Graph Diffusion Node Classification Algorithm Based on Ollivier-Ricci Curvature
Graph neural networks (GNNs) have emerged as powerful tools for analyzing structured data, particularly in tasks such as node classification, link prediction, and graph classification. Among various GNN approaches, graph diffusion methods have gained attention for their ability to capture long-range dependencies in graphs by propagating information across multiple hops. However, traditional graph diffusion techniques often struggle with complex edge relationships, leading to reduced accuracy in node classification tasks. To address this limitation, a novel curvature-based graph diffusion neural network has been proposed, leveraging Ollivier-Ricci curvature to enhance the geometric understanding of graph structures and improve message aggregation.
Introduction
Graphs serve as fundamental data structures for modeling complex relationships in various domains, including social networks, biological systems, and recommendation systems. Node classification, a crucial task in graph analysis, requires effective utilization of both node attributes and topological information. While conventional GNNs like Graph Convolutional Networks (GCNs) and Graph Attention Networks (GATs) rely on local neighborhood aggregation, they often fail to capture distant node relationships and suffer from over-smoothing—where node representations become indistinguishable with increasing network depth.
Graph diffusion methods, such as Personalized PageRank (PPR) and heat kernel diffusion, overcome some of these limitations by incorporating multi-hop neighborhood information. These approaches smooth graph signals and reduce high-frequency noise while preserving structural information. However, existing diffusion-based models like Graph Diffusion Convolution (GDC) still face challenges in distinguishing important edges from noisy ones, particularly in densely connected graphs. This limitation motivates the integration of geometric curvature measures to refine edge weighting and enhance information propagation.
Methodology
The proposed Curvature Graph Diffusion Convolution (CGDC) framework combines graph diffusion with Ollivier-Ricci curvature to optimize message aggregation. The algorithm consists of four key steps: computing the graph diffusion matrix, extracting curvature features, adjusting edge weights based on curvature, and integrating the curvature-enhanced matrix into the diffusion process.
Graph Diffusion Matrix
The diffusion matrix captures node interactions by simulating information propagation across the graph. It is constructed using a weighted combination of transition matrices, where each step represents information passing through increasingly distant neighbors. The symmetric transition matrix ensures stability and convergence, while self-loop weights help retain node-specific features during diffusion. Two primary diffusion mechanisms are employed: PPR, which balances local and global information through a restart probability, and heat kernel diffusion, which models information spread as a heat dissipation process.
Ollivier-Ricci Curvature
Ollivier-Ricci curvature quantifies the geometric properties of edges by measuring the structural similarity between connected nodes. In graph theory, curvature indicates how much a local neighborhood deviates from a “flat” structure. High positive curvature suggests tightly connected node pairs, while negative curvature indicates bridging edges between loosely connected communities. By computing the Wasserstein distance between probability distributions over node neighborhoods, Ollivier-Ricci curvature provides a nuanced understanding of edge importance, enabling more informed adjustments to diffusion weights.
Curvature-Enhanced Diffusion
The curvature matrix, derived from edge-wise Ollivier-Ricci values, is combined with the diffusion matrix through element-wise multiplication. This operation ensures that edges with high curvature—indicating strong structural connections—receive greater emphasis during information propagation. Conversely, edges with low or negative curvature are downweighted to reduce noise and irrelevant connections. The resulting curvature-adjusted diffusion matrix is then used to update node features, enhancing the model’s ability to distinguish between meaningful and spurious relationships.
Experimental Results
The performance of CGDC was evaluated on six benchmark datasets spanning academic citation networks, co-authorship graphs, and e-commerce networks. Comparisons with baseline models, including GAT-PPR, MoNet, and GDE, demonstrated consistent improvements in classification accuracy.
Performance Analysis
CGDC achieved significant accuracy gains over traditional diffusion methods, particularly in datasets with complex edge structures. For instance, on the CiteSeer dataset, CGDC-PPR improved accuracy by 2 percentage points compared to the original GDC model. Similarly, on the Computers dataset, both CGDC variants outperformed GDE by approximately 2.5 percentage points. These improvements highlight the benefits of curvature-aware diffusion in capturing meaningful graph topology.
Visualization using t-SNE further confirmed that CGDC produces more separable node embeddings compared to GCN and GAT, particularly in high-edge-density graphs like Photo. The curvature-based weighting helped cluster nodes more distinctly, reducing overlap between different classes.
Ablation Studies
Ablation experiments comparing Ollivier-Ricci curvature with Forman curvature underscored the superiority of the former in enhancing diffusion performance. The optimal transport foundation of Ollivier-Ricci curvature proved more effective in distinguishing structural relationships, leading to better classification outcomes. Additionally, parameter sensitivity analyses revealed that CGDC maintains robust performance across a range of diffusion coefficients, with heat kernel diffusion showing greater stability than PPR in some scenarios.
Conclusion
The integration of Ollivier-Ricci curvature into graph diffusion networks represents a significant advancement in node classification. By incorporating geometric insights into message passing, CGDC effectively addresses the limitations of conventional diffusion methods in handling complex edge relationships. Experimental results across diverse datasets validate its ability to improve classification accuracy while maintaining computational efficiency.
Future research directions include exploring alternative curvature measures, optimizing curvature computation for large-scale graphs, and extending the framework to dynamic graph settings. The success of CGDC underscores the potential of geometric approaches in advancing graph representation learning, offering a reliable foundation for future innovations in the field.
doi.org/10.19734/j.issn.1001-3695.2024.06.0192
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