PSMC: Provable and Scalable Algorithms for Motif Conductance Based Graph Clustering

Higher-order graph clustering aims to partition the graph using frequently occurring subgraphs. Motif conductance is one of the most promising higher-order graph clustering models due to its strong interpretability. However, existing motif conductance based graph clustering algorithms are mainly limited by a seminal two-stage reweighting computing framework, needing to enumerate all motif instances to obtain an edge-weighted graph for partitioning. However, such a framework has two-fold vital defects: (1) It can only provide a quadratic bound for the motif with three vertices, and whether there is provable clustering quality for other motifs is still an open question. (2) The enumeration procedure of motif instances incurs prohibitively high costs against large motifs or large dense graphs due to combinatorial explosions. Besides, expensive spectral clustering or local graph diffusion on the edge-weighted graph also makes existing methods unable to handle massive graphs with millions of nodes. To overcome these dilemmas, we propose a Provable and Scalable Motif Conductance algorithm PSMC, which has a fixed and motif-independent approximation ratio for any motif. Specifically, PSMC first defines a new vertex metric Motif Resident based on the given motif, which can be computed locally. Then, it iteratively deletes the vertex with the smallest motif resident value very efficiently using novel dynamic update technologies. Finally, it outputs the locally optimal result during the above iterative process. To further boost efficiency, we propose several effective bounds to estimate the motif resident value of each vertex, which can greatly reduce computational costs. Empirical results show that our proposed algorithms achieve 3.2-32 times speedup and improve the quality by at least 12 times than the baselines.