Hypergraph neural networks (HGNNs) are widely used models for analyzing higher-order relational data. HGNNs suffer from the rapid performance degradation with increasing layers. Hypergraph dynamic system (HDS) is a potential way to deal with this challenge. However, hypergraph dynamic system is confined to a time-continuous isotropic model, lacking positional information in the structural space of the hypergraph. In contrast, anisotropic diffusion can capture structural space differences among vertices, providing a more precise representation of the information propagation process in hypergraph structures than isotropic diffusion. In this paper, we introduce HGNNv2, a stable hypergraph neural network, which is built as a hypergraph dynamic system with partial differential equation (PDE). This model incorporates a position-aware anisotropic diffusion term and an external control term. We further present the vertex-rooted subtree method to determine anisotropic diffusion intensity. HGNNv2 has properties that vertices occupying equivalent positions in the structural space share equivalent structural labels and positional features. Experiments on 6 hypergraph datasets and 3 graph datasets reveal that HGNNv2 outperforms all 12 compared methods. HGNNv2 is capable of achieving stable final representations and task accuracy even under noisy conditions. HGNNv2 achieves stable performance with fewer layers than hypergraph dynamic systems employing isotropic diffusion. We provide feature visualizations to illustrate the evolution of representations.

Existing methods for whole-slide histopathological image analysis commonly depend on patch-level sampling and deep feature extraction based on ImageNet pre-trained models. However, this strategy frequently fails to fully capture the diversity and complex associations within the tissue microenvironment. To address this problem, we propose a HyperGraph Self-supervised Learning (HGSL) method to utilize both local and global information in whole-slide images. Specifically, HGSL consists of two key modules: a Transformer-based self-supervised pretraining (TranSSL) module and a pathology hypergraph neural network (PathHGNN) module. TranSSL uses a mask reconstruction task to mine latent structures and patterns from unlabeled data, thus yielding pathology semantic deep representations. PathHGNN builds a hypergraph structure based on these representations to describe high-order relationships among patches, enabling multi-level feature aggregation and propagation to integrate local microenvironmental information with the overall topological correlations. The proposed HGSL was validated on the TCGA-GBM and TCGA-KIRC datasets, and the experimental results show that HGSL significantly outperforms all existing comparative methods in survival prediction tasks. Furthermore, the KM curve demonstrates that the proposed HGSL significantly distinguishes between high-and low-risk groups, indicating its potential for future clinical applications.

Recently, hypergraph neural networks (HGNNs) exhibit the potential to tackle tasks with high-order correlations and have achieved success in many tasks. However, existing evolution on the hypergraph has poor controllability and lacks sufficient theoretical support (like dynamic systems), thus yielding sub-optimal performance. One typical scenario is that only one or two layers of HGNNs can achieve good results and more layers lead to degeneration of performance. Under such circumstances, it is important to increase the controllability of HGNNs. In this paper, we first introduce hypergraph dynamic systems (HDS), which bridge hypergraphs and dynamic systems and characterize the continuous dynamics of representations. We then propose a control-diffusion hypergraph dynamic system by an ordinary differential equation (ODE). We design a multi-layer HDS$^{ode}$ as a neural implementation, which contains control steps and diffusion steps. HDS$^{ode}$ has the properties of controllability and stabilization and is allowed to capture long-range correlations among vertices. Experiments on $9$ datasets demonstrate HDS$^{ode}$ beat all compared methods. HDS$^{ode}$ achieves stable performance with increased layers and solves the poor controllability of HGNNs. We also provide the feature visualization of the evolutionary process to demonstrate the controllability and stabilization of HDS$^{ode}$.