Connectivity-Guided Sparsification of 2-FWL GNNs: Preserving Full Expressivity with Improved Efficiency

Abstract

Higher-order Graph Neural Networks (HOGNNs) based on the 2-FWL test achieve superior expressivity by modeling 2-node and 3-node interactions, but incur cubic computational cost. Existing efficiency methods typically reduce this burden at the expense of expressivity. We propose Co-Sparsify, a connectivity-aware sparsification framework that eliminates provably redundant computations while preserving full 2-FWL expressive power. Our key insight is that 3-node interactions are expressively necessary only within biconnected components, namely, maximal subgraphs where every node pair lies on a cycle. Outside these components, structural relationships are fully captured via 2-node message passing and graph readouts, rendering higher-order modeling unnecessary. Co-Sparsify restricts 2-node message passing to connected components and 3-node interactions to biconnected components, eliminating redundant computation without approximation or sampling. We prove that Co-Sparsified GNNs match the expressivity of the 2-FWL test. Empirically, when applied to PPGN, Co-Sparsify matches or exceeds accuracy on synthetic substructure counting tasks and achieves state-of-the-art performance on real-world benchmarks (ZINC, QM9 and TUD). This study demonstrates that high expressivity and scalability are not mutually exclusive: principled, topology-guided sparsification enables powerful, efficient GNNs with theoretical guarantees.