Nonparametric regression for multiple heterogeneous networks

Abstract: We study nonparametric methods for the setting where multiple distinct networks are observed on the same set of nodes. Such samples may arise in the form of replicated networks drawn from a common distribution, or in the form of heterogeneous networks, with the network generating process varying from one network to another, e.g.~dynamic and cross-sectional networks. Nonparametric methods for undirected networks have focused on estimation of the graphon model. While the graphon model accounts for nodal heterogeneity, it does not account for network heterogeneity, a feature specific to applications where multiple networks are observed. To address this setting of multiple networks, we propose a multi-graphon model which allows node-level as well as network-level heterogeneity. We show how information from multiple networks can be leveraged to enable estimation of the multi-graphon via standard nonparametric regression techniques, e.g. kernel regression, orthogonal series estimation. We study theoretical properties of the proposed estimator establishing recovery of the latent nodal positions up to negligible error, and convergence of the multi-graphon estimator to the normal distribution. Finite sample performance are investigated in a simulation study and application to two real-world networks---a dynamic contact network of ants and a collection of structural brain networks from different subjects---illustrate the utility of our approach.