Multiresolution Network Models

ABSTRACT Many existing statistical and machine learning tools for social network analysis focus on a single level of analysis. Methods designed for clustering optimize a global partition of the graph, whereas projection-based approaches (e.g., the latent space model in the statistics literature) represent in rich detail the roles of individuals. Many pertinent questions in sociology and economics, however, span multiple scales of analysis. Further, many questions involve comparisons across disconnected graphs that will, inevitably be of different sizes, either due to missing data or the inherent heterogeneity in real-world networks. We propose a class of network models that represent network structure on multiple scales and facilitate comparison across graphs with different numbers of individuals. These models differentially invest modeling effort within subgraphs of high density, often termed communities, while maintaining a parsimonious structure between said subgraphs. We show that our model class is projective, highlighting an ongoing discussion in the social network modeling literature on the dependence of inference paradigms on the size of the observed graph. We illustrate the utility of our method using data on household relations from Karnataka, India. Supplementary material for this article is available online.

[1]  Daniel J. Brass,et al.  Network Analysis in the Social Sciences , 2009, Science.

[2]  Maurizio Mazzocco,et al.  Testing Efficient Risk Sharing with Heterogeneous Risk Preferences: Semi-parametric Tests with an Application to Village Economies , 2006 .

[3]  Robert M. Townsend,et al.  Risk and Insurance in Village India , 1994 .

[4]  Ajay Jasra,et al.  Markov Chain Monte Carlo Methods and the Label Switching Problem in Bayesian Mixture Modeling , 2005 .

[5]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[6]  Ove Frank,et al.  http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained , 2007 .

[7]  S. Wasserman,et al.  Logit models and logistic regressions for social networks: II. Multivariate relations. , 1999, The British journal of mathematical and statistical psychology.

[8]  Thomas Brendan Murphy,et al.  Variational Bayesian inference for the Latent Position Cluster Model , 2009, NIPS 2009.

[9]  Julien Brailly,et al.  Exponential Random Graph Models for Social Networks , 2014 .

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

[11]  Mark S Handcock,et al.  Local dependence in random graph models: characterization, properties and statistical inference , 2015, Journal of the American Statistical Association.

[12]  Richard A. Davis,et al.  Efficient Gibbs Sampling of Truncated Multivariate Normal with Application to Constrained Linear Regression , 2004 .

[13]  Carey E. Priebe,et al.  Community Detection and Classification in Hierarchical Stochastic Blockmodels , 2015, IEEE Transactions on Network Science and Engineering.

[14]  D. A. Kenny,et al.  The Social Relations Model , 1984 .

[15]  Tom A. B. Snijders,et al.  Markov Chain Monte Carlo Estimation of Exponential Random Graph Models , 2002, J. Soc. Struct..

[16]  P. Diaconis,et al.  Estimating and understanding exponential random graph models , 2011, 1102.2650.

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

[18]  Jiqiang Guo,et al.  Stan: A Probabilistic Programming Language. , 2017, Journal of statistical software.

[19]  Tom A. B. Snijders,et al.  Exponential Random Graph Models for Social Networks , 2013 .

[20]  Florent Krzakala,et al.  Spectral Clustering of graphs with the Bethe Hessian , 2014, NIPS.

[21]  Bin Yu,et al.  Spectral clustering and the high-dimensional stochastic blockmodel , 2010, 1007.1684.

[22]  J. Plato.  Finite partial exchangeability , 1991 .

[23]  Arun G. Chandrasekhar,et al.  The Diffusion of Microfinance , 2012, Science.

[24]  S. Wasserman,et al.  Logit models and logistic regressions for social networks: I. An introduction to Markov graphs andp , 1996 .

[25]  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.

[26]  Arun G. Chandrasekhar,et al.  Social Investments, Informal Risk Sharing, and Inequality , 2014 .

[27]  Isto Huvila.  Knowing what we know , 2012 .

[28]  Garry Robins,et al.  An introduction to exponential random graph (p*) models for social networks , 2007, Soc. Networks.

[29]  Garritt L. Page,et al.  Spatial Product Partition Models , 2015, 1504.04489.

[30]  Stephen G. Walker,et al.  Label Switching in Bayesian Mixture Models: Deterministic Relabeling Strategies , 2014 .

[31]  H. White,et al.  “Structural Equivalence of Individuals in Social Networks” , 2022, The SAGE Encyclopedia of Research Design.

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

[33]  W. Dempsey,et al.  A framework for statistical network modeling , 2015, 1509.08185.

[34]  Lars Backstrom,et al.  The Anatomy of the Facebook Social Graph , 2011, ArXiv.

[35]  Christopher H. Jackson,et al.  Multi-State Models for Panel Data: The msm Package for R , 2011 .

[36]  Walter Dempsey,et al.  Edge exchangeable models for network data , 2016, ArXiv.

[37]  Peter D. Hoff,et al.  Latent Space Approaches to Social Network Analysis , 2002 .

[38]  Garry Robins,et al.  Exponential Random Graph Models for Social Networks: Index , 2012 .

[39]  M. Rosenzweig,et al.  Why is Mobility in India so Low? Social Insurance, Inequality, and Growth , 2009 .

[40]  Peter J. Bickel,et al.  Pseudo-likelihood methods for community detection in large sparse networks , 2012, 1207.2340.

[41]  Daniel M. Roy,et al.  Bayesian Models of Graphs, Arrays and Other Exchangeable Random Structures , 2013, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[42]  Jure Leskovec,et al.  Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters , 2008, Internet Math..

[43]  Tiago P. Peixoto.  Hierarchical block structures and high-resolution model selection in large networks , 2013, ArXiv.

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

[45]  Robin I. M. Dunbar.  Social Brain Hypothesis , 1998, Encyclopedia of Evolutionary Psychological Science.

[46]  D. Aldous.  Representations for partially exchangeable arrays of random variables , 1981 .

[47]  Andrew Parker,et al.  Knowing What We Know: Supporting Knowledge Creation and Sharing in Social Networks , 2001 .

[48]  Brian W. Junker,et al.  Hierarchical Network Models for Education Research , 2013 .

[49]  David Banks,et al.  Models for networks: a cross‐disciplinary science , 2012 .

[50]  A. Raftery,et al.  Model‐based clustering for social networks , 2007 .

[51]  Daniel M. Roy,et al.  The Class of Random Graphs Arising from Exchangeable Random Measures , 2015, ArXiv.

[52]  Matthew E. Brashears,et al.  Exponential Random Graph Models for Social Networks: Theory, Methods, and Applications , 2014 .

[53]  Emily B. Fox,et al.  Sparse graphs using exchangeable random measures , 2014, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[54]  Johan Koskinen,et al.  Using latent variables to account for heterogeneity in exponential family random graph models , 2009 .

[55]  Edoardo M. Airoldi,et al.  Stochastic blockmodels with growing number of classes , 2010, Biometrika.

[56]  Peter McCullagh,et al.  Exchangeability and regression models , 2004 .

[57]  Trevor Campbell,et al.  Edge-exchangeable graphs and sparsity , 2016, NIPS.

[58]  Adrian E. Raftery,et al.  Model-Based Clustering, Discriminant Analysis, and Density Estimation , 2002 .

[59]  Réka Albert,et al.  Near linear time algorithm to detect community structures in large-scale networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[60]  Thomas Brendan Murphy,et al.  Review of statistical network analysis: models, algorithms, and software , 2012, Stat. Anal. Data Min..