Graph Neural Networks for Edge Signals: Orientation Equivariance and Invariance

2024-10-22

arXiv.org

AI Summary

Existing topological methods struggle to jointly model directed (e.g., water flow) and undirected (e.g., pipe diameter) edge signals, while failing to distinguish the intrinsic directionality of edges themselves. To address this, we propose a direction-aware edge signal modeling paradigm that formally defines and simultaneously satisfies both direction equivariance and direction invariance—establishing the first principled framework for unified edge-level representation learning. Based on this, we introduce EIGN, the first general-purpose edge-level topological graph neural network, equipped with an algebraic-topology-inspired, direction-aware edge-level graph shift operator that rigorously preserves theoretical guarantees. Extensive experiments across traffic, hydrological, and power systems demonstrate consistent superiority over state-of-the-art methods: EIGN achieves up to 23.5% reduction in RMSE for flow simulation tasks.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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%.

Problem

• Modeling directed and undirected edge signals in graphs.
• Distinguishing between directed and undirected edges in topological models.
• Developing a GNN architecture for edge-level tasks with improved performance.

Innovation

• Revised orientation equivariance for direction-aware models
• Introduced orientation invariance for undirected signals
• Developed EIGN with novel graph shift operators