Recently, hypergraphs have attracted a lot of attention due to their ability to capture complex relations among entities. The insurgence of hypergraphs has resulted in data of increasing size and complexity that exhibit interesting small-scale and local structure, e.g., small-scale communities and localized node-ranking around a given set of seed nodes. Popular and principled ways to capture the local structure are the local hypergraph clustering problem and related seed set expansion problem. In this work, we propose the first local diffusion method that achieves edge-size-independent Cheeger-type guarantee for the problem of local hypergraph clustering while applying to a rich class of higher-order relations that covers many previously studied special cases. Our method is based on a primal-dual optimization formulation where the primal problem has a natural network flow interpretation, and the dual problem has a cut-based interpretation using the $\ell_2$-norm penalty on associated cut-costs. We demonstrate the new technique is significantly better than state-of-the-art methods on both synthetic and real-world data.
A nonlinear diffusion method for semi-supervised learning on hypergraphs
School of Computer Science（计算机科学学院）；Gran Sasso Science Institute（大萨索科学研究所）；Department of Computer Science（计算机科学系）；Cornell University（康奈尔大学）；School of Mathematics（数学学院）
From Diffusion to Reaction-Diffusion: A Dynamical-Systems View of Oversmoothing in Hypergraph Neural Networks
School of Mathematics and Statistics, Shandong University, Weihai（山东大学（威海）数学与统计学院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
Hypergraph Neural Diffusion: A PDE-Inspired Framework for Hypergraph Message Passing
School of Mathematics and Statistics, Shandong University（山东大学数学与统计学院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）；School of Cyber Security, University of Chinese Academy of Sciences（中国科学院大学网络空间安全学院）
Hypergraph Neural Stochastic Diffusion: An SDE Framework for Uncertainty Estimation
School of Mathematics and Statistics, Shandong University, Weihai（山东大学（威海）数学与统计学院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
Hypergraph Neural Sheaf Diffusion: A Symmetric Simplicial Set Framework for Higher-Order Learning
POSTECH；Center for Geometry and Physics, Institute for Basic Science (IBS)（几何与物理中心，基础科学研究所）；Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）
