The β-model — Maximum Likelihood, Cramér-Rao Bounds, and Hypothesis Testing

We study the maximum likelihood estimator in a setting where the dependent variable is a random graph and covariates are available on a graph-level. The model generalizes the well-known β-model for random graphs by replacing the constant model parameters with regression functions. CramérRao bounds are derived for special cases of the undirected βmodel, the directed β-model, and the covariate-based β-model. The corresponding maximum likelihood estimators are compared to the bounds by means of simulations. Moreover, examples are given on how to use the presented maximum likelihood estimators to test for directionality and significance. Last, the applicability of the model is demonstrated using temporal social network data describing communication among healthcare workers.

[1]  T. Yan,et al.  A note on asymptotic distributions in directed exponential random graph models with bi-degree sequences , 2017 .

[2]  T. Yan,et al.  Asymptotics of score test in the generalized β-model for networks , 2016 .

[3]  Stephen E. Fienberg,et al.  Statistical Inference in a Directed Network Model With Covariates , 2016, Journal of the American Statistical Association.

[4]  Elena Deych,et al.  Gibbs distribution for statistical analysis of graphical data with a sample application to fcMRI brain images , 2016, Statistics in medicine.

[5]  M. Bolla,et al.  Estimating Parameters of a Probabilistic Heterogeneous Block Model via the EM Algorithm , 2015 .

[6]  Thomas E. Nichols,et al.  A positive-negative mode of population covariation links brain connectivity, demographics and behavior , 2015, Nature Neuroscience.

[7]  Nitesh V. Chawla,et al.  Inferring user demographics and social strategies in mobile social networks , 2014, KDD.

[8]  Chenlei Leng,et al.  Asymptotics in directed exponential random graph models with an increasing bi-degree sequence , 2014, 1408.1156.

[9]  Yin Li,et al.  Assessing Goodness of Fit of Exponential Random Graph Models , 2013 .

[10]  A. Barrat,et al.  Estimating Potential Infection Transmission Routes in Hospital Wards Using Wearable Proximity Sensors , 2013, PloS one.

[11]  Hong Qin,et al.  Asymptotic normality in the maximum entropy models on graphs with an increasing number of parameters , 2013, J. Multivar. Anal..

[12]  Morten Mørup,et al.  Nonparametric Bayesian modeling of complex networks: an introduction , 2013, IEEE Signal Processing Magazine.

[13]  C. Hillar,et al.  Maximum entropy distributions on graphs , 2013, 1301.3321.

[14]  Hartwig R. Siebner,et al.  Identifying modular relations in complex brain networks , 2012, 2012 IEEE International Workshop on Machine Learning for Signal Processing.

[15]  T. Suesse.  Marginalized Exponential Random Graph Models , 2012 .

[16]  Vishesh Karwa,et al.  Inference using noisy degrees: Differentially private $\beta$-model and synthetic graphs , 2012, 1205.4697.

[17]  T. Yan,et al.  A central limit theorem in the β-model for undirected random graphs with a diverging number of vertices , 2012, 1202.3307.

[18]  Bruce A. Desmarais,et al.  Inferential Network Analysis with Exponential Random Graph Models , 2011, Political Analysis.

[19]  Jari Saramäki,et al.  Temporal Networks , 2011, Encyclopedia of Social Network Analysis and Mining.

[20]  A. Barabasi,et al.  Interactome Networks and Human Disease , 2011, Cell.

[21]  Hisayuki Hara,et al.  Graver basis for an undirected graph and its application to testing the beta model of random graphs , 2011, 1102.2583.

[22]  Allan Sly,et al.  Random graphs with a given degree sequence , 2010, 1005.1136.

[23]  Stephen E. Fienberg,et al.  Algebraic Statistics for a Directed Random Graph Model with Reciprocation , 2009, 0909.0073.

[24]  Peng Wang,et al.  Recent developments in exponential random graph (p*) models for social networks , 2007, Soc. Networks.

[25]  R. Sharan,et al.  Network-based prediction of protein function , 2007, Molecular systems biology.

[26]  G. Fagiolo.  Directed or Undirected? A New Index to Check for Directionality of Relations in Socio-Economic Networks , 2006, physics/0612017.

[27]  T. Snijders,et al.  p2: a random effects model with covariates for directed graphs , 2004 .

[28]  S. Schaal,et al.  Computational approaches to motor learning by imitation. , 2003, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[29]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[30]  Yuchung J. Wang,et al.  Stochastic Blockmodels for Directed Graphs , 1987 .

[31]  David Strauss.  On a general class of models for interaction , 1986 .

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

[33]  P. Holland,et al.  An Exponential Family of Probability Distributions for Directed Graphs , 1981 .

[34]  F. Massey.  The Kolmogorov-Smirnov Test for Goodness of Fit , 1951 .

[35]  S. S. Wilks.  The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses , 1938 .

[36]  Bryan S. Graham,et al.  An econometric model of link formation with degree heterogeneity , 2014 .

[37]  J. Komlos,et al.  When the degree sequence is a sufficient statistic , 2012 .

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

[39]  T. Minka.  A comparison of numerical optimizers for logistic regression , 2004 .

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

[41]  S. Fienberg,et al.  Categorical Data Analysis of Single Sociometric Relations , 1981 .

[42]  G. Rasch.  On General Laws and the Meaning of Measurement in Psychology , 1961 .

[43]  E. S. Pearson,et al.  On the Problem of the Most Efficient Tests of Statistical Hypotheses , 1933 .