Local Clustering on Complex Graphs and Complex Hypergraphs

Local/seeded clustering aims to find a compact cluster near the given starting instances. While most existing studies on graph clustering assume a discrete graph setting (i.e., unweighted, undirected graphs without self-loops), real-world graphs can be more complex. In this paper, we extend the classic non-approximating Andersen-Chung-Lang (ACL) clustering algorithm beyond discrete graphs and generalize its quadratic optimality to a wider range of complex graphs, including weighted, directed, and self-looped graphs and hypergraphs with edge-dependent vertex weights. Specifically, by leveraging PageRank, we propose two algorithms: GeneralACL for graphs and HyperACL for hypergraphs. We prove that, under two mild conditions, both algorithms can identify a quadratically optimal cluster in terms of conductance 1 with at least 1 2 probability. Additionally, we provide experiments to validate our theoretical findings. Our code is available at https://github.com/iDEA-iSAIL- Lab-UIUC/HyperACL.