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

Jicheng Ma; Yunyan Yang; Juan Zhao; Liang Zhao

cs.LGMar 27, 2026

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

Authors: Jicheng Ma, Yunyan Yang, Juan Zhao, Liang Zhao

Organizations: School of Mathematics, Renmin University of China, Beijing, China. · School of Mathematical Sciences, Key Laboratory of Mathematics and Complex Systems of MOE, Beijing Normal University, Beijing, China.

Abstract

We introduce the Geometric Evolution Graph Convolutional Network (GEGCN), a novel framework that enhances graph representation learning through explicit modeling of geometric evolution on graph structures. Specifically, GEGCN leverages a Long Short-Term Memory (LSTM) network to capture the dynamic structural sequence generated by discrete Ricci flow, and infuses the learned dynamic representations into a graph convolutional network. Extensive experiments demonstrate that GEGCN achieves excellent performance on classification tasks across various benchmark datasets, including homophilic/heterophilic graphs, filtered graphs, and large-scale graphs.

Figures & tables

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.

	Cora	Citeseer	Pubmed	Coauthor CS	Amazon Photos
MLP	55.1 $\pm$ 1.4	59.1 $\pm$ 1.2	71.4 $\pm$ 0.8	88.3 $\pm$ 0.7	69.6 $\pm$ 3.8
MoNet	81.7 $\pm$ 0.4	71.2 $\pm$ 0.7	78.6 $\pm 0.5$	90.8 $\pm$ 0.6	91.2 $\pm$ 1.3
GCN	81.5 $\pm$ 0.5	70.9 $\pm$ 0.5	79.0 $\pm$ 0.3	91.1 $\pm$ 0.5	91.2 $\pm$ 1.2
GraphSAGE	79.2 $\pm$ 7.7	71.6 $\pm$ 1.9	77.4 $\pm$ 2.2	91.3 $\pm$ 2.8	91.4 $\pm$ 1.4
GAT	83.0 $\pm$ 0.7	72.5 $\pm$ 0.7	79.0 $\pm$ 0.3	90.5 $\pm$ 0.6	85.1 $\pm$ 2.3
JKNet	83.7 $\pm$ 0.7	72.5 $\pm$ 0.4	82.6 $\pm$ 0.5	91.1 $\pm$ 0.3	86.1 $\pm$ 1.1

Table 1: Mean test set classification accuracy (in %) and standard deviation across five homophilic benchmarks.

	Cornell	Texas	Wisconsin	Chameleon	Actor
GCN	52.69 $\pm$ 0.21	61.19 $\pm$ 0.49	54.60 $\pm$ 0.86	41.33 $\pm$ 0.18	23.84 $\pm$ 0.43
Undirected	53.20 $\pm$ 0.53	63.38 $\pm$ 0.87	51.37 $\pm$ 1.15	42.02 $\pm$ 0.30	21.45 $\pm$ 0.47
+FA	58.29 $\pm$ 0.49	64.82 $\pm$ 0.29	55.48 $\pm$ 0.62	42.67 $\pm$ 0.17	24.14 $\pm$ 0.43
DIGL	58.26 $\pm$ 0.50	62.03 $\pm$ 0.43	49.53 $\pm$ 0.27	42.02 $\pm$ 0.13	24.77 $\pm$ 0.32
DIGL + Undirected	59.54 $\pm$ 0.64	63.54 $\pm$ 0.38	52.23 $\pm$ 0.54	42.68 $\pm$ 0.12	25.45 $\pm$ 0.30
SDRF	54.60 $\pm$ 0.39	64.46 $\pm$ 0.38	55.51 $\pm$ 0.27	42.73 $\pm$ 0.15	28.42 $\pm$ 0.75

Table 2: Mean test set classification accuracy (%) and standard deviation across heterophilic benchmarks.

	Roman-empire	Amazon-ratings	Minesweeper	Questions	Squirrel-f	Chameleon-f
ResNet	65.88 $\pm$ 0.38	45.90 $\pm$ 0.52	50.89 $\pm$ 1.39	70.34 $\pm$ 0.76	36.55 $\pm$ 1.82	36.73 $\pm$ 4.71
GAT	80.87 $\pm$ 0.30	49.09 $\pm$ 0.63	92.01 $\pm$ 0.68	77.43 $\pm$ 1.20	35.62 $\pm$ 2.06	39.21 $\pm$ 3.08
H2GCN	60.11 $\pm$ 0.52	36.47 $\pm$ 0.23	89.71 $\pm$ 0.31	63.59 $\pm$ 1.46	35.10 $\pm$ 1.15	26.75 $\pm$ 3.64
GPR-GNN	64.85 $\pm$ 0.27	44.88 $\pm$ 0.34	86.24 $\pm$ 0.61	55.48 $\pm$ 0.91	35.11 $\pm$ 1.24	25.90 $\pm$ 3.58
GloGNN	59.63 $\pm$ 0.69	36.89 $\pm$ 0.14	51.08 $\pm$ 1.23	65.74 $\pm$ 1.19	35.51 $\pm$ 1.65	39.61 $\pm$ 2.60
FSGNN	79.92 $\pm$ 0.56	52.74 $\pm$ 0.83	90.08 $\pm$ 0.70	78.86 $\pm$ 0.92	35.92 $\pm$ 1.32	40.61 $\pm$ 2.97

Table 3: Mean test set classification accuracy (%) and standard deviation across filtered benchmarks.

