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We construct Bayesian and frequentist finite-sample goodness-of-fit tests for three different variants of the stochastic blockmodel for network data. Since all of the stochastic blockmodel variants are log-linear in form when block assignments are known, the tests for the latent block model versions combine a block membership estimator with the algebraic statistics machinery for testing goodness-of-fit in log-linear models. We describe Markov bases and marginal polytopes of the variants of the stochastic blockmodel and discuss how both facilitate the development of goodness-of-fit tests and understanding of model behaviour. The general testing methodology developed here extends to any finite mixture of log-linear models on discrete data, and as such is the first application of the algebraic statistics machinery for latent-variable models.

 

Abstract Exponential random graph models, or ERGMs, are a flexible and general class of models for modeling dependent data. While the early literature has shown them to be powerful in capturing many network features of interest, recent work highlights difficulties related to the models’ ill behavior, such as most of the probability mass being concentrated on a very small subset of the parameter space. This behavior limits both the applicability of an ERGM as a model for real data and inference and parameter estimation via the usual Markov chain Monte Carlo algorithms. To address this problem, we propose a new exponential family of models for random graphs that build on the standard ERGM framework. Specifically, we solve the problem of computational intractability and “degenerate” model behavior by an interpretable support restriction. We introduce a new parameter based on the graph-theoretic notion of degeneracy, a measure of sparsity whose value is commonly low in real-world networks. The new model family is supported on the sample space of graphs with bounded degeneracy and is called degeneracy-restricted ERGMs, or DERGMs for short. Since DERGMs generalize ERGMs—the latter is obtained from the former by setting the degeneracy parameter to be maximal—they inherit good theoretical properties, while at the same time place their mass more uniformly over realistic graphs. The support restriction allows the use of new (and fast) Monte Carlo methods for inference, thus making the models scalable and computationally tractable. We study various theoretical properties of DERGMs and illustrate how the support restriction improves the model behavior. We also present a fast Monte Carlo algorithm for parameter estimation that avoids many issues faced by Markov Chain Monte Carlo algorithms used for inference in ERGMs. 

Differential privacy has emerged as a popular model to provably limit privacy risks associated with a given data release. However releasing high dimensional synthetic data under differential privacy remains a challenging problem. In this paper, we study the problem of releasing synthetic data in the form of a high dimensional histogram under the constraint of differential privacy.We develop an $(\epsilon, \delta)$-differentially private categorical data synthesizer called \emph{Stability Based Hashed Gibbs Sampler} (SBHG). SBHG works by combining a stability based sparse histogram estimation algorithm with Gibbs sampling and feature selection to approximate the empirical joint distribution of a discrete dataset. SBHG offers a competitive alternative to state-of-the art synthetic data generators while preserving the sparsity structure of the original dataset, which leads to improved statistical utility as illustrated on simulated data. Finally, to study the utility of the resulting synthetic data sets generated by SBHG, we also perform logistic regression using the synthetic datasets and compare the classification accuracy with those from using the original dataset. 

Statistics computed from data are viewed as random variables. When they are used for tasks like hypothesis testing and confidence intervals, their true finite sample distributions are often replaced by approximating distributions that are easier to work with (for example, the Gaussian, which results from using approximations justified by the Central Limit Theorem). When data are perturbed by differential privacy, the approximating distributions also need to be modified. Prior work provided various competing methods for creating such approximating distributions with little formal justification beyond the fact that they worked well empirically. In this paper, we study the question of how to generate statistical approximating distributions for differentially private statistics, provide finite sample guarantees for the quality of the approximations.