Title: An estimator for the tail-index of graphex processes,
Abstract: Sparse exchangeable graphs 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 sparse exchangeable 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. Accordingly, the
estimator is practically useful for estimation of sparse exchangeable models.
We also derive the analogous results for the bipartite sparse exchangeable
case.