Learning Graphons via Structured Gromov-Wasserstein Barycenters

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-ofthe-art methods, and outperforms them on both synthetic and real-world data. The code is available at https: //github. com/ HongtengXu/SGWB-Graphon.

Authors

• Hongteng Xu Renmin University of China
• Dixin Luo School of Computer Science and Technology, Beijing Institute of Technology
• Lawrence Carin
• Hongyuan Zha Shenzhen Institute of Artificial Intelligence and Robotics for Society The Chinese University of Hong Kong, Shenzhen

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