Beyond Geometric Blindness: Leveraging Ollivier-Ricci Curvature for Effective Knowledge Graph Completion

Introduction

In the submitted work, the authors propose a novel relational graph neural network that integrates geometric (curvature) and semantic information within relational graph attention layers, specifically during the message aggregation phase. In particular, they leverage Ollivier–Ricci curvature to downweight redundant (high-curvature) edges and to amplify bridge (low-curvature) edges, thereby contributing to the research field that integrates curvature information into knowledge graph embeddings, with a focus on the link prediction task.

Methodology: Edge Aggregation via Parallel Resistance Analogy

The authors propose a methodology to compute Ollivier–Ricci curvature on multigraphs (knowledge graphs) by transforming the multigraph into a weighted directed graph, following a formulation that closely resembles that of [2], although it is neither cited nor acknowledged and is instead presented as novel. Even if the formulation is elegant, given the unit resistance assumption and the fact that the resulting expression essentially collapses to 1/(number of non-loop edges), the formalization could be presented in a clearer and more explicit manner.

Methodology: Attention-Based Aggregation

The proposed message passing neural network consists of two layers, both of which employ attention mechanisms during the aggregation phase to weight different members of a node’s neighborhood.

In the first layer, neighbors are aggregated using attention weights computed as a softmax over the Ollivier–Ricci curvature values obtained in the previous phase. This mechanism effectively downweights redundant edges and favors the aggregation of bridge edges.

In the second layer, neighbors are aggregated using attention weights computed as the softmax over dissimilarity scores between the node embedding and the neighbor embeddings produced by the first layer. However, this module is not clearly motivated, and additional explanation or justification should be provided.

The authors use the term “evolutionary” to describe the progression from geometric attention in shallow layers to semantic attention in deeper layers. However, this terminology is potentially misleading, as it is strongly associated with a different research field. A change in terminology or a clearer reformulation is therefore recommended.

Experiments

The authors provide a standard evaluation for knowledge graph completion tasks based on ranking metrics (MRR, Hits@K), comparing the proposed model against several sota methods, including translational, semantic, MLP-based, GNN-based, and attention-based approaches. However, the experimental comparison lacks methods from the literature that explicitly leverage curvature information (e.g., [1]), which closely resemble the experimental setup and choice of scoring function and would therefore represent a particularly relevant baseline.

The effectiveness of the proposed approach is further supported by an extensive ablation study, which demonstrates the individual contributions of both the geometric and semantic aggregation components. Additionally, structural analyses provide insights into the model’s performance on specific datasets, thereby strengthening the experimental results and clarifying the scenarios in which the model is likely to perform better. Further analysis examining the impact of the proposed Ollivier–Ricci curvature–based methodology, specifically, selecting the top k% of edges, removing them, and then applying state-of-the-art models, reinforces the effectiveness of the semantic-aware module.

However, the experimental section lacks an analysis of training and evaluation times. Such results would be important to assess the computational complexity and overhead introduced by curvature computation and semantic similarity calculations, and to enable a more complete comparison with existing models.

Long-term stable URL for resources

The provided repository does not include a README file or any usage documentation. The accompanying Python code contains minimal docstrings and lacks clear usage explanations. In addition, the data files are not well organized, making it difficult to directly identify the purpose of each file. As a result, reproducing the experiments and models based on the proposed methodology is challenging. The absence of documentation significantly hinders usability, and the inclusion of a README, basic usage instructions, tutorials, or example scripts is strongly recommended. Nevertheless, the provided datasets themselves appear to be complete.

Conclusion

In conclusion, the proposed methodology could represent a valuable addition to the field, provided that the text is revised for clarity, additional information on training and evaluation times is included, and the accompanying resources are supported by more comprehensive and detailed documentation.

Questions

1. The semantic-aware aggregation module is not sufficiently motivated. Could the authors provide additional explanation regarding its role and intended effect? In particular, is this component designed to mitigate oversmoothing effects in GNNs, as discussed in the introduction?

2. In the implementation of the proposed semantic-aware aggregation module, have any optimization strategies been introduced to reduce the number of required computations? Specifically, how is the computation of similarity between non-connected nodes avoided, and how does the approach scale to semantic web datasets with a large number of entities?

3. With reference to Section 4.4, and in particular Table 3, does the removal of the top-k edges have an impact on model training time? Furthermore, could this methodology be leveraged to improve scalability when applied to large-scale knowledge graphs?

References

[1] Guo, D., Su, M., Cao, C., Yuan, F., Zhang, X., Liu, Y., and Fu, J. (2023). Curvature-driven knowledge graph embedding for link prediction. In Proceedings of the 26th International Conference on Computer Supported Cooperative Work in Design (CSCWD), pages 1226–1231. IEEE.

[2] Klein, D. J., and Randić, M. (1993). Resistance distance. Journal of Mathematical Chemistry, 12(1), 81–95.

First, the novelty relative to existing curvature-based GNN work is not clearly articulated. Prior methods such as CGNN, CurvGN, CurvDrop, and BORF already integrate Ