EigenSpokes: Surprising Patterns and Scalable Community Chipping in Large Graphs

We report a surprising, persistent pattern in an important class of large sparse social graphs, which we term EigenSpokes. We focus on large Mobile Call graphs, spanning hundreds of thousands of nodes and edges, and find that the singular vectors of these graphs exhibit a striking EigenSpokes pattern wherein, when plotted against each other, they have clear, separate lines that often neatly align along specific axes (hence the term "spokes"). We show this phenomenon to be persistent across both temporal and geographic samples of Mobile Call graphs. Through experiments on synthetic graphs, EigenSpokes are shown to be associated with the presence of community structure in these social networks. This is further verified by analysing the eigenvectors of the Mobile Call graph, which yield nodes that form tightly-knit communities. The presence of such patterns in the singular spectra has useful applications, and could potentially be used to design simple, efficient community extraction algorithms.

[1]  M E J Newman,et al.  Modularity and community structure in networks. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[2]  Christos Faloutsos,et al.  Fully automatic cross-associations , 2004, KDD.

[3]  Ankur Teredesai,et al.  Extracting Social Networks from Instant Messaging Populations , 2004 .

[4]  Noga Alon,et al.  Finding a large hidden clique in a random graph , 1998, SODA '98.

[5]  Inderjit S. Dhillon,et al.  Information-theoretic co-clustering , 2003, KDD '03.

[6]  Corinna Cortes,et al.  Communities of interest , 2001, Intell. Data Anal..

[7]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[8]  Sougata Mukherjea,et al.  On the structural properties of massive telecom call graphs: findings and implications , 2006, CIKM '06.

[9]  Inderjit S. Dhillon,et al.  Weighted Graph Cuts without Eigenvectors A Multilevel Approach , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

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

[11]  Vipin Kumar,et al.  A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs , 1998, SIAM J. Sci. Comput..

[12]  A. Barabasi,et al.  Quantifying social group evolution , 2007, Nature.

[13]  Padhraic Smyth,et al.  A Spectral Clustering Approach To Finding Communities in Graph , 2005, SDM.

[14]  Pietro Perona,et al.  A Factorization Approach to Grouping , 1998, ECCV.

[15]  A-L Barabási,et al.  Structure and tie strengths in mobile communication networks , 2006, Proceedings of the National Academy of Sciences.

[16]  Jure Leskovec,et al.  Statistical properties of community structure in large social and information networks , 2008, WWW.

[17]  Xiaowei Ying,et al.  On Randomness Measures for Social Networks , 2009, SDM.

[18]  Jian Pei,et al.  On mining cross-graph quasi-cliques , 2005, KDD '05.

[19]  C. Lee Giles,et al.  Efficient identification of Web communities , 2000, KDD '00.

[20]  Srinivasan Parthasarathy,et al.  Scalable graph clustering using stochastic flows: applications to community discovery , 2009, KDD.

[21]  Ulrike von Luxburg,et al.  A tutorial on spectral clustering , 2007, Stat. Comput..

[22]  Christos Faloutsos,et al.  Mobile call graphs: beyond power-law and lognormal distributions , 2008, KDD.

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

[24]  A. Barabasi,et al.  Network effects in service usage , 2006, physics/0611177.

[25]  Kim L. Boyer,et al.  Quantitative Measures of Change Based on Feature Organization: Eigenvalues and Eigenvectors , 1998, Comput. Vis. Image Underst..

[26]  Srinivasan Keshav,et al.  Why cell phones will dominate the future internet , 2005, CCRV.

[27]  M. Newman,et al.  Finding community structure in very large networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Mikhail Belkin,et al.  Data spectroscopy: learning mixture models using eigenspaces of convolution operators , 2008, ICML '08.

[29]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[30]  M. Newman,et al.  Finding community structure in networks using the eigenvectors of matrices. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  G. Strang.  Introduction to Linear Algebra , 1993 .

[32]  Kevin J. Lang.  Fixing two weaknesses of the Spectral Method , 2005, NIPS.