[Paper Review] Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matching

Hongteng Xu, Dixin Luo+1 more|arXiv (Cornell University)|May 18, 2019

Advanced Graph Neural NetworksComputer Science54 references35 citations

TL;DR

Proposes S-GWL, a scalable framework using Gromov-Wasserstein discrepancy (GW) to unify and accelerate graph partitioning and matching, including multi-graph analysis via GW barycenters.

ABSTRACT

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.

Motivation & Objective

• Motivate a unified GW-based approach for both graph partitioning and matching.
• Develop a scalable algorithm to apply GW discrepancy on large graphs.
• Introduce a recursive K-partition strategy to reduce complexity.
• Incorporate a regularized proximal gradient method with sparsity to accelerate computation.
• Enable multi-graph partitioning and matching via GW barycenters.

Proposed method

• Use GW discrepancy as a pseudometric to relate graphs and obtain node correspondences or partitions via optimal transport.
• Formulate graph partitioning as d_gw(G, G_dc) where G_dc is a disconnected K-node graph, yielding a K-way partition.
• Extend to multi-graph analysis with a GW barycenter GWB that minimizes weighted GW discrepancy to observed graphs.
• Apply a regularized proximal gradient method (p=2) to compute d_gw efficiently, with a Sinkhorn-based solver and KL proximal term.
• Incorporate node-distribution-based priors and a node-regularization cost to guide transport toward topology-aware matchings.
• Introduce a recursive K-partition mechanism to decompose large graphs into smaller sub-graphs and perform local GW matching, reducing overall complexity to O(K(E+V) log_K V).

Experimental results

Research questions

• RQ1 Can GW discrepancy be used to unify graph partitioning and graph matching in a single optimization framework?
• RQ2 How can GW computations be scaled to large graphs while preserving accuracy?
• RQ3 Can a learned GW barycenter enable robust multi-graph matching and partitioning?
• RQ4 Does a recursive K-partition strategy significantly reduce runtime without sacrificing matching/partitioning quality?
• RQ5 What priors or regularizations improve stability and performance of GW-based graph analysis?

Key findings

• S-GWL achieves scalable graph partitioning and matching with time complexity O(K(E+V) log_K V) and memory O(E+VK).
• GW discrepancy acts as a natural, relational pseudometric that links node correspondences to partitions via optimal transport.
• Learning a GW barycenter enables coherent multi-graph matching and partitioning by aligning clusters across graphs.
• Regularized proximal gradient with KL term improves convergence and numerical stability compared to entropy-based methods.
• Empirical results show S-GWL and GWL outperform state-of-the-art baselines in accuracy and/or efficiency on synthetic and real networks, and S-GWL accelerates GWL significantly.
• The method remains effective when applied to synthetic Gaussian/Barabási–Albert graphs and real-world networks, including PPI and organizational communication data.