Uniform estimation of a class of random graph functionals

We consider estimation of certain functionals of random graphs. The random graph is generated by a possibly sparse stochastic block model (SBM). The number of classes is fixed or grows with the number of vertices. Minimax lower and upper bounds of estimation along specific submodels are derived. The results are nonasymptotic and imply that uniform estimation of a single connectivity parameter is much slower than the expected asymptotic pointwise rate. Specifically, the uniform quadratic rate does not scale as the number of edges, but only as the number of vertices. The lower bounds are local around any possible SBM. An analogous result is derived for functionals of a class of smooth graphons