  We propose a novel and principled method to learn a nonparametric graph model
called graphon, which is defined in an infinite-dimensional space and
represents arbitrary-size graphs. Based on the weak regularity lemma from the
theory of graphons, we leverage a step function to approximate a graphon. We
show that the cut distance of graphons can be relaxed to the Gromov-Wasserstein
distance of their step functions. Accordingly, given a set of graphs generated
by an underlying graphon, we learn the corresponding step function as the
Gromov-Wasserstein barycenter of the given graphs. Furthermore, we develop
several enhancements and extensions of the basic algorithm, $e.g.$, the
smoothed Gromov-Wasserstein barycenter for guaranteeing the continuity of the
learned graphons and the mixed Gromov-Wasserstein barycenters for learning
multiple structured graphons. The proposed approach overcomes drawbacks of
prior state-of-the-art methods, and outperforms them on both synthetic and
real-world data. The code is available at
https://github.com/HongtengXu/SGWB-Graphon.
