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Homology-Preserving Multi-Scale Graph Skeletonization Using Mapper on Graphs (1804.11242v5)

Published 3 Apr 2018 in cs.SI and stat.ML

Abstract: Node-link diagrams are a popular method for representing graphs that capture relationships between individuals, businesses, proteins, and telecommunication endpoints. However, node-link diagrams may fail to convey insights regarding graph structures, even for moderately sized data of a few hundred nodes, due to visual clutter. We propose to apply the mapper construction -- a popular tool in topological data analysis -- to graph visualization, which provides a strong theoretical basis for summarizing the data while preserving their core structures. We develop a variation of the mapper construction targeting weighted, undirected graphs, called {\mog}, which generates homology-preserving skeletons of graphs. We further show how the adjustment of a single parameter enables multi-scale skeletonization of the input graph. We provide a software tool that enables interactive explorations of such skeletons and demonstrate the effectiveness of our method for synthetic and real-world data.

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References (110)
  1. ASK-GraphView: A large scale graph visualization system. IEEE Trans. on Visual. and Comp. Graph., 12(5), 2006.
  2. Topolayout: Multilevel graph layout by topological features. IEEE Trans. on Visual. and Comp. Graph., 13(2), 2007.
  3. GrouseFlocks: Steerable exploration of graph hierarchy space. IEEE Trans. on Visual. and Comp. Graph., 14(4), 2008.
  4. Tugging graphs faster: Efficiently modifying path-preserving hierarchies for browsing paths. IEEE Trans. on Visual. and Comp. Graph., 17(3), 2010.
  5. The readability of path-preserving clusterings of graphs. Comp. Graph. Forum, 29(3), 2010.
  6. Towards unambiguous edge bundling: Investigating confluent drawings for network visualization. IEEE Trans. on Visual. and Comp. Graph., 23(1), 2017.
  7. M. Bampasidou and T. Gentimis. Modeling collaborations with persistent homology. arXiv preprint, 2014.
  8. Gephi: an open source software for exploring and manipulating networks. In AAAI Web and Social Media, 2009.
  9. Visual analysis of large graphs using (x, y)-clustering and hybrid visualizations. IEEE Trans. on Visual. and Comp. Graph., 17(11), 2010.
  10. Reeb graphs for shape analysis and applications. Theoretical Comp. Sci., 392, 2008.
  11. An overview on properties and efficacy of topological skeletons in shape modelling. Shape Modeling International, 2003.
  12. Fast unfolding of communities in large networks. J. of Statistical Mechanics, 2008.
  13. Deep graph mapper: Seeing graphs through the neural lens. Frontiers in Big Data, 4, 2021.
  14. Design and update of a classification system: The ucsd map of science. PloS One, 7(7), 2012.
  15. On modularity clustering. IEEE Trans. on Knowledge and Data Engineering, 20(2), 2007.
  16. Experiments on graph clustering algorithms. In European Symp. on Algorithms, 2003.
  17. U. Brandes and C. Pich. Eigensolver methods for progressive multidimensional scaling of large data. In Graph Drawing, 2007.
  18. S. Brin and L. Page. The anatomy of a large-scale hypertextual web search engine. Computer Networks and ISDN Systems, 30(1-7), 1998.
  19. Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology (APCT), 5:99–140, 2021.
  20. Graph bisection algorithms with good average case behavior. Combinatorica, 7(2), 1987.
  21. G. Carlsson. Topological pattern recognition for point cloud data. Acta Numerica, 23, 2014.
  22. Statistical analysis and parameter selection for mapper. J. of Machine Learning Research, 19(12), 2018.
  23. M. Carriére and S. Oudot. Structure and stability of the one-dimensional mapper. Foundations of Computational Mathematics, 18(6), 2018.
  24. C. Carstens and K. Horadam. Persistent homology of collaboration networks. Mathematical Problems in Engineering, 2013.
  25. Brain activity: Conditional dissimilarity and persistent homology. IEEE Symp. on Biomedical Imaging, 2015.
  26. Adaptive covers for mapper graphs using information criteria. IEEE Big Data, 2021.
  27. Hot spots conjecture and its application to modeling tubular structures. In Workshop on Machine Learning in Medical Imaging, 2011.
  28. G. Cordasco and L. Gargano. Community detection via semi-synchronous label propagation algorithms. In IEEE Workshop on Business Applications of Social Network Analysis, 2010.
  29. Geometry-based edge clustering for graph visualization. IEEE Trans. on Visual. and Comp. Graph., 14(6), 2008.
  30. A topological paradigm for hippocampal spatial map formation using persistent homology. PLoS Computational Biology, 8(8), 2012.
  31. T. Davis and Y. Hu. The University of Florida sparse matrix collection. ACM Trans. on Mathematical Software, 38(1), 2011.
