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Motif-aware Riemannian Graph Neural Network with Generative-Contrastive Learning

Li Sun, Zhenhao Huang, Zixi Wang, Feiyang Wang, Hao Peng, Philip Yu

TL;DR

This paper tackles limitations of existing Riemannian graph learning approaches, notably the use of a single curvature, numerical instability from Exp/Log maps, and limited motif modeling. It proposes MotifRGC, a motif-aware Riemannian framework that combines a Diverse-curvature GCN (D-GCN) with a stable gyrovector-kernel mapping and a self-supervised Generative-Contrastive objective. Key contributions include constructing a diverse-curvature product manifold, replacing unstable maps with a kernel-based transform, and introducing a motif generator/discriminator along with motif-aware contrast for robust, label-free representation learning. Experimental results across multiple datasets demonstrate improved link prediction and node classification performance, enhanced numerical stability, and the model’s capacity to generate and exploit graph motifs such as triangles in a principled, curvature-aware manner.

Abstract

Graphs are typical non-Euclidean data of complex structures. In recent years, Riemannian graph representation learning has emerged as an exciting alternative to Euclidean ones. However, Riemannian methods are still in an early stage: most of them present a single curvature (radius) regardless of structural complexity, suffer from numerical instability due to the exponential/logarithmic map, and lack the ability to capture motif regularity. In light of the issues above, we propose the problem of \emph{Motif-aware Riemannian Graph Representation Learning}, seeking a numerically stable encoder to capture motif regularity in a diverse-curvature manifold without labels. To this end, we present a novel Motif-aware Riemannian model with Generative-Contrastive learning (MotifRGC), which conducts a minmax game in Riemannian manifold in a self-supervised manner. First, we propose a new type of Riemannian GCN (D-GCN), in which we construct a diverse-curvature manifold by a product layer with the diversified factor, and replace the exponential/logarithmic map by a stable kernel layer. Second, we introduce a motif-aware Riemannian generative-contrastive learning to capture motif regularity in the constructed manifold and learn motif-aware node representation without external labels. Empirical results show the superiority of MofitRGC.

Motif-aware Riemannian Graph Neural Network with Generative-Contrastive Learning

TL;DR

This paper tackles limitations of existing Riemannian graph learning approaches, notably the use of a single curvature, numerical instability from Exp/Log maps, and limited motif modeling. It proposes MotifRGC, a motif-aware Riemannian framework that combines a Diverse-curvature GCN (D-GCN) with a stable gyrovector-kernel mapping and a self-supervised Generative-Contrastive objective. Key contributions include constructing a diverse-curvature product manifold, replacing unstable maps with a kernel-based transform, and introducing a motif generator/discriminator along with motif-aware contrast for robust, label-free representation learning. Experimental results across multiple datasets demonstrate improved link prediction and node classification performance, enhanced numerical stability, and the model’s capacity to generate and exploit graph motifs such as triangles in a principled, curvature-aware manner.

Abstract

Graphs are typical non-Euclidean data of complex structures. In recent years, Riemannian graph representation learning has emerged as an exciting alternative to Euclidean ones. However, Riemannian methods are still in an early stage: most of them present a single curvature (radius) regardless of structural complexity, suffer from numerical instability due to the exponential/logarithmic map, and lack the ability to capture motif regularity. In light of the issues above, we propose the problem of \emph{Motif-aware Riemannian Graph Representation Learning}, seeking a numerically stable encoder to capture motif regularity in a diverse-curvature manifold without labels. To this end, we present a novel Motif-aware Riemannian model with Generative-Contrastive learning (MotifRGC), which conducts a minmax game in Riemannian manifold in a self-supervised manner. First, we propose a new type of Riemannian GCN (D-GCN), in which we construct a diverse-curvature manifold by a product layer with the diversified factor, and replace the exponential/logarithmic map by a stable kernel layer. Second, we introduce a motif-aware Riemannian generative-contrastive learning to capture motif regularity in the constructed manifold and learn motif-aware node representation without external labels. Empirical results show the superiority of MofitRGC.
Paper Structure (25 sections, 16 equations, 2 figures, 5 tables, 1 algorithm)

This paper contains 25 sections, 16 equations, 2 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Riemannian Geometry
  4. Euclidean & Riemannian GCNs
  5. Problem Formulation
  6. Notations.
  7. Our Approach
  8. A New Formulation of D-GCN
  9. Product Layer
  10. Kernel Layer
  11. Network Architecture
  12. Motif-aware Generative-Contrastive Learning on Riemannian Manifold
  13. Riemannian Motif Generator
  14. Motif Discriminator
  15. Motif-aware Riemannian Contrast
  16. ...and 10 more sections

Figures (2)

  • Figure 1: Overall architecture of MotifRGC. The proposed D-GCN consists of product layer, kernel layer and convolution layer. We learn the curvatures of the product manifold in the motif-aware Riemannian generative learning (blue box), and learn the node representations in the motif-aware Riemannian contrastive learning (olive box).
  • Figure 2: AUC (%) of Triangle Generation/Prediction.

Theorems & Definitions (2)

  • proof
  • proof