Provable Higher-Order Graph Clustering: The Power of Peeling-Based Approaches

Higher-order graph clustering partitions graphs use frequently occurring subgraphs instead of edges, proving effective in community detection and knowledge discovery. Motif conductance, known for its strong interpretability, is a leading model. However, existing motif conductance algorithms are hindered by a two-stage reweighting framework that requires enumerating motif instances to generate an edge-weighted graph for partitioning. This framework has two major drawbacks: (1) It provides only a quadratic bound for three-vertex motifs, with no provable approximation guarantees for other motifs. (2) Enumerating motif instances is computationally prohibitive for large motifs or dense graphs due to combinatorial explosions. Besides, costly spectral clustering or local graph diffusion on the edge-weighted graph limits their scalability. In this paper, we propose a novel peeling-based clustering framework, PSMC, offering a motif-independent approximation ratio for any motif. Specifically, PSMC first defines a new locally computable vertex metric Motif Resident based on the given motif. Then, it iteratively deletes vertices with the smallest motif resident using efficient dynamic update techniques, outputting a locally optimal result with approximation guarantees. Besides, we introduce several powerful optimization techniques to further reduce computational costs. Empirical results on real-world and synthetic datasets showcase our proposed solutions' superiority over ten competitors.