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Hypergraph Convolution and Hypergraph Attention
Song Bai, Feihu Zhang, Philip H. S. Torr
TL;DR
Existing graph neural networks are not applicable to high-order structures, motivating methods for irregular data. The paper proposes hypergraph convolution and hypergraph attention as trainable operators, with attention mathematically serving as a special case of convolution.
Problem
Existing graph neural networks are not applicable to high-order structures, while irregular data structures motivate further research.
Method
The paper proposes hypergraph convolution and hypergraph attention as end-to-end trainable operators, with attention acting as an optional module on the incidence matrix.
Results
Hypergraph attention is mathematically a special case of hypergraph convolution.
Takeaways & Limitations
The two operators extend graph neural networks to high-order graph-structured data.
Takeaways & Limitations
The experiments equally weight hyperedges; a proper weighting mechanism is promising when extra knowledge is available.
Abstract
from arXiv · showhide
Recently, graph neural networks have attracted great attention and achieved prominent performance in various research fields. Most of those algorithms have assumed pairwise relationships of objects of interest. However, in many real applications, the relationships between objects are in higher-order, beyond a pairwise formulation. To efficiently learn deep embeddings on the high-order graph-structured data, we introduce two end-to-end trainable operators to the family of graph neural networks, i.e., hypergraph convolution and hypergraph attention. Whilst hypergraph convolution defines the basic formulation of performing convolution on a hypergraph, hypergraph attention further enhances the capacity of representation learning by leveraging an attention module. With the two operators, a graph neural network is readily extended to a more flexible model and applied to diverse applications where non-pairwise relationships are observed. Extensive experimental results with semi-supervised node classification demonstrate the effectiveness of hypergraph convolution and hypergraph attention.
1. Introduction
The introduction motivates extending graph neural networks to hypergraphs because many applications involve higher-order relationships, then presents hypergraph convolution and attention as trainable operators for such data.
- Most existing graph-based methods assume pairwise relationships, with each graph edge connecting only two vertices.
- Many real applications involve relationships more complex than pairwise, including triadic, tetradic, and higher-order affinities.Recommendation systems provide an example in which one rating edge may connect multiple items through multiple users.
- Hypergraph convolution defines a basic convolutional operator that propagates information efficiently between vertices while exploiting high-order relationships and local clustering.
- Hypergraph attention adds an attention mechanism that learns dynamic hyperedge connections and gathers information from task-relevant graph parts to generate more discriminative node embeddings.
- Both operators are end-to-end trainable and can be inserted into most graph neural network variants when non-pairwise relationships occur.
- Extensive experiments on benchmark datasets demonstrate the efficacy of the proposed methods for semi-supervised node classification.
2. Related Work
Related work covers graph neural networks and graph convolution methods for pairwise graph data, while positioning this work as differentiable hypergraph operators for relationships beyond pairwise.
- Graph neural networks learn deep models with graph data and have been applied across graph-based research areas.
- Existing graph convolution approaches define operators through spectral filters, localized expansions, spatial aggregation, diffusion, or message passing.
- Graph convolutional network methods simplify filtering by using first-order neighbors and demonstrate efficiency and effectiveness on semi-supervised classification tasks.
- Most existing graph neural network variants assume pairwise relationships, whereas this work operates on high-order hypergraphs with beyond-pairwise relationships.
- Hypergraph learning methods differ in hypergraph structure and Laplacian definitions, including clique expansion, star expansion, and Chebyshev-based approaches.
- The paper directly defines two differentiable operators—hypergraph convolution and hypergraph attention—to learn more discriminative deep embeddings.
3. Proposed Approach
The proposed approach extends graph neural networks to hypergraphs, where hyperedges model relationships involving more than two vertices. Hypergraph convolution propagates vertex embeddings using weighted incidence structure, while hypergraph attention learns a dynamic incidence matrix to refine connectivity.
- Hypergraphs represent higher-order relationships because a hyperedge can connect more than two vertices, unlike simple graph edges that link only two vertices.
- Hypergraph Convolution: Hypergraph convolution propagates vertex embeddings through shared hyperedges, with larger hyperedge weights receiving greater propagation confidence.The operator is differentiable with respect to the vertex features and layer weights, enabling gradient-based model training.
- Hypergraph Convolution: Symmetric normalization constrains the maximum eigenvalue of D^-1/2HWB^-1HTD^-1/2 to at most 1, addressing scale changes and instability across stacked layers.Without constrained spectral radius, repeated layers can cause numerical instabilities and increase exploding or vanishing gradient risk.
- Hypergraph Attention: Hypergraph attention is feasible when vertices and hyperedges belong to, or can be projected into, the same homogeneous domain; heterogeneous domains make direct attention over H problematic.The applicability therefore depends on how the hypergraph is constructed.
- Hypergraph Attention: Hypergraph attention adds a learnable module to the incidence matrix, assigning non-binary connectivity values and propagating gradients to H as well as X^(l) and P.Its goal is to learn a dynamic incidence matrix and transition matrix that better reveal relationships between vertices.
- Relationship with Graph Convolution: Graph convolution is mathematically a special case of hypergraph convolution when each hypergraph edge links two vertices, with equal hyperedge weights.This establishes a bridge for developing higher-order counterparts of other graph neural network variants for non-pairwise relationships.
4. Experiments
The experiments evaluate hypergraph convolution and attention on citation and text datasets, showing that modeling higher-order relationships generally improves classification over pairwise baselines. Results also indicate that structural fit, skip connections, and representation length affect performance.
- Datasets: Experiments use Cora, Citeseer, and Pubmed citation networks, plus a modified 20-newsgroup text-categorization dataset.The citation datasets support comparison with prior methods, while the text dataset contains word-posting relationships suitable for hypergraphs.
- Experimental setup: Each dataset trains on 20 nodes per category, validates on 500 nodes, and evaluates performance on 1000 nodes.
- Baseline comparisons: Hypergraph attention improves over hypergraph convolution by 0.42 on Cora and 0.53 on Citeseer.
- Representation analysis: Performance increases with hidden-representation length and peaks at length 16, while hypergraph convolution consistently beats GCN* across tested lengths.The comparison attributes the gain to the graph structure used to establish relationships between objects.
5. Conclusion
The work contributes hypergraph convolution and hypergraph attention as end-to-end trainable operators for learning on high-order relationships. The methods generalize representative graph neural networks, support diverse non-pairwise applications, and show efficacy in semi-supervised node classification, while several limitations remain.
- The paper contributes two end-to-end trainable operators, hypergraph convolution and hypergraph attention, to the family of graph neural networks.
- Hypergraph convolution and hypergraph attention handle non-pairwise relationships modeled in high-order hypergraphs.
- Graph convolution networks and graph attention networks are theoretically shown to be special cases of the proposed methods.
- The proposed operators are more flexible for arbitrary relationship orders and diverse applications involving non-pairwise relationships.
- Experimental results with semi-supervised node classification demonstrate the efficacy of the proposed methods.
- Applying the method requires abstracting non-pairwise relationships from data to define hyperedges, and the implementation handles directed and undirected hypergraphs similarly.
- The paper identifies inherited challenges including training substantially deeper models, handling dynamic structures, and batch-wise model training.
- The current hypergraph attention implementation cannot operate when vertex and hyperedge sets come from two heterogeneous domains.