Ordinal embedding for network estimation via graphon

Many applications routinely record multiple networks on the same set of nodes. In such applications, interest often lies in understanding differences in interaction behavior at the node level as well as the variability in structure across networks. In this paper, we focus on estimation of the underlying network structure at the node-level, via a graph limit object, known as the graphon function. The graphon determines network structure via probabilities of interaction between node pairs, and interest lies in its estimation. We propose a two-step graphon estimation procedure. First, we establish data sampling regimes which allow consistent recovery of the latent node positions via ordinal embedding. Second, we show how estimated node positions lead to a plug-in graphon estimator via classical nonparametric regression techniques such as series estimation. We derive the rate at which the proposed plug-in series graphon estimator converges to the corresponding classical series graphon estimator based on the true node positions. Finite sample performance investigated in a simulation study show significant improvements over existing approaches. Application to a dynamic contact network of ants and a collection of structural brain networks illustrate the utility of our approach.