Understanding the Representation Power of Graph Neural Networks in Learning Graph Topology

TL;DR

This paper analyzes the limitations of standard GCNs in capturing graph topology, proposing modular designs with varied propagation rules and residual connections to enhance their ability to distinguish complex graph structures.

Contribution

It introduces a modular GCN architecture with diverse propagation rules and residual connections, significantly improving graph representation power and ability to differentiate graph models.

Findings

Deeper GCNs better learn higher order graph moments.

Modular GCNs with different propagation rules outperform standard models.

Depth has a greater impact than width on GCN expressiveness.

Abstract

To deepen our understanding of graph neural networks, we investigate the representation power of Graph Convolutional Networks (GCN) through the looking glass of graph moments, a key property of graph topology encoding path of various lengths. We find that GCNs are rather restrictive in learning graph moments. Without careful design, GCNs can fail miserably even with multiple layers and nonlinear activation functions. We analyze theoretically the expressiveness of GCNs, concluding a modular GCN design, using different propagation rules with residual connections could significantly improve the performance of GCN. We demonstrate that such modular designs are capable of distinguishing graphs from different graph generation models for surprisingly small graphs, a notoriously difficult problem in network science. Our investigation suggests that, depth is much more influential than width, with deeper GCNs being more capable of learning higher order graph moments. Additionally, combining GCN modules with different propagation rules is critical to the representation power of GCNs.