Estimation of Local Degree Distributions via Local Weighted Averaging and Monte Carlo Cross-Validation

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.