Expander Graph Propagation (2210.02997v2)

Abstract: Deploying graph neural networks (GNNs) on whole-graph classification or regression tasks is known to be challenging: it often requires computing node features that are mindful of both local interactions in their neighbourhood and the global context of the graph structure. GNN architectures that navigate this space need to avoid pathological behaviours, such as bottlenecks and oversquashing, while ideally having linear time and space complexity requirements. In this work, we propose an elegant approach based on propagating information over expander graphs. We leverage an efficient method for constructing expander graphs of a given size, and use this insight to propose the EGP model. We show that EGP is able to address all of the above concerns, while requiring minimal effort to set up, and provide evidence of its empirical utility on relevant graph classification datasets and baselines in the Open Graph Benchmark. Importantly, using expander graphs as a template for message passing necessarily gives rise to negative curvature. While this appears to be counterintuitive in light of recent related work on oversquashing, we theoretically demonstrate that negatively curved edges are likely to be required to obtain scalable message passing without bottlenecks. To the best of our knowledge, this is a previously unstudied result in the context of graph representation learning, and we believe our analysis paves the way to a novel class of scalable methods to counter oversquashing in GNNs.

• The paper proposes Expander Graph Propagation (EGP), a novel framework using expander graphs to address bottlenecks and oversquashing in Graph Neural Networks (GNNs) for whole-graph tasks.
• EGP leverages the high connectivity and low diameter of expander graphs to facilitate scalable message-passing, theoretically handling negative curvature properties to enable efficient global communication.
• Empirical validation shows EGP alleviates oversquashing and improves performance on graph classification benchmarks compared to conventional GNN architectures.

Expander Graph Propagation

Introduction

Graph Neural Networks (GNNs) are pivotal in learning representations over graph-structured data, with applications spanning virtual drug screening, traffic prediction, and combinatorial chip design. Despite their versatility, GNNs face significant challenges in tasks requiring whole-graph classification or regression. These tasks demand that node features efficiently capture both local interactions in their neighborhood and the graph's global structure. Common GNN architectures often suffer from pathological behaviors such as bottlenecks and oversquashing. The paper proposes a novel approach to circumvent these issues using expander graphs as a propagation framework, introducing the Expander Graph Propagation (EGP) model.

Expander Graph Propagation (EGP)

Expander graphs are employed in message-passing paradigms to mitigate bottlenecks and oversquashing. Their intrinsic properties include high connectivity and low diameter, allowing nodes to communicate efficiently without encountering bottlenecks, thus facilitating scalable message-passing. The proposed EGP model harnesses expander graphs to ensure linear time and space complexity.

Theoretical Insights

The paper meticulously examines the theoretical foundations of expander graphs, highlighting their favorable characteristics such as sparsity, high Cheeger constants, and logarithmic diameter. Notably, expander graphs inherently possess negatively curved edges, which might intuitively seem counterproductive. However, the authors assert that these edges could be necessary for scalable communication without bottlenecks in GNNs. This hypothesis forms a central, novel contribution to graph representation learning, challenging existing assumptions about curvature and oversquashing.

Empirical Validation

Empirical evidence underscores EGP's efficacy in various graph classification tasks. The model demonstratively alleviates oversquashing, evidenced by its performance on benchmarks such as the Open Graph Benchmark datasets. Compared with conventional GNN architectures, EGP exhibits improved average precision and accuracy, reinforcing its practical utility.

Implications and Future Directions

The research opens avenues for new methodologies in countering oversquashing by utilizing expander graphs. Practically, this holds promise for GNNs deployed in complex scenarios requiring global interaction awareness. The work suggests potential future explorations into varied expander graph applications or aligning expander structures more cohesively with input graph topologies.

Conclusion

The paper proposes EGP, a theoretically grounded, efficient framework for facilitating global communication in GNNs, while sidestepping issues of oversquashing and bottlenecks. By employing expander graphs, this model serves as a beacon for scalable graph representation learning in tasks demanding nuanced global context awareness. Future research can further refine this framework, enhancing its applicability across broader domains in AI and graph theory.