layout Method for Visualizing Multiple-Category Graphs,” Proc. IEEE Pacific Visualization Symp. (PacificVis '09), pp. 121-128, 2009. [36] Drawing Graphs, M. Kaufmann and D. Wagner, eds. Springer Verlag, 2001. [37] J. Kratochvíl , “String Graphs II - Recognizing String Graphs Is NP-Hard,” J. Combinatorial Theory, Series B, vol. 52, no. 1, pp. 67-78, 1991. [38] J. Lamping and R. Rao , “The Hyperbolic Browser: A Focus + Context Technique for Visualizing Large Hierarchies,” J. Visual Languages and Computing, vol. 7, no. 1, pp. 33-55, 1996. [39] H. Le , V. Le , and H. Müller , “Splitting a Graph into Disjoint Induced Paths or Cycles,” Discrete Applied Math., vol. 131, pp. 199-212, 2003. [40] M. Ley , “The DBLP Computer Science Bibliography,” http://www.informatik.uni-trier.de/~leydb /, 2011. [41] D. Lichtenstein , “Planar Formulae and Their Uses,” SIAM J. Computing, vol. 11, pp. 185-225, 1982. [42] T. Łuczak , “Size and Connectivity of the K-Core of a Random Graph,” Discrete Math., vol. 91, no. 1, pp. 61-68, 1991. [43] T. Munzner , “H3: Laying Out Large Directed Graphs in 3D Hyperbolic Space,” Proc. IEEE Symp. Information Visualization (INFOVIS), pp. 2-10, 1997. [44] T. Pattison , R. Vernik , and M. Phillips , “Information Visualisation Using Composable Layouts and Visual Sets,” Proc. Australian Symp. Information Visualisation, pp. 1-10, 2001. [45] H.C. Purchase , “Which Aesthetic Has the Greatest Effect on Human Understanding?,” Proc. Graph Drawing (GD '97), pp. 248-261, 1998. [46] H.C. Purchase , D.A. Carrington , and J.-A. Allder , “Empirical Evaluation of Aesthetics-Based Graph Layout,” Empirical Software Eng., vol. 7, no. 3, pp. 233-255, 2002. [47] D. Schaffer , Z. Zuo , S. Greenberg , L. Bartram , J. Dill , S. Dubs , and M. Roseman , “Navigating Hierarchically Clustered Networks through Fisheye and Full-Zoom Methods,” ACM Trans. Computer-Human Interaction, vol. 3, no. 2, pp. 162-188, 1996. [48] F. Schreiber and K. Skodinis , “NP-Completeness of Some Tree-Clustering Problems,” Proc. Sixth Int'l Symp. Graph Drawing (GD '98), pp. 288-301, 1998. [49] S.B. Seidman , “Network Structure and Minimum Degree,” Social Networks, vol. 5, pp. 269-287, 1983. [50] L. Shi , N. Cao , S. Liu , W. Qian , L. Tan , G. Wang , J. Sun , and C. Lin , “HiMap: Adaptive Visualization of Large-Scale Online Social Networks,” Proc. IEEE Pacific Visualization Symp. (PacificVis '09), pp. 41-48, 2009. [51] B. Shneiderman , “The Eyes Have It: A Task by Data Type Taxonomy for Information Visualizations,” Proc. Visual Languages Conf., pp. 336-343, 1996. [52] G. Sindre , B. Gulla , and H.G. Jokstad , “Onion Graphs: Aesthetics and Layout,” Proc. IEEE Symp. Visual Languages, pp. 287-291, 1993. [53] R. Tamassia , “On Embedding a Graph in the Grid with the Minimum Number of Bends,” SIAM J. Computing, vol. 16, no. 3, pp. 421-444, 1987. [54] F. van Ham and J.J. van Wijk , “Interactive Visualization of Small World Graphs,” Proc. IEEE Symp. Information Visualization (INFOVIS '04), pp. 199-206, 2004. [55] S. Wuchty and E. Almaas , “Peeling the Yeast Protein Network,” Proteomics, vol. 5, no. 2, pp. 444-449, 2005. [56] S. Zhao , M.J. McGuffin , and M.H. Chignell , “Elastic Hierarchies: Combining Treemaps and Node-Link Diagrams,” Proc. IEEE Symp. Information Visualization (INFOVIS '05), pp. 57-64, 2005.