Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-Euclidean Spatial Graph Neural Network

Zheng Zhang, Sirui Li, Jingcheng Zhou, Junxiang Wang, Abhinav Angirekula, Allen Zhang, Liang Zhao

TL;DR

This work tackles learning representations for spatial networks embedded on non-Euclidean manifolds by introducing MSGNN, which discretizes the embedded surface with a triangular mesh and encodes each edge’s geodesic path as a sequence of mesh-unit features processed by an edge-level RNN. The edge embeddings are then integrated with node features via graph neural message passing, yielding representations that are invariant to SE(3) transformations while preserving enough geometric detail to distinguish complex spatial patterns. The authors provide theoretical guarantees of rotation/translation invariance and geometry-preserving properties, along with an approximation-bound analysis for mesh discretization. Empirically, MSGNN achieves state-of-the-art performance across synthetic and real-world datasets (brain networks, airline networks, and 3D shapes), shows robustness to manifold irregularity, and demonstrates the practical value of non-Euclidean geometry in representation learning.

Abstract

Spatial networks are networks whose graph topology is constrained by their embedded spatial space. Understanding the coupled spatial-graph properties is crucial for extracting powerful representations from spatial networks. Therefore, merely combining individual spatial and network representations cannot reveal the underlying interaction mechanism of spatial networks. Besides, existing spatial network representation learning methods can only consider networks embedded in Euclidean space, and can not well exploit the rich geometric information carried by irregular and non-uniform non-Euclidean space. In order to address this issue, in this paper we propose a novel generic framework to learn the representation of spatial networks that are embedded in non-Euclidean manifold space. Specifically, a novel message-passing-based neural network is proposed to combine graph topology and spatial geometry, where spatial geometry is extracted as messages on the edges. We theoretically guarantee that the learned representations are provably invariant to important symmetries such as rotation or translation, and simultaneously maintain sufficient ability in distinguishing different geometric structures. The strength of our proposed method is demonstrated through extensive experiments on both synthetic and real-world datasets.

Non-Euclidean Spatial Graph Neural Network

TL;DR

This work tackles learning representations for spatial networks embedded on non-Euclidean manifolds by introducing MSGNN, which discretizes the embedded surface with a triangular mesh and encodes each edge’s geodesic path as a sequence of mesh-unit features processed by an edge-level RNN. The edge embeddings are then integrated with node features via graph neural message passing, yielding representations that are invariant to SE(3) transformations while preserving enough geometric detail to distinguish complex spatial patterns. The authors provide theoretical guarantees of rotation/translation invariance and geometry-preserving properties, along with an approximation-bound analysis for mesh discretization. Empirically, MSGNN achieves state-of-the-art performance across synthetic and real-world datasets (brain networks, airline networks, and 3D shapes), shows robustness to manifold irregularity, and demonstrates the practical value of non-Euclidean geometry in representation learning.

Abstract

Spatial networks are networks whose graph topology is constrained by their embedded spatial space. Understanding the coupled spatial-graph properties is crucial for extracting powerful representations from spatial networks. Therefore, merely combining individual spatial and network representations cannot reveal the underlying interaction mechanism of spatial networks. Besides, existing spatial network representation learning methods can only consider networks embedded in Euclidean space, and can not well exploit the rich geometric information carried by irregular and non-uniform non-Euclidean space. In order to address this issue, in this paper we propose a novel generic framework to learn the representation of spatial networks that are embedded in non-Euclidean manifold space. Specifically, a novel message-passing-based neural network is proposed to combine graph topology and spatial geometry, where spatial geometry is extracted as messages on the edges. We theoretically guarantee that the learned representations are provably invariant to important symmetries such as rotation or translation, and simultaneously maintain sufficient ability in distinguishing different geometric structures. The strength of our proposed method is demonstrated through extensive experiments on both synthetic and real-world datasets.
Paper Structure (26 sections, 5 theorems, 4 equations, 5 figures, 3 tables)

This paper contains 26 sections, 5 theorems, 4 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Method
  5. Framework.
  6. Manifold-constrained spatial curve representation.
  7. Complexity analysis
  8. Experimental Results
  9. Experimental Settings.
  10. Synthetic datasets.
  11. Real-world datasets
  12. Comparison models
  13. Effectiveness Results
  14. Synthetic datasets results.
  15. Real-world datasets results.
  16. ...and 11 more sections

Key Result

Theorem 4.1

Here the distances $d \in [0,\infty)$, angle $\theta \in [0, \pi)$, torsions $\phi\in [-\pi, \pi)$, and relative orientation angle $\varphi\in [0, \pi)$ are rigorously invariant under all rotation and translation transformations $\mathcal{T}\in \mathrm{SE(3)}$.

Figures (5)

  • Figure 1: Spatial network contains not only network topology information but also their interaction with the embedded spatial surface.
  • Figure 2: Two spatial networks with different connectivity mechanisms on a holomorphic manifold. The left figure reflects that nodes tend to be connected by the shortest distance (called the first law of geography tobler1970computer), while the right figure reflects the spatial pattern in which nodes tend to be connected by circuitous lines. Distinguishing these two spatial networks requires new approaches to jointly consider the spatial curves on the manifold and network topology.
  • Figure 3: Illustration of the overall proposed framework. (a) The discretization process of the continuous manifold and convolutional neural networks for passing and aggregating the geometric information on spatial curves. (b) The RNN module extracts the geometric information along the irregular spatial curves between nodes.
  • Figure 4: Accuracy trend results for our proposed MSGNN model and all competing models against varying degree of manifold irregularity. The performance of models at the lowest degree of irregularity ($\omega=1$) is set as the base value.
  • Figure 5: Robustness test on rotation and translation invariance by augmenting the data on the test set. The x-axis corresponds to the magnitude of the rotation angle (left) and translation distance (right) while y-axis shows the accuracy scores.

Theorems & Definitions (7)

  • Theorem 4.1
  • Theorem 4.2
  • Lemma A .1
  • Definition 1: Plane Angle
  • Definition 2: Circumradius
  • Theorem 1.1: Angle Bound
  • Lemma A .2