  We propose a scalable Gromov-Wasserstein learning (S-GWL) method and
establish a novel and theoretically-supported paradigm for large-scale graph
analysis. The proposed method is based on the fact that Gromov-Wasserstein
discrepancy is a pseudometric on graphs. Given two graphs, the optimal
transport associated with their Gromov-Wasserstein discrepancy provides the
correspondence between their nodes and achieves graph matching. When one of the
graphs has isolated but self-connected nodes ($i.e.$, a disconnected graph),
the optimal transport indicates the clustering structure of the other graph and
achieves graph partitioning. Using this concept, we extend our method to
multi-graph partitioning and matching by learning a Gromov-Wasserstein
barycenter graph for multiple observed graphs; the barycenter graph plays the
role of the disconnected graph, and since it is learned, so is the clustering.
Our method combines a recursive $K$-partition mechanism with a regularized
proximal gradient algorithm, whose time complexity is $\mathcal{O}(K(E+V)\log_K
V)$ for graphs with $V$ nodes and $E$ edges. To our knowledge, our method is
the first attempt to make Gromov-Wasserstein discrepancy applicable to
large-scale graph analysis and unify graph partitioning and matching into the
same framework. It outperforms state-of-the-art graph partitioning and matching
methods, achieving a trade-off between accuracy and efficiency.
