Finding and Visualizing Graph Clusters Using PageRank Optimization

Abstract We give algorithms for finding graph clusters and drawing graphs, highlighting local community structure within the context of a larger network. For a given graph G, we use the personalized PageRank vectors to determine a set of clusters, by optimizing the jumping parameter α subject to several cluster variance measures in order to capture the graph structure according to PageRank. We then give a graph visualization algorithm for the clusters using PageRank-based coordinates. Several drawings of real-world data are given, illustrating the partition and local community structure.

[1]  J. MacQueen.  Some methods for classification and analysis of multivariate observations , 1967 .

[2]  Satu Elisa Schaeffer,et al.  Graph Clustering , 2017, Encyclopedia of Machine Learning and Data Mining.

[3]  Andrei Z. Broder,et al.  Graph structure in the Web , 2000, Comput. Networks.

[4]  Albert-László Barabási,et al.  Internet: Diameter of the World-Wide Web , 1999, Nature.

[5]  D. Lusseau,et al.  The bottlenose dolphin community of Doubtful Sound features a large proportion of long-lasting associations , 2003, Behavioral Ecology and Sociobiology.

[6]  Jennifer Widom,et al.  Scaling personalized web search , 2003, WWW '03.

[7]  Michael I. Jordan,et al.  On Spectral Clustering: Analysis and an algorithm , 2001, NIPS.

[8]  Michalis Faloutsos,et al.  On power-law relationships of the Internet topology , 1999, SIGCOMM '99.

[9]  Venkatesan Guruswami,et al.  The complexity of the covering radius problem on lattices and codes , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[10]  Andreas Noack,et al.  Modularity clustering is force-directed layout , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Peter Eades,et al.  Multilevel Visualization of Clustered Graphs , 1996, GD.

[12]  Emden R. Gansner,et al.  An open graph visualization system and its applications to software engineering , 2000, Softw. Pract. Exp..

[13]  Pierre Hansen,et al.  NP-hardness of Euclidean sum-of-squares clustering , 2008, Machine Learning.

[14]  Vladimir Batagelj,et al.  Exploratory Social Network Analysis with Pajek , 2005 .

[15]  A. Frieze,et al.  A simple heuristic for the p-centre problem , 1985 .

[16]  Venkatesan Guruswami,et al.  The complexity of the covering radius problem , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[17]  Fan Chung Graham,et al.  A Sharp PageRank Algorithm with Applications to Edge Ranking and Graph Sparsification , 2010, WAW.

[18]  Charu C. Aggarwal,et al.  Graph Clustering , 2010, Encyclopedia of Machine Learning and Data Mining.

[19]  Allan Kuchinsky,et al.  Integrating user-perceived quality into Web server design , 2000, Comput. Networks.

[20]  James Moody,et al.  Peer influence groups: identifying dense clusters in large networks , 2001, Soc. Networks.

[21]  David Harel,et al.  Graph Drawing by High-Dimensional Embedding , 2002, J. Graph Algorithms Appl..

[22]  Anton J. Enright,et al.  An efficient algorithm for large-scale detection of protein families. , 2002, Nucleic acids research.

[23]  F. Chung.  PageRank as a discrete Green ’ s function , 2008 .

[24]  Fan Chung Graham,et al.  Distributing Antidote Using PageRank Vectors , 2009, Internet Math..

[25]  Emden R. Gansner,et al.  Bunch: a clustering tool for the recovery and maintenance of software system structures , 1999, Proceedings IEEE International Conference on Software Maintenance - 1999 (ICSM'99). 'Software Maintenance for Business Change' (Cat. No.99CB36360).

[26]  Miklós Simonovits,et al.  Random Walks in a Convex Body and an Improved Volume Algorithm , 1993, Random Struct. Algorithms.

[27]  Fan Chung Graham,et al.  Detecting Sharp Drops in PageRank and a Simplified Local Partitioning Algorithm , 2007, TAMC.

[28]  Sergey Brin,et al.  The Anatomy of a Large-Scale Hypertextual Web Search Engine , 1998, Comput. Networks.

[29]  Colin Ware,et al.  Visualization of Large Nested Graphs in 3D: Navigation and Interaction , 1998, J. Vis. Lang. Comput..

[30]  Steven Fortune,et al.  A sweepline algorithm for Voronoi diagrams , 1986, SCG '86.

[31]  M E J Newman,et al.  Community structure in social and biological networks , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[32]  Jitendra Malik,et al.  Normalized cuts and image segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[33]  W. Zachary,et al.  An Information Flow Model for Conflict and Fission in Small Groups , 1977, Journal of Anthropological Research.

[34]  Ulrik Brandes,et al.  Journal of Graph Algorithms and Applications Visual Ranking of Link Structures , 2022 .

[35]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[36]  M E J Newman,et al.  Finding and evaluating community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  Mark Rudelson,et al.  Sampling from large matrices: An approach through geometric functional analysis , 2005, JACM.

[38]  Fan Chung Graham,et al.  Local Graph Partitioning using PageRank Vectors , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[39]  Satoru Kawai,et al.  An Algorithm for Drawing General Undirected Graphs , 1989, Inf. Process. Lett..