[Paper Review] COPT: Coordinated Optimal Transport for Graph Sketching

Yihe Dong, Will Sawin|arXiv (Cornell University)|Mar 9, 2020

Complex Network Analysis Techniques51 references4 citations

TL;DR

COPT introduces a novel unsupervised graph distance metric based on coordinated optimal transport, simultaneously optimizing vertex and spectral transport plans to preserve global graph structure. It enables high-accuracy graph sketching and classification, achieving 97.8% retrieval accuracy on synthetic graphs—15.7 percentage points above spectral projections—while being 2000× faster than Gromov-Wasserstein distance.

ABSTRACT

We introduce COPT, a novel distance metric between graphs defined via an optimization routine, computing a coordinated pair of optimal transport maps simultaneously. This gives an unsupervised way to learn general-purpose graph representation, applicable to both graph sketching and graph comparison. COPT involves simultaneously optimizing dual transport plans, one between the vertices of two graphs, and another between graph signal probability distributions. We show theoretically that our method preserves important global structural information on graphs, in particular spectral information, and analyze connections to existing studies. Empirically, COPT outperforms state of the art methods in graph classification on both synthetic and real datasets.

Motivation & Objective

• To develop a new unsupervised graph distance metric that preserves global structural and spectral information.
• To enable efficient graph sketching that maintains representational fidelity for downstream tasks like classification and retrieval.
• To address limitations of existing graph distance methods, such as high computational cost and inability to handle graphs of different sizes.
• To unify vertex-level and spectral-level comparison via coordinated transport plans for improved graph representation.

Proposed method

• COPT defines a distance metric using a coordinated optimization over two transport plans: one mapping vertices between graphs and another mapping their Laplacian spectral distributions.
• The method minimizes a joint loss function that couples vertex transport and spectral transport, ensuring alignment between structural and spectral similarity.
• It leverages optimal transport theory to compute a metric that is zero if and only if graphs are isomorphic, satisfying key metric properties.
• The algorithm computes a sketch by reducing the original graph to a smaller one via optimal transport, preserving key spectral and structural features.
• The sketch is represented as a compressed Laplacian matrix, with node labels derived from the transport plan to reflect original node groupings.
• The approach enables efficient similarity search using l1 distance on sketches, significantly accelerating retrieval compared to full graph comparison.

Experimental results

Research questions

• RQ1 Can a unified graph distance metric be designed that jointly optimizes vertex and spectral-level correspondence between graphs?
• RQ2 How well can COPT preserve global spectral and structural information during graph sketching compared to existing methods?
• RQ3 To what extent does COPT improve retrieval accuracy and speed in graph classification pipelines?
• RQ4 Can COPT be used effectively in real-world graph datasets with varying sizes and topologies?
• RQ5 How does COPT compare to state-of-the-art methods like Gromov-Wasserstein and spectral projection in terms of accuracy and efficiency?

Key findings

• COPT achieved 97.8% ± 1.1% retrieval accuracy on a synthetic dataset of 600 graphs, outperforming spectral projection by 15.7 percentage points.
• COPT sketches enabled retrieval accuracy within 2.0% of the full Gromov-Wasserstein distance on original graphs, while being 2000× faster (1.81 ms vs. 3.69 s).
• In a hybrid pipeline, COPT+GW achieved 98.7% ± 0.3% accuracy when filtering to top 3 candidates, surpassing spectral projection+GW (89.4% ± 2.4%).
• On the real BZR_MD dataset, COPT achieved 57.2% ± 4.9% accuracy when reduced to 7-node sketches, outperforming OTC (52.2% ± 5.5%).
• Visualizations confirmed that COPT preserves global structures and maps structurally similar nodes in the original graph to the same or nearby nodes in the sketch.
• The method demonstrated robustness across diverse graph types, including barbell, wheel, ladder, and random geometric graphs, with consistent preservation of spectral and topological features.