Probabilistic framework for network partition.

Given a large and complex network, we would like to find the partition of this network into a small number of clusters. This question has been addressed in many different ways. In a previous paper, we proposed a deterministic framework for an optimal partition of a network as well as the associated algorithms. In this paper, we extend this framework to a probabilistic setting, in which each node has a certain probability of belonging to a certain cluster. Two classes of numerical algorithms for such a probabilistic network partition are presented and tested. Application to three representative examples is discussed.

[1]  Julia Martin,et al.  Pattern , 2005, The Fairchild Books Dictionary of Fashion.

[2]  Tiejun Li,et al.  Optimal partition and effective dynamics of complex networks , 2008, Proceedings of the National Academy of Sciences.

[3]  Chris H Wiggins,et al.  Bayesian approach to network modularity. , 2007, Physical review letters.

[4]  David Gfeller,et al.  Spectral coarse graining of complex networks. , 2007, Physical review letters.

[5]  E A Leicht,et al.  Mixture models and exploratory analysis in networks , 2006, Proceedings of the National Academy of Sciences.

[6]  Ann B. Lee,et al.  Diffusion maps and coarse-graining: a unified framework for dimensionality reduction, graph partitioning, and data set parameterization , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[7]  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.

[8]  Leon Danon,et al.  Comparing community structure identification , 2005, cond-mat/0505245.

[9]  D. Ruppert.  The Elements of Statistical Learning: Data Mining, Inference, and Prediction , 2004 .

[10]  Dla Polski,et al.  EURO , 2004 .

[11]  M. Newman.  Fast algorithm for detecting community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Claudio Castellano,et al.  Defining and identifying communities in networks. , 2003, Proceedings of the National Academy of Sciences of the United States of America.

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

[14]  Fang Wu,et al.  Finding communities in linear time: a physics approach , 2003, ArXiv.

[15]  Mathew D. Penrose,et al.  Random Geometric Graphs , 2003 .

[16]  Haijun Zhou.  Distance, dissimilarity index, and network community structure. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  A. Chorin.  Conditional Expectations and Renormalization , 2002, Multiscale Model. Simul..

[18]  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.

[19]  Robert Tibshirani,et al.  The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd Edition , 2001, Springer Series in Statistics.

[20]  Trevor Hastie,et al.  The Elements of Statistical Learning , 2001 .

[21]  Alexandre J. Chorin,et al.  Unresolved Computation and Optimal Predictions , 1999 .

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

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

[24]  Gerardo Beni,et al.  A Validity Measure for Fuzzy Clustering , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[25]  Robert Trappl,et al.  Cybernetics and Systems ’86 , 1986, Springer Netherlands.

[26]  James C. Bezdek,et al.  Pattern Recognition with Fuzzy Objective Function Algorithms , 1981, Advanced Applications in Pattern Recognition.

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

[28]  J. C. Dunn,et al.  A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters , 1973 .

[29]  Elmer S. West.  From the U. S. A. , 1965 .

[30]  K. N. Dollman,et al.  - 1 , 1743 .