Chaining 2-FWL GNNs for Combinatorial Graph Alignment

For the combinatorial graph alignment problem (GAP) -- finding the node correspondence that maximizes the number of common edges (nce) between two unlabeled graphs -- properly initialized FAQ remains a strong classical baseline, while existing GNN approaches struggle in the purely structural setting. We introduce a chaining procedure: a sequence of Folklore-type (2-FWL) GNNs in which each network is trained with cross-entropy after decoding the previous network's similarity matrix and ranking nodes by their current alignment quality. This non-differentiable ranking step injects discrete combinatorial feedback at every link; at inference, we iterate the final network and keep the candidate with highest observed nce. On sparse Erdos-Renyi graphs at noise level 0.25, chained FGNNs with FAQ post-processing reach 85% accuracy versus 13% for FAQ initialized from the convex relaxation, and essentially 0% for prior GNN methods. On correlated regular graphs, where MPNNs with constant features produce identical node embeddings (1-WL fails to refine) and FAQ's convex initialization is degenerate, chaining is the only method we know that recovers a non-trivial alignment. On three real-world benchmarks (yeast PPI, coauthorship, and road networks), we show that recent comparisons underestimate FAQ by initializing it from a uniform doubly stochastic matrix; once FAQ is initialized from the convex relaxation it already surpasses prior reported numbers, and dataset-specific chained FGNNs further improve on this strengthened baseline.