Towards Metric-Faithful Neural Graph Matching

TL;DR

This paper develops a theoretical framework linking encoder geometry to neural graph edit distance estimation quality, demonstrating that geometry-aware encoders improve GED prediction and ranking across benchmarks.

Contribution

It introduces a geometry-aware perspective for neural graph matching, showing that bi-Lipschitz encoders enhance GED estimation and ranking stability, and provides empirical validation with a new encoder variant.

Findings

Bi-Lipschitz encoders improve GED surrogate control.

Geometry-aware variants outperform baselines on benchmarks.

Encoder geometry is a key design principle for neural graph matching.

Abstract

Graph Edit Distance (GED) is a fundamental, albeit NP-hard, metric for structural graph similarity. Recent neural graph matching architectures approximate GED by first encoding graphs with a Graph Neural Network (GNN) and then applying either a graph-level regression head or a matching-based alignment module. Despite substantial architectural progress, the role of encoder geometry in neural GED estimation remains poorly understood. In this paper, we develop a theoretical framework that connects encoder geometry to GED estimation quality for two broad classes of neural GED estimators: graph similarity predictors and alignment-based methods. On fixed graph collections, where the doubly-stochastic metric $d_{\mathrm{DS}}$ is comparable to GED, we show that graph-level bi-Lipschitz encoders yield controlled GED surrogates and improved ranking stability; for matching-based estimators, node-level bi-Lipschitz geometry propagates to encoder-induced alignment costs and the resulting optimized alignment objective. We instantiate this perspective using FSW-GNN, a bi-Lipschitz WL-equivalent encoder, as a drop-in replacement in representative neural GED architectures. Across representative baselines and benchmark datasets, the resulting geometry-aware variants significantly improve GED prediction and ranking metrics. A faithfulness case study of untrained encoders, together with ablations and transfer experiments, supports the view that these gains arise from improved representation geometry, positioning encoder geometry as a useful design principle for neural graph matching.