Exponential random graph models for networks with community structure

Although the community structure organization is an important characteristic of real-world networks, most of the traditional network models fail to reproduce the feature. Therefore, the models are useless as benchmark graphs for testing community detection algorithms. They are also inadequate to predict various properties of real networks. With this paper we intend to fill the gap. We develop an exponential random graph approach to networks with community structure. To this end we mainly built upon the idea of blockmodels. We consider both the classical blockmodel and its degree-corrected counterpart and study many of their properties analytically. We show that in the degree-corrected blockmodel, node degrees display an interesting scaling property, which is reminiscent of what is observed in real-world fractal networks. A short description of Monte Carlo simulations of the models is also given in the hope of being useful to others working in the field.

[1]  S. Wasserman,et al.  Building stochastic blockmodels , 1992 .

[2]  Kathryn B. Laskey,et al.  Stochastic blockmodels: First steps , 1983 .

[3]  P. Shannon,et al.  Cytoscape: a software environment for integrated models of biomolecular interaction networks. , 2003, Genome research.

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

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

[6]  M. Newman,et al.  Statistical mechanics of networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Gerard T. Barkema,et al.  Monte Carlo Methods in Statistical Physics , 1999 .

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

[9]  A. Rinaldo,et al.  CONSISTENCY UNDER SAMPLING OF EXPONENTIAL RANDOM GRAPH MODELS. , 2011, Annals of statistics.

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

[11]  S. Havlin,et al.  Self-similarity of complex networks , 2005, Nature.

[12]  Agata Fronczak,et al.  Fluctuation-dissipation relations in complex networks. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  S. Wasserman,et al.  Blockmodels: Interpretation and evaluation , 1992 .

[14]  E. Todeva.  Networks , 2007 .

[15]  Tiago P. Peixoto.  The entropy of stochastic blockmodel ensembles , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[17]  Alessandro Vespignani,et al.  Dynamical Processes on Complex Networks , 2008 .

[18]  Steve Gregory,et al.  Ordered community structure in networks , 2011, 1104.0923.

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

[20]  Shlomo Havlin,et al.  Origins of fractality in the growth of complex networks , 2005, cond-mat/0507216.

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

[22]  Xueqi Cheng,et al.  Exploring the structural regularities in networks , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  M. Newman.  Communities, modules and large-scale structure in networks , 2011, Nature Physics.

[24]  M. Walsh,et al.  An Introduction , 2002, The Counseling Psychologist.

[25]  Tamara G. Kolda,et al.  Community structure and scale-free collections of Erdös-Rényi graphs , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  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.

[27]  T. Snijders,et al.  Estimation and Prediction for Stochastic Blockmodels for Graphs with Latent Block Structure , 1997 .

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

[29]  H. J. Mclaughlin,et al.  Learn , 2002 .

[30]  W. Marsden.  I and J , 2012 .

[31]  C. Ross.  Found , 1869, The Dental register.