Estimation of local degree distributions via local weighted averaging and Monte Carlo cross-validation

Owing to their capability of summarising the interactions between the elements of a system, networks have become a common type of data across a broad range of scientific fields. As networks can be heterogeneous – in the sense that different regions of the network may exhibit different topologies – an important topic concerns the study of their local properties. A method to estimate the local degree distribution of a vertex in a heterogeneous network is developed. The contributions are twofold: firstly, the proposal of an estimator based on local weighted averaging and secondly, the set up of a Monte Carlo cross-validation procedure to pick the parameters of this estimator. The method is illustrated by several numerical experiments, showing in particular that the approach considerably improves upon the natural, empirical estimator.