Bootstrap estimators for the tail-index and for the count statistics of graphex processes

TL;DR

This paper introduces bootstrap-based estimators for key functionals of graphex processes, addressing the gap in nonparametric inference methods for these models which are important for modeling sparse and projective random graphs.

Contribution

It proposes novel bootstrap estimators for graphon functionals in graphex processes, a significant step towards nonparametric inference in this area.

Findings

Estimators are consistent and asymptotically normal.

Method effectively captures tail-index and count statistics.

Provides a new bootstrap framework for graphex process inference.

Abstract

Graphex processes resolve some pathologies in traditional random graph models, notably, providing models that are both projective and allow sparsity. Most of the literature on graphex processes study them from a probabilistic point of view. Techniques for inferring the parameter of these processes -- the so-called \textit{graphon} -- are still marginal; exceptions are a few papers considering parametric families of graphons. Nonparametric estimation remains unconsidered. In this paper, we propose estimators for a selected choice of functionals of the graphon. Our estimators originate from the subsampling theory for graphex processes, hence can be seen as a form of bootstrap procedure.