Non-parametric regression for networks

Network data are becoming increasingly available, and so there is a need to develop suitable methodology for statistical analysis. Networks can be represented as graph Laplacian matrices, which are a type of manifold-valued data. Our main objective is to estimate a regression curve from a sample of graph Laplacian matrices conditional on a set of Euclidean covariates, for example in dynamic networks where the covariate is time. We develop an adapted Nadaraya-Watson estimator which has uniform weak consistency for estimation using Euclidean and power Euclidean metrics.  
 
We apply the methodology to a study of peptide shape variation from molecular dynamics simulations, where networks are formed from the correlations between atoms. We investigate nonparametric regression of the networks versus time, and also versus a predictor measuring the change in size of the peptide. Further applications are given to an email corpus to model smooth trends in monthly networks and highlight anomalous networks. A final motivating application is given in corpus linguistics, which explores trends in an author’s writing style over time based on word co-occurrence networks. 
 
This is joint work with Katie Severn and Simon Preston.