Effective Structural Encodings via Local Curvature Profiles

TL;DR

This paper proposes a novel structural encoding based on discrete Ricci curvature (Local Curvature Profiles, short LCP) and shows that it significantly outperforms existing encoding approaches and compares different encoding types with (curvature-based) rewiring techniques.

Abstract

Structural and Positional Encodings can significantly improve the performance of Graph Neural Networks in downstream tasks. Recent literature has begun to systematically investigate differences in the structural properties that these approaches encode, as well as performance trade-offs between them. However, the question of which structural properties yield the most effective encoding remains open. In this paper, we investigate this question from a geometric perspective. We propose a novel structural encoding based on discrete Ricci curvature (Local Curvature Profiles, short LCP) and show that it significantly outperforms existing encoding approaches. We further show that combining local structural encodings, such as LCP, with global positional encodings improves downstream performance, suggesting that they capture complementary geometric information. Finally, we compare different encoding types with (curvature-based) rewiring techniques. Rewiring has recently received a surge of interest due to its ability to improve the performance of Graph Neural Networks by mitigating over-smoothing and over-squashing effects. Our results suggest that utilizing curvature information for structural encodings delivers significantly larger performance increases than rewiring.

Figures (4)

• Figure 1: Computing ORC.
• Figure 2: Illustration of optimal transport plans for computing the ORC of $(u,v)$ in the 4x4 Rooke (left) and Shrikhande (right) graphs. Edges along which a mass of $\frac{1}{deg(u)}=\frac{1}{deg(v)}=\frac{1}{6}$ is moved are marked in red. We see that $\kappa(u,v)=\frac{1}{3}$ in the Rooke and $\kappa(u,v)=0$ in the Shrikhande graph.
• Figure 3: Graph classification accuracy with increasing number of GCN layers. Dashed lines show accuracies using BORF, normal lines accuracy using the LCP.
• Figure 4: Example networks from the mutag, enzymes, imdb, and proteins datasets, which we use for graph classification (top row). The middle row shows the same example networks with their edges colored according to their ORC values. We also depict the adjusted graphs after rewiring using BORF (bottom row).

Theorems & Definitions (3)

• Theorem 3.1
• proof
• Theorem A.1