Asymptotic analysis of statistical estimators related to MultiGraphex processes under misspecification

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 (J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 (2017) 1295–1366). The analysis is however performed here under very weak assumptions on the underlying data generating process, which may be different from the model of (J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 (2017) 1295–1366) 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, satisfy our assumptions.