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

Abstract

Graphex processes resolve some pathologies in traditional random graph models, notably, providing models that are both projective and allow sparsity. In a recent paper, Caron and Rousseau (2017) show that for a large class of graphex models, the sparsity behaviour is governed by a single parameter: the tail-index of the function (the graphon) that parameterizes the model. We propose an estimator for this parameter and quantify its risk. Our estimator is a simple, explicit function of the degrees of the observed graph. In many situations of practical interest, the risk decays polynomially in the size of the observed graph. We illustrate the importance of a good estimator for the tail-index through the graph analogue of the unseen species problem. We also derive the analogous results for the bipartite graphex processes.