Theoretical Expressive Power and Design Space of Higher-Order Graph Transformers

Core Concepts

This paper provides a systematic study of the theoretical expressive power of order-k graph transformers and their sparse variants. It shows that a plain order-k graph transformer without additional structural information is less expressive than the k-Weisfeiler Lehman (k-WL) test, but adding explicit tuple indices can make it as expressive as k-WL. The paper then explores strategies to sparsify and enhance the higher-order graph transformers, aiming to improve both their efficiency and expressiveness.

Abstract

The paper starts by introducing a natural formulation of order-k transformers Ak. It shows that without "indices" information of k-tuples, Ak is strictly less expressive than k-WL. However, when augmented with the indices information, its expressive power is at least that of k-WL.
The paper then explores strategies to improve the efficiency of higher-order graph transformers while maintaining strong expressive power. It proposes several sparse high-order transformers, including:

Neighbor attention ANgbh
k: This mechanism computes attention only with the k-neighbors of each tuple, and is as expressive as k-WL while having lower complexity (O(nk+1kd) vs O(n2kd) for plain Ak).

Local neighbor attention ALN
k: This is a more sparse variant of neighbor attention, which only considers local neighbors. It is at least as powerful as δ-k-LWL, a stronger variant of k-WL.

Virtual tuple attention AVT
k: This model introduces a virtual tuple that computes attention with all other real tuples, while each real tuple only computes attention with the virtual tuple. It can approximate the plain Ak efficiently.

The paper also discusses simplicial attention as a way to further reduce the number of input k-tuples.
Experiments on synthetic and real-world datasets verify the theoretical properties and empirical effectiveness of the proposed sparse higher-order transformers.

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Key Insights Distilled From

On the Theoretical Expressive Power and the Design Space of Higher-Order Graph Transformers

by Cai Zhou,Ros... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.03380.pdf

Deeper Inquiries

What are some other potential strategies to enhance the expressiveness of higher-order graph transformers beyond the k-WL hierarchy

One potential strategy to enhance the expressiveness of higher-order graph transformers beyond the k-WL hierarchy is to incorporate domain-specific knowledge or constraints into the model architecture. This can involve designing specialized attention mechanisms that capture specific graph structures or relationships relevant to the task at hand. For example, introducing attention mechanisms that focus on capturing long-range dependencies or hierarchical relationships within the graph can enhance the model's ability to represent complex patterns. Additionally, integrating external knowledge graphs or ontologies into the model can provide valuable context and improve the model's understanding of the data.

How can we theoretically analyze the trade-offs between the expressive power and computational efficiency of different higher-order graph transformer variants

The trade-offs between the expressive power and computational efficiency of different higher-order graph transformer variants can be theoretically analyzed by considering the complexity of the model in terms of both time and space. Expressiveness can be evaluated based on the model's ability to capture complex graph structures and patterns, while computational efficiency can be assessed by analyzing the model's scalability and resource requirements. By comparing the theoretical properties of different variants, such as their time complexity, space complexity, and expressive power, researchers can determine the optimal balance between model complexity and computational efficiency for a given task.

Are there any real-world applications where the proposed higher-order graph transformers can significantly outperform standard graph neural networks and transformers

Higher-order graph transformers have the potential to significantly outperform standard graph neural networks and transformers in real-world applications that require capturing long-range dependencies, hierarchical relationships, or complex graph structures. For tasks such as molecular property prediction, social network analysis, or bioinformatics, where understanding global interactions and structural patterns in graphs is crucial, higher-order graph transformers can offer superior performance. By leveraging their enhanced expressive power and ability to capture intricate relationships within graphs, these models can provide more accurate predictions and insights compared to traditional graph neural networks.