Graph Neural Networks for Edge Signals: Orientation Equivariance and Invariance

TL;DR

This work addresses edge-level learning with signals that may have intrinsic direction or be directionless, and edges that themselves may be directed or undirected. It formalizes joint orientation-equivariance and joint orientation-invariance, and introduces EIGN, a direction-aware edge-level GNN built from Magnetic Edge Laplacians and inter-modality fusion to model both signal modalities and edge directions. EIGN provably satisfies the proposed desiderata and demonstrates superior performance across synthetic and real-world tasks, including electrical circuits, with notable RMSE reductions. The approach offers a general, scalable framework for edge-level inference that can leverage both orientation-sensitive and orientation-insensitive information in graphs containing mixed edge types.

Abstract

Many applications in traffic, civil engineering, or electrical engineering revolve around edge-level signals. Such signals can be categorized as inherently directed, for example, the water flow in a pipe network, and undirected, like the diameter of a pipe. Topological methods model edge signals with inherent direction by representing them relative to a so-called orientation assigned to each edge. These approaches can neither model undirected edge signals nor distinguish if an edge itself is directed or undirected. We address these shortcomings by (i) revising the notion of orientation equivariance to enable edge direction-aware topological models, (ii) proposing orientation invariance as an additional requirement to describe signals without inherent direction, and (iii) developing EIGN, an architecture composed of novel direction-aware edge-level graph shift operators, that provably fulfills the aforementioned desiderata. It is the first general-purpose topological GNN for edge-level signals that can model directed and undirected signals while distinguishing between directed and undirected edges. A comprehensive evaluation shows that EIGN outperforms prior work in edge-level tasks, for example, improving in RMSE on flow simulation tasks by up to 23.5%.

Figures (8)

• Figure 1: EIGN models an arbitrary combination of orientation-equivariant and -invariant edge-level inputs or targets. In this example, the car flow is equivariant and represented relative to two different (top and bottom) arbitrary direction-consistent orientations${\mathfrak{O}}$ and ${\hat{{\mathfrak{O}}}}$ (notice the sign of equivariant signals), while speed limits are invariant. EIGN makes consistent predictions for ${\mathfrak{O}}$ and ${\hat{{\mathfrak{O}}}}$: It outputs the same invariant signals while the sign of equivariant outputs is determined by the orientation.
• Figure 2: Two scenarios (top, bottom) that differ in the direction of one edge but model different situations (flame in bottom left). Their representations are indistinguishable for models that are orientation-equivariant for directed edges.
• Figure 2: Modelling capabilities of all architectures. "-" denotes that the modality is modeled without satisfying orientation invariance/equivariance.
• Figure 3: EIGN architecture: In each layer, message passing using novel Laplacians is performed within and between orientation-equivariant and orientation-invariant signals. The two aggregated modalities ${{\bm{Z}}^{(l)}_{\text{equ}}}$ and then ${{\bm{Z}}^{(l)}_\text{inv}}$ are then fused.
• Figure 4: The Equivariant Magnetic Edge Laplacian ${ {{\bm{L}}^{(\IfNoValueTF{-NoValue-}{q}{-NoValue-})}_\text{equ}} }$ induces a complex phase shift of $\pi q$ for signals of directed edges that are aggregated by the black undirected edge. Signals of misaligned edges are re-oriented.

Theorems & Definitions (37)

• Definition 4.1: Joint Orientation Equivariance
• Definition 4.2: Joint Orientation Invariance
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Theorem 4.1
• Proposition A.1
• proof
• Lemma A.1
• proof