On Spectral Analysis of Signed and Dispute Graphs: Application to Community Structure

This paper presents a spectral analysis of signed networks from both theoretical and practical aspects. On the theoretical aspect, we conduct theoretical studies based on results from matrix perturbation for analyzing community structures of complex signed networks and show how the negative edges affect distributions and patterns of node spectral coordinates in the spectral space. We prove and demonstrate that node spectral coordinates form orthogonal clusters for two types of signed networks: graphs with dense inter-community mixed sign edges and <inline-formula> <tex-math notation="LaTeX">$k$</tex-math><alternatives><inline-graphic xlink:href="wu-ieq1-2684809.gif"/> </alternatives></inline-formula>-dispute graphs where inner-community connections are absent or very sparse but inter-community connections are dense with negative edges. The cluster orthogonality pattern is different from the line orthogonality pattern (i.e., node spectral coordinates form orthogonal lines) observed in the networks with <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives> <inline-graphic xlink:href="wu-ieq2-2684809.gif"/></alternatives></inline-formula>-block structure. We show why the line orthogonality pattern does not hold in the spectral space for these two types of networks. On the practical aspect, we have developed a clustering method to study signed networks and <inline-formula><tex-math notation="LaTeX"> $k$</tex-math><alternatives><inline-graphic xlink:href="wu-ieq3-2684809.gif"/></alternatives> </inline-formula>-dispute networks. Empirical evaluations on both synthetic networks (with up to one million nodes) and real networks show our algorithm outperforms existing clustering methods on signed networks in terms of accuracy and efficiency.

[1]  Anna R. Karlin,et al.  Spectral analysis of data , 2001, STOC '01.

[2]  W. Kahan,et al.  The Rotation of Eigenvectors by a Perturbation. III , 1970 .

[3]  F. Harary,et al.  Structural Models in Anthropology , 1986 .

[4]  Takehiro Inohara,et al.  Characterization of clusterability of signed graph in terms of Newcomb's balance of sentiments , 2002, Appl. Math. Comput..

[5]  Matei Zaharia,et al.  Matrix Computations and Optimization in Apache Spark , 2015, KDD.

[6]  Jure Leskovec,et al.  {SNAP Datasets}: {Stanford} Large Network Dataset Collection , 2014 .

[7]  Andrew B. Kahng,et al.  New spectral methods for ratio cut partitioning and clustering , 1991, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[8]  Nagarajan Natarajan,et al.  Prediction and clustering in signed networks: a local to global perspective , 2013, J. Mach. Learn. Res..

[9]  Ichigaku Takigawa,et al.  A spectral clustering approach to optimally combining numericalvectors with a modular network , 2007, KDD '07.

[10]  Krishna P. Gummadi,et al.  On the evolution of user interaction in Facebook , 2009, WOSN '09.

[11]  Paolo Avesani,et al.  Trust-Aware Collaborative Filtering for Recommender Systems , 2004, CoopIS/DOA/ODBASE.

[12]  Xintao Wu,et al.  Analysis of Spectral Space Properties of Directed Graphs Using Matrix Perturbation Theory with Application in Graph Partition , 2015, 2015 IEEE International Conference on Data Mining.

[13]  Ling Huang,et al.  Spectral Clustering with Perturbed Data , 2008, NIPS.

[14]  A. Seary,et al.  Spectral methods for analyzing and visualizing networks : an introduction , 2000 .

[15]  Zhi-Hua Zhou,et al.  Spectral Analysis of k-Balanced Signed Graphs , 2011, PAKDD.

[16]  Christos Faloutsos,et al.  Spectral Analysis for Billion-Scale Graphs: Discoveries and Implementation , 2011, PAKDD.

[17]  V. Traag,et al.  Community detection in networks with positive and negative links. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[19]  Ahmet Izmirlioglu,et al.  The Correlates of War Dataset , 2017 .

[20]  Ravi Kumar,et al.  Structure and evolution of online social networks , 2006, KDD '06.

[21]  Santosh S. Vempala,et al.  Latent semantic indexing: a probabilistic analysis , 1998, PODS '98.

[22]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[23]  Zhi-Hua Zhou,et al.  Examining spectral space of complex networks with positive and negative links , 2012, Int. J. Soc. Netw. Min..

[24]  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.

[25]  Xintao Wu,et al.  Spectrum-Based Network Visualization for Topology Analysis , 2013, IEEE Computer Graphics and Applications.

[26]  Leting Wu,et al.  A Spectrum-Based Framework for Quantifying Randomness of Social Networks , 2011, IEEE Transactions on Knowledge and Data Engineering.

[27]  Alex Pothen,et al.  PARTITIONING SPARSE MATRICES WITH EIGENVECTORS OF GRAPHS* , 1990 .

[28]  Martine D. F. Schlag,et al.  Spectral K-Way Ratio-Cut Partitioning and Clustering , 1993, 30th ACM/IEEE Design Automation Conference.

[29]  Michalis Vazirgiannis,et al.  Clustering and Community Detection in Directed Networks: A Survey , 2013, ArXiv.

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

[31]  Zhi-Hua Zhou,et al.  A spectral approach to detecting subtle anomalies in graphs , 2013, Journal of Intelligent Information Systems.

[32]  Anirban Dasgupta,et al.  Spectral analysis of random graphs with skewed degree distributions , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

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

[34]  Inderjit S. Dhillon,et al.  Low rank modeling of signed networks , 2012, KDD.

[35]  Frank McSherry,et al.  Spectral partitioning of random graphs , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[36]  Donald W. Bouldin,et al.  A Cluster Separation Measure , 1979, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[37]  Faten Ghosn,et al.  The MID3 Data Set, 1993—2001: Procedures, Coding Rules, and Description , 2004 .

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

[39]  Inderjit S. Dhillon,et al.  Scalable clustering of signed networks using balance normalized cut , 2012, CIKM.

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

[41]  Jure Leskovec,et al.  Signed networks in social media , 2010, CHI.

[42]  Zhi-Hua Zhou,et al.  Line Orthogonality in Adjacency Eigenspace with Application to Community Partition , 2011, IJCAI.

[43]  Mikhail Belkin,et al.  Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering , 2001, NIPS.

[44]  Xiaowei Ying,et al.  Spectrum based fraud detection in social networks , 2011, ICDE.

[45]  Christos Faloutsos,et al.  EigenSpokes: Surprising Patterns and Scalable Community Chipping in Large Graphs , 2010, PAKDD.

[46]  Chris H. Q. Ding,et al.  A min-max cut algorithm for graph partitioning and data clustering , 2001, Proceedings 2001 IEEE International Conference on Data Mining.

[47]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[48]  Sahin Albayrak,et al.  Spectral Analysis of Signed Graphs for Clustering, Prediction and Visualization , 2010, SDM.