A Lovász-Simonovits Theorem for Hypergraphs with Application to Local Clustering

Abstract

We present the first analysis of diffusion on hypergraphs based on the Lovász-Simonovits theory. We demonstrate that an averaging-based diffusion operator is the appropriate generalization of the lazy random walk diffusion on 2-graphs because the diffusion rapidly converges to its stationary state from any initial state. By proving a Lovász-Simonovits-like theorem for this diffusion, we show that the diffusion rate depends on the hypergraph's conductance. To use averaging-based diffusion for clustering, we define a generalization of personalized page rank for hypergraphs, which we call ''Averaging-based Personalized Page Rank for Hypergraphs'' (APPRH). The fact that averaging-based diffusion is linear, unlike previous hypergraph diffusions used for clustering in the literature, allows us to use the Forward Push algorithm to compute APPRH efficiently. Using this method, we obtain theoretical bounds for the conductance of our clustering that are at least a constant times better than the best-known bounds in the literature. We compare our algorithm A- HyperCut against baselines on million-scale hypergraphs and find that our method is an order of magnitude faster while being competitive regarding the conductance of the local clusters produced.