	Test Accuracy
MLP	57.65 $\pm$ 0.12
Node2vec	71.29 $\pm$ 0.13
GraphSAGE	72.77 $\pm$ 0.16
GCN	73.00 $\pm$ 0.17
GAT	73.30 $\pm$ 0.16
UniMP	74.50 $\pm$ 0.05

Table 4: Mean test set classification accuracy (%) and standard deviation on OGBN-arXiv.

Figure 2: Ablation Study. The bar chart compares the mean test accuracy (%) of the baseline GCN, five ablation variants that replace the LSTM module for modeling Ricci curvature evolution with First, Last, Mean Pooling, Max Pooling, or MLP-based aggregation, and the full GEGCN model. The consistent performance gains of GEGCN across all datasets demonstrate the effectiveness of LSTM-based modeling of 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.

	Train (s)	Inf. (s)
Chameleon-f	34.62	0.078
Minesweeper	137.90	0.574
Squirrel-f	153.96	0.597
Roman-empire	158.84	0.353
Amazon-ratings	244.22	1.357
Questions	457.05	1.759

Table 5: The average training and inference time of GEGCN across seven datasets.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Notation	Description
$\mathcal{G}=(\mathcal{V},\mathcal{E})$	Graph with node set $\mathcal{V}$ and edge set $\mathcal{E}$
$w_{e}$	Weight of edge $e\in\mathcal{E}$
$\kappa_{e}$	Ricci curvature of edge $e\in\mathcal{E}$
$\rho_{e}$	Distance between nodes $u$ and $v$ of an edge $e=(u,v)$
$\mathcal{N}(v)$	Neighborhood of node $v$
$m_{v}$	Probability measure on neighbors of $v$

Appendix

Table 10

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.

Hyperparameter	Search Range / Values	Distribution
Optimization Parameters
Learning rate ( $\eta$ )	$[10^{-5},10^{-2}]$	Log-uniform
Weight decay ( $\lambda$ )	$[10^{-6},10^{-2}]$	Log-uniform
Dropout rate ( $p$ )	$[0.01,0.99]$	Uniform
Hidden dimension ( $d$ )	$\{32,64,128\}$	Categorical
Curvature & Flow Parameters

Appendix

Table 6: Hyperparameter Search Space and Optimization Details.

Dataset	Nodes	Edges	Features	Classes	Homophily ( $\mathcal{H}(G)$ )
Cora	2,708	5,278	1,433	7	0.81
Citeseer	3,327	4,552	3,703	6	0.74
Pubmed	19,717	44,324	500	3	0.80
Photo	7,650	119,081	745	8	0.83
Coauthor CS	18,333	81,894	6,805	15	0.81
Cornell	183	277	1,703	5	0.12

Appendix

Table 7: Statistics of Datasets.

Dataset	Nodes	Edges	Time (s)
Cora	2,708	5,278	34.90
Citeseer	3,327	4,552	2.59
Pubmed	19,717	44,324	851.70
Photo	7,650	119,081	1831.04
Coauthor CS	18,333	81,894	1079.04
Cornell	183	277	4.84

Appendix

Table 8: Efficiency of offline pre-computation. We report the estimated execution time (in seconds) for computing

T=10

steps of Discrete Ricci Flow.

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Message passing is a core mechanism in Graph Neural Networks (GNNs), enabling the iterative update of node embeddings by aggregating information from neighboring nodes. Graph Convolutional Networks (GCNs) exemplify this approach by adapting convolutional operations for graph structures, allowing features from adjacent nodes to be combined effectively. However, GCNs encounter challenges with complex or dynamic data. Capturing long-range dependencies often requires deeper layers, which not only increase computational costs but also lead to over-smoothing, where node embeddings become indistinguishable. To overcome these challenges, reservoir computing has been integrated into GNNs, leveraging iterative message-passing dynamics for stable information propagation without extensive parameter tuning. Despite its promise, existing reservoir-based models lack structured convolutional mechanisms, limiting their ability to accurately aggregate multi-hop neighborhood information. To address these limitations, we propose RGC-Net (Reservoir-based Graph Convolutional Network), which integrates reservoir dynamics with structured graph convolution. Key contributions include: (i) a reimagined convolutional framework with fixed-random reservoir weights and a leaky integrator to enhance feature retention; (ii) a robust, adaptable model for graph classification; and (iii) an RGC-Net-powered transformer for graph generation with application to dynamic brain connectivity. Extensive experiments show RGC-Net achieves state-of-the-art performance in classification and generative tasks, including brain graph evolution, with faster convergence and mitigated over-smoothing. Our source code is available at https://github.com/basiralab/RGC-Net.

Abstract

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