Adaptive Expansion for Hypergraph Learning

TL;DR

This work tackles the information loss and task-agnostic nature of classical hypergraph expansions by introducing Adaptive Expansion (AdE), which learns how to best convert a hypergraph into a weighted graph for downstream GNNs. AdE combines a Global Simulation Network (GSi-Net) to adaptively select two representative nodes per hyperedge with a distance-aware kernel that assigns edge weights based on feature similarity, producing a graph $\mathcal G_a$ whose structure and weights are tailored to the downstream task. The approach is theoretically grounded in WL-type isomorphism tests (1-WL and 1-GWL) and is empirically validated on seven benchmark hypergraphs, showing superior performance over traditional expansions and HyGNNs across both homophilic and heterophilic settings. Overall, AdE provides a flexible, principled framework for high-order hypergraph learning by learning task-specific graph structures during expansion, enabling more effective node representations for classification tasks.

Abstract

Hypergraph, with its powerful ability to capture higher-order relationships, has gained significant attention recently. Consequently, many hypergraph representation learning methods have emerged to model the complex relationships among hypergraphs. In general, these methods leverage classic expansion methods to convert hypergraphs into weighted or bipartite graphs, and further employ message passing mechanisms to model the complex structures within hypergraphs. However, classical expansion methods are designed in straightforward manners with fixed edge weights, resulting in information loss or redundancy. In light of this, we design a novel clique expansion-based Adaptive Expansion method called AdE to adaptively expand hypergraphs into weighted graphs that preserve the higher-order structure information. Specifically, we introduce a novel Global Simulation Network to select two representative nodes for adaptively symbolizing each hyperedge and connect the rest of the nodes within the same hyperedge to the corresponding selected nodes. Afterward, we design a distance-aware kernel function, dynamically adjusting edge weights to ensure similar nodes within a hyperedge are connected with larger weights. Extensive theoretical justifications and empirical experiments over seven benchmark hypergraph datasets demonstrate that AdE has excellent rationality, generalization, and effectiveness compared to classic expansion models.

Figures (4)

• Figure 1: Illustration of classic hypergraph expansion methods.
• Figure 2: The overall framework of AdE: given a hypergraph $\mathcal{H}=(\mathcal{V}, \mathcal{E}, \mathcal{X})$ with attribute feature $\mathbf{X}$, (i) AdE first feeds the attribute feature $\mathbf{X}$ into a global pooling layer, and leverages simulation network (Si-Net) to learn the importance of each feature dimension, further obtaining the weight matrix $W_g$. Afterward, AdE scales the attribute feature $\mathbf{X}_\text{a}$ with the learnable weight matrix $W_g$ and identifies a representative node pair $(v_{e^-}, v_{e^+})$ for each hyperedge $e$. (ii) Using representative node pairs, AdE employs a distance-aware kernel function with prior distance information $\mathcal{U}$ to learn the edge weights adaptively and further constructs an adaptive weighted graph $\mathcal{G}_\text{a}$.
• Figure 3: Performances of different model variants over DBLP, Cora, Citeseer, and Pumbed.
• Figure 4: Embeddings visualization for AdE-GIN, HyperGCN, LE-GIN, and HCHA over two citation hypergraphs.

Theorems & Definitions (10)

• Definition 1: Hypergraph
• Definition 2: Graph Neural Networks
• Proposition 1
• Proposition 2
• Proposition 3
• Definition 3: Weisfeiler-Lehman test (1-WL test)
• Definition 4: Generalized Weisfeiler-Lehman test (1-GWL test)
• Lemma 1
• Theorem 1
• Theorem 2