Geometric Evolution Graph Convolutional Networks: Enhancing Graph Representation Learning via Ricci Flow

Abstract

We introduce the Geometric Evolution Graph Convolutional Network (GEGCN), a novel framework that enhances graph representation learning by modeling geometric evolution on graphs. Specifically, GEGCN employs a Long Short-Term Memory to model the structural sequence generated by discrete Ricci flow, and the learned dynamic representations are infused into a Graph Convolutional Network. Extensive experiments demonstrate that GEGCN achieves state-of-the-art performance on classification tasks across various benchmark datasets, with its performance being particularly outstanding on heterophilic graphs.

Figures (5)

• Figure 1: Overview of the GEGCN framework. It comprises three phases: (i) Geometric Evolution Generator via Discrete Ricci Flow; (ii) Structural Dynamics Encoder for capturing temporal dynamics and constructing the edge importance matrix via LSTM; and (iii) Feature Fusion for integrating geometric insights into curvature-aware GCN layers.
• Figure 2: Ablation Study. The bar chart compares the mean test accuracy (%) of the baseline GCN, three non-LSTM variants (utilizing First, Last, and Mean Ricci curvatures), and the full GEGCN model. The results demonstrate that GEGCN consistently outperforms the baselines across all datasets, validating the effectiveness of modeling the Ricci curvature evolution sequence.
• Figure 3: Oversmoothing analysis on Cora. We report the mean test accuracy across various numbers of layers $L \in \{2, 4, 8, 16, 32, 64\}$. All models eventually collapse to the chance level ($30.82\%$), but GEGCN maintains significantly higher accuracy at $L=4$ compared to the vanilla GCN.
• Figure 4: Sensitivity analysis of evolution steps (T). The node classification accuracy across three heterophilic datasets (Actor, Wisconsin, and Cornell) generally improves as the Ricci flow evolution length $T$ increases, demonstrating the advantage of capturing long-term geometric refinement. The performance tends to stabilize after $T=6$, indicating a saturation of geometric information extraction.
• Figure 5: Comparison of Ollivier-Ricci curvature in continuous manifolds and discrete graphs. (Top row) Continuous manifolds: (Left) Hyperbolic paraboloid with negative curvature ($\kappa < 0$); (Middle) Plane with zero curvature ($\kappa = 0$); (Right) Elliptic paraboloid with positive curvature ($\kappa > 0$). (Bottom row) Discrete graphs: (Left) Two dense clusters connected by a single edge with negative curvature; (Middle) Regular 3×3 grid with zero curvature; (Right) Complete graph $K_7$ with positive curvature. All edge weights are set to 1.