  32. Exploratory social network analysis with Pajek: Revised and expanded edition for updated software, vol. 46. Cambridge University Press, 2018.
  33. Topological analysis of nerves, reeb spaces, mappers, and multiscale mappers. arXiv preprint, 2017.
  34. I. Dhillon. Co-clustering documents and words using bipartite spectral graph partitioning. In ACM SIGKDD Knowledge Discovery and Data Mining, 2001.
  35. A min-max cut algorithm for graph partitioning and data clustering. IEEE Data Mining, 2001.
  36. Compressed adjacency matrices: untangling gene regulatory networks. IEEE Trans. on Visual. and Comp. Graph., 18(12), 2012.
  37. Decimation of fast states and weak nodes: topological variation via persistent homology. European Conf. on Complex Systems, 2012.
  38. Untangling force-directed layouts using persistent homology. In Topological Data Analysis and Visualization (TopoInVis), 2022.
  39. C. Dunne and B. Shneiderman. Motif simplification. ACM SIGCHI Human Factors in Computing Systems, 2013.
  40. Edge compression techniques for visualization of dense directed graphs. IEEE Trans. on Visual. and Comp. Graph., 19(12), 2013.
  41. G. Ellis and A. Dix. A taxonomy of clutter reduction for information visualisation. IEEE Trans. on Visual. and Comp. Graph., 13(6), 2007.
  42. Graphviz— open source graph drawing tools. Graph Drawing, 2002.
  43. Skeleton-based edge bundling for graph visualization. IEEE Trans. on Visual. and Comp. Graph., 17(12), 2011.
  44. A density-based algorithm for discovering clusters in large spatial databases with noise. In ACM SIGKDD Knowledge Discovery and Data Mining, 1996.
  45. M. Fiedler. Algebraic connectivity of graphs. Czechoslovak Mathematical Lournal, 23(2), 1973.
  46. T. Fruchterman and E. Reingold. Graph drawing by force-directed placement. Software: Practice and Exp., 21(11), 1991.
  47. Multilevel agglomerative edge bundling for visualizing large graphs. In IEEE PacificVis, 2011.
  48. Graph drawing by stress majorization. In Graph Drawing, 2005.
  49. Topological fisheye views for visualizing large graphs. IEEE Trans. on Visual. and Comp. Graph., 11(4), 2005.
  50. A technique for drawing directed graphs. IEEE Trans. on Software Eng., 19(3), 1993.
  51. Simplifying tuggraph using zipping algorithms. Pattern Recognition, 103, 2020.
  52. V. Grolmusz. A note on the pagerank of undirected graphs. arXiv preprint, 2012.
  53. Exploring network structure, dynamics, and function using networkx. Technical report, Los Alamos National Lab.(LANL), Los Alamos, NM (United States), 2008.
  54. MOG: Mapper on graphs for relationship preserving clustering. arXiv preprint, 2018.
  55. Visual detection of structural changes in time-varying graphs using persistent homology. IEEE PacificVis, 2018.
  56. Analyzing social media networks with NodeXL: Insights from a connected world. Morgan Kaufmann, 2010.
  57. D. Holten and J. van Wijk. Force-directed edge bundling for graph visualization. Comp. Graph. Forum, 28(3), 2009.
  58. Persistent homology of complex networks. J. of Statistical Mechanics: Theory and Experiment, 2009.
  59. Y. Hu. Efficient, high-quality force-directed graph drawing. Mathematica Journal, 10(1), 2005.
  60. G. Jeh and J. Widom. SimRank: a measure of structural-context similarity. In ACM SIGKDD Knowledge Discovery and Data Mining, 2002.
  61. Graphprism: Compact visualization of network structure. In Adv. Visual Interfaces, 2012.
  62. Detecting divergent subpopulations in phenomics data using interesting flares. In ACM Bioinfo., Computational Biology, and Health Informatics, 2018.
  63. Drawing large graphs by low-rank stress majorization. Comp. Graph. Forum, 31, 2012.
  64. Möbius transformations for global intrinsic symmetry analysis. Comp. Graph. Forum, 29(5), 2010.
  65. Y. Koren. On spectral graph drawing. In Computing and Combinatorics. Springer, 2003.
  66. Ace: A fast multiscale eigenvectors computation for drawing huge graphs. IEEE Symp. on Information Visualization, 2002.
  67. Semi-supervised graph clustering: a kernel approach. Machine Learning, 74(1), 2009.
  68. S. Lafon and A. B. Lee. Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning, and data set parameterization. IEEE Trans. on Pattern Analysis and Machine Intelligence, 28(9), 2006.
  69. Computing the shape of brain networks using graph filtration and gromov-hausdorff metric. Conference on Medical Image Computing and Computer Assisted Intervention, 2011.
  70. Discriminative persistent homology of brain networks. IEEE Symp. on Biomedical Imaging, 2011.
  71. Persistent brain network homology from the perspective of dendrogram. IEEE Trans. on Medical Imaging, 31(12), 2012.
  72. Weighted functional brain network modeling via network filtration. NIPS Workshop on Algebraic Topology and Machine Learning, 2012.
  73. J. Leskovec and A. Krevl. SNAP Datasets: Stanford large network dataset collection, 2014.
  74. B. Lévy. Laplace-Beltrami eigenfunctions towards an algorithm that understands geometry. In IEEE Shape Modeling and App., 2006.
  75. Visualizing high-dimensional data: Advances in the past decade. IEEE Trans. on Visual. and Comp. Graph., 23(3), 2017.
  76. Extracting insights from the shape of complex data using topology. Scientific Reports, 3, 2013.
  77. U. V. Luxburg. A tutorial on spectral clustering. Statistics and Computing, 17(4), 2007.
  78. F. McGee and J. Dingliana. An empirical study on the impact of edge bundling on user comprehension of graphs. In Advanced Visual Interfaces, 2012.
  79. E. Munch and B. Wang. Convergence between categorical representations of Reeb space and mapper. Symp. on Comp. Geometry, 51, 2016.
  80. M. Newman. The structure of scientific collaboration networks. Nat. Academy of Sci., 98(2), 2001.
  81. M. Newman. Properties of highly clustered networks. Physical Review E, 68(2), 2003.
  82. M. Newman. Fast algorithm for detecting community structure in networks. Physical Review E, 69(6), 2004.
  83. On spectral clustering: Analysis and an algorithm. In Advances in Neural Info. Processing Systems, 2002.
  84. Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Nat. Academy of Sci., 108(17), 2011.
  85. Pagerank citation ranking: Bringing order to the web. Technical report, Stanford, 1999.
  86. Networks and cycles: A persistent homology approach to complex networks. European Conf. on Complex Systems, 2013.
  87. Topological strata of weighted complex networks. PloS One, 8(6), 2013.
  88. A topological study of repetitive co-activation networks in in vitro cortical assemblies. Physical Biology, 12(1), 2015.
  89. P. Pons and M. Latapy. Computing communities in large networks using random walks. In Symp. on Computer and Info. Sciences, 2005.
  90. L. Pretto. Analysis of web link analysis algorithms: The mathematics of ranking. In Information Access through Search Engines and Digital Libraries. Springer, 2008.
  91. Near linear time algorithm to detect community structures in large-scale networks. Physical Review E, 76(3), 2007.
  92. Discrete laplace–beltrami operators for shape analysis and segmentation. Computers & Graphics, 33(3), 2009.
  93. R. Rossi and N. Ahmed. The network data repository with interactive graph analytics and visualization. In AAAI Artificial Intelligence, vol. 29, 2015.
  94. S. Schaeffer. Graph clustering. Computer Sci. Review, 1(1), 2007.
  95. Divided edge bundling for directional network data. IEEE Trans. on Visual. and Comp. Graph., 17(12), 2011.
  96. The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Processing Magazine, 30(3), 2013.
  97. B. Silverman. Density estimation for statistics and data analysis, vol. 26. CRC press, 1986.
  98. Topological methods for the analysis of high dimensional data sets and 3D object recognition. Eurographics Symp. on Point-Based Graphics, 22, 2007.
  99. Persistent homology guided force-directed graph layouts. IEEE Trans. on Visual. and Comp. Graph., 26(1), 2019.
  100. Efficient aggregation for graph summarization. In ACM SIGKDD Knowledge Discovery and Data Mining, 2008.
  101. Tracking resilience to infections by mapping disease space. PLoS Biology, 14(4), 2016.
  102. W. Tutte. How to draw a graph. London Mathematical Society, s3-13(1), 1963.
  103. Interactive visualization of small world graphs. In IEEE Symp. on Information Visualization, 2004.
  104. Visualizing group structures in graphs: A survey. Comp. Graph. Forum, 36(6), 2017.
  105. Visual analysis of large graphs: State-of-the-art and future research challenges. Comp. Graph. Forum, 30(6), 2011.
  106. S. White and P. Smyth. A spectral clustering approach to finding communities in graphs. In SIAM Data Mining, 2005.
  107. P. Wills and F. Meyer. Metrics for graph comparison: a practitioner’s guide. PloS One, 15(2), 2020.
  108. Pheno-Mapper: an interactive toolbox for the visual exploration of phenomics data. ACM Bioinformatics, Computational Biology, and Health Informatics, 2021.
  109. Topological simplifications of hypergraphs. IEEE Trans. on Visual. and Comp. Graph., 2022.
  110. BioSNAP Datasets: Stanford biomedical network dataset collection, 2018.
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Authors (3)
  1. Paul Rosen (41 papers)
  2. Mustafa Hajij (51 papers)
  3. Bei Wang (102 papers)
Citations (5)
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