Concentration and consistency results for canonical and curved exponential-family models of random graphs

Abstract

Statistical inference for exponential-family models of random graphs with dependent edges is challenging. We stress the importance of additional structure and show that additional structure facilitates statistical inference. A simple example of a random graph with additional structure is a random graph with neighborhoods and local dependence within neighborhoods. We develop the first concentration and consistency results for maximum likelihood and $M$-estimators of a wide range of canonical and curved exponential-family models of random graphs with local dependence. All results are non-asymptotic and applicable to random graphs with finite populations of nodes, although asymptotic consistency results can be obtained as well. In addition, we show that additional structure can facilitate subgraph-to-graph estimation, and present concentration results for subgraph-to-graph estimators. As an application, we consider popular curved exponential-family models of random graphs, with local dependence induced by transitivity and parameter vectors whose dimensions depend on the number of nodes.

References in corpus (15)

• High-dimensional Ising model selection using ${\ell }_1$-regularized logistic regression
• Modeling social networks from sampled data
• Exponential-Family Random Graph Models for Valued Networks
• Estimating and understanding exponential random graph models
• Random graphs with a given degree sequence
• Consistency under sampling of exponential random graph models
• Adjusting for Network Size and Composition Effects in Exponential-Family Random Graph Models
• Concentration inequalities for dependent Random variables via the martingale method
• Maximum lilkelihood estimation in the $\beta$-model
• Asymptotics in directed exponential random graph models with an increasing bi-degree sequence
• On the Question of Effective Sample Size in Network Modeling: An Asymptotic Inquiry
• On Graphical Models via Univariate Exponential Family Distributions
• Estimation in spin glasses: A first step
• A framework for statistical network modeling
• Statistics of the two-star ERGM

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• Signal Detection in Degree Corrected ERGMs
• A Structural Model of Business Card Exchange Networks