Title: Estimation of Local Degree Distributions via Local Weighted Averaging and Monte Carlo Cross-Validation,
Abstract: Owing to their capability of summarising interactions between elements of a
system, networks have become a common type of data in many fields. As networks
can be inhomogeneous, in that different regions of the network may exhibit
different topologies, an important topic concerns their local properties. This
paper focuses on the estimation of the local degree distribution of a vertex in
an inhomogeneous network. The contributions are twofold: we propose an
estimator based on local weighted averaging, and we set up a Monte Carlo
cross-validation procedure to pick the parameters of this estimator. Under a
specific modelling assumption we derive an oracle inequality that shows how the
model parameters affect the precision of the estimator. We illustrate our
method by several numerical experiments, on both real and synthetic data,
showing in particular that the approach considerably improves upon the natural,
empirical estimator.