Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications

In this paper we extend our previous work on the stochastic block model, a commonly used generative model for social and biological networks, and the problem of inferring functional groups or communities from the topology of the network. We use the cavity method of statistical physics to obtain an asymptotically exact analysis of the phase diagram. We describe in detail properties of the detectability-undetectability phase transition and the easy-hard phase transition for the community detection problem. Our analysis translates naturally into a belief propagation algorithm for inferring the group memberships of the nodes in an optimal way, i.e., that maximizes the overlap with the underlying group memberships, and learning the underlying parameters of the block model. Finally, we apply the algorithm to two examples of real-world networks and discuss its performance.

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

[2]  Tobias J. Hagge,et al.  Physics , 1929, Nature.

[3]  W. Kauzmann.  The Nature of the Glassy State and the Behavior of Liquids at Low Temperatures. , 1948 .

[4]  L. Tippett,et al.  Applied Statistics. A Journal of the Royal Statistical Society , 1952 .

[5]  E. Parzen.  Annals of Mathematical Statistics , 1962 .

[6]  H. Kesten,et al.  Additional Limit Theorems for Indecomposable Multidimensional Galton-Watson Processes , 1966 .

[7]  H. Kesten,et al.  Limit theorems for decomposable multi-dimensional Galton-Watson processes , 1967 .

[8]  R. Cox,et al.  Journal of the Royal Statistical Society B , 1972 .

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

[10]  D. Thouless,et al.  Stability of the Sherrington-Kirkpatrick solution of a spin glass model , 1978 .

[11]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[12]  Martin E. Dyer,et al.  The Solution of Some Random NP-Hard Problems in Polynomial Expected Time , 1989, J. Algorithms.

[13]  H. Nishimori.  Optimum Decoding Temperature for Error-Correcting Codes , 1993 .

[14]  N. Sourlas.  Spin Glasses, Error-Correcting Codes and Finite-Temperature Decoding , 1994 .

[15]  Y. Iba.  The Nishimori line and Bayesian statistics , 1998, cond-mat/9809190.

[16]  H. Nishimori.  Statistical Physics of Spin Glasses and Information Processing , 2001 .

[17]  M. Mézard,et al.  The Bethe lattice spin glass revisited , 2000, cond-mat/0009418.

[18]  R. Karp,et al.  Algorithms for graph partitioning on the planted partition model , 2001 .

[19]  M. Mézard,et al.  A ferromagnet with a glass transition , 2001, cond-mat/0103026.

[20]  西森 秀稔.  Statistical physics of spin glasses and information processing : an introduction , 2001 .

[21]  T. Snijders,et al.  Estimation and Prediction for Stochastic Blockstructures , 2001 .

[22]  Kazuyuki Tanaka.  Statistical-mechanical approach to image processing , 2002 .

[23]  Gerhard Lakemeyer,et al.  Exploring artificial intelligence in the new millennium , 2003 .

[24]  M. Luise.  European Transactions on Telecommunications , 2003 .

[25]  Elchanan Mossel,et al.  Robust reconstruction on trees is determined by the second eigenvalue , 2004, math/0406447.

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

[27]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[28]  宁北芳,et al.  疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A , 2005 .

[29]  A. Arenas,et al.  Community detection in complex networks using extremal optimization. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  M. Mézard,et al.  Reconstruction on Trees and Spin Glass Transition , 2005, cond-mat/0512295.

[31]  Hendrik B. Geyer,et al.  Journal of Physics A - Mathematical and General, Special Issue. SI Aug 11 2006 ?? Preface , 2006 .

[32]  M. Hastings.  Community detection as an inference problem. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[34]  Florent Krzakala,et al.  Phase Transitions in the Coloring of Random Graphs , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Andrea Montanari,et al.  Gibbs states and the set of solutions of random constraint satisfaction problems , 2006, Proceedings of the National Academy of Sciences.

[36]  M. Newman,et al.  Hierarchical structure and the prediction of missing links in networks , 2008, Nature.

[37]  F. Radicchi,et al.  Benchmark graphs for testing community detection algorithms. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Michele Leone,et al.  (Un)detectable cluster structure in sparse networks. , 2007, Physical review letters.

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

[40]  Edoardo M. Airoldi,et al.  Mixed Membership Stochastic Blockmodels , 2007, NIPS.

[41]  Andrea Montanari,et al.  Estimating random variables from random sparse observations , 2007, Eur. Trans. Telecommun..

[42]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[43]  Lenka Zdeborová,et al.  Belief propagation for graph partitioning , 2009, ArXiv.

[44]  Santo Fortunato,et al.  Community detection in graphs , 2009, ArXiv.

[45]  Andrea Lancichinetti,et al.  Community detection algorithms: a comparative analysis: invited presentation, extended abstract , 2009, VALUETOOLS.

[46]  P. Bickel,et al.  A nonparametric view of network models and Newman–Girvan and other modularities , 2009, Proceedings of the National Academy of Sciences.

[47]  Roger Guimerà,et al.  Missing and spurious interactions and the reconstruction of complex networks , 2009, Proceedings of the National Academy of Sciences.

[48]  Florent Krzakala,et al.  Hiding Quiet Solutions in Random Constraint Satisfaction Problems , 2009, Physical review letters.

[49]  Benjamin H. Good,et al.  Performance of modularity maximization in practical contexts. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[50]  F. Krzakala,et al.  Generalization of the cavity method for adiabatic evolution of Gibbs states , 2010, 1003.2748.

[51]  Cristopher Moore,et al.  Phase transition in the detection of modules in sparse networks , 2011, Physical review letters.

[52]  Mark E. J. Newman,et al.  Stochastic blockmodels and community structure in networks , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[53]  Mark E. J. Newman,et al.  An efficient and principled method for detecting communities in networks , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  Physics Reports , 2022 .