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 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. On correlated regular graphs, where constant-feature MPNNs collapse to uninformative similarities and FAQ's convex initialization is degenerate, chaining recovers non-trivial alignments among the methods we evaluate. On three real-world benchmarks (yeast PPI, coauthorship, and road networks), dataset-specific chained FGNNs provide modest gains over a properly initialized FAQ baseline.