Uncertainty Quantification in Graphon Estimation Using Generalized Fiducial Inference

Yi Su, Jan Hannig, Thomas C. M. Lee

Abstract: Network data can be modeled as an exchangeable graph model (ExGM), and graphon is a two-dimensional function that generates an ExGM. The problem of graphon estimation has been popular in recent years, and several consistent estimation methods have been proposed. However, statistical inference on graphon has not been intensively studied. In this paper, we propose applying the generalized fiducial inference (GFI) methodology to the framework of graphon and perform the uncertainty quantification task. GFI is a branch of inference methods that utilizes the “switching principle” of the parameter and the data, and it seeks for a distribution estimator of the parameters without the need of a prior. We propose an easy-to-implement algorithm to generate fiducial samples of a graphon, which are then used to construct confidence sets. We establish theoretical guarantees of the GFI confidence intervals, and use synthetic graphons to demonstrate its empirical performance for finite sample size. When the labels are unknown, we extend our algorithm and discuss its asymptotic properties. We also apply the proposed method to Facebook social network data and unveil some interesting patterns.