Asymptotic Analysis of Statistical Estimators related to MultiGraphex Processes under Misspecification

TL;DR

This work analyzes the asymptotic behavior of statistical estimators for parameters linked to graphex/multigraphex graph models under misspecification, within a broad sparse-graph setting that allows unbounded degrees. It develops a likelihood-based framework where the log-likelihood is well-approximated by a tractable surrogate $\mathcal{Q}_t$, enabling a unique, consistent MLE $\hat{\phi}_t=(\hat{\sigma}_t,\hat{\tau}_t,\hat{s}_t)$ with $\hat{\sigma}_t\to\alpha_0$, $\hat{\tau}_t\to\tau_*$ and $\hat{s}_t/s_{*,t}\to1$. In the Bayesian setting, the posterior concentrates and satisfies a Bernstein–von Mises type result in the sparse regime, while in the dense regime the posterior on $\sigma$ concentrates on non-positive values; well-specified Caron–Fox models yield faster convergence and credible regions with correct frequentist coverage. The paper also proves that multigraphex processes exhibit sparsity and power-law degree properties, and that such processes satisfy the required degree-distribution concentration assumptions, enabling the general results to apply to a broad class of graphs, including several classical models. Overall, the results provide a rigorous link between network sparsity, degree distributions, and inferential properties of estimators for graphex/MultiGraphex parameters, with implications for model misspecification and robust inference in large sparse networks.

Abstract

This article studies the asymptotic properties of Bayesian or frequentist estimators of a vector of parameters related to structural properties of sequences of graphs. The estimators studied originate from a particular class of graphex model introduced by Caron and Fox. The analysis is however performed here under very weak assumptions on the underlying data generating process, which may be different from the model of Caron and Fox or from a graphex model. In particular, we consider generic sparse graph models, with unbounded degree, whose degree distribution satisfies some assumptions. We show that one can relate the limit of the estimator of one of the parameters to the sparsity constant of the true graph generating process. When taking a Bayesian approach, we also show that the posterior distribution is asymptotically normal. We discuss situations where classical random graphs models such as configuration models, sparse graphon models, edge exchangeable models or graphon processes satisfy our assumptions.

Theorems & Definitions (37)

• Theorem 1
• Theorem 2: Bayes, Sparse regime
• Theorem 3: Bayes, dense regime
• Theorem 4
• Lemma 1
• Lemma 2
• Remark 1
• Proposition 1
• Theorem 5: Asymptotic properties of the multigraphex process
• Corollary 1: Sparsity and power-law degree distribution