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 the Enron email corpus to model smooth trends in monthly networks and highlight anomalous networks. Another motivating application is given in corpus linguistics, which explores trends in an author's writing style over time based on word co-occurrence networks

Topological Learning for Brain Networks

This paper proposes a novel topological learning framework that can integrate networks of different sizes and topology through persistent homology. This is possible through the introduction of a new topological loss function that enables such challenging task. The use of the proposed loss function bypasses the intrinsic computational bottleneck associated with matching networks. We validate the method in extensive statistical simulations with ground truth to assess the effectiveness of the topological loss in discriminating networks with different topology. The method is further applied to a twin brain imaging study in determining if the brain network is genetically heritable. The challenge is in overlaying the topologically different functional brain networks obtained from the resting-state functional MRI (fMRI) onto the template structural brain network obtained through the diffusion MRI (dMRI)