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Workshop: Symmetry and Geometry in Neural Representations

Cayley Graph Propagation

Joseph Wilson · Petar Veličković

Abstract: In spite of the plethora of success stories with graph neural networks (GNNs) on modelling graph-structured data, they are notoriously vulnerable to tasks which necessitate mixing of information between distant pairs of nodes, especially in the presence of bottlenecks in the graph. For this reason, a significant body of research has dedicated itself to discovering or pre-computing graph structures which ameliorate such bottlenecks. Bottleneck-free graphs are well-known in the mathematical community as *expander graphs*, with prior work—Expander Graph Propagation (EGP)—proposing the use of a well-known expander graph family—the Cayley graphs of the $\mathrm{SL}(2,\mathbb{Z}_n)$ special linear group—as a computational template for GNNs. However, despite its solid theoretical grounding, the actual computational graphs used by EGP are *truncated* Cayley graphs, which causes them to lose expansion properties. In this work, we propose to use the full Cayley graph within EGP, recovering significant improvements on datasets from the Open Graph Benchmark (OGB). Our empirical evidence suggests that the retention of the nodes in the expander graph can provide benefit for graph representation learning, which may provide valuable insight for future models.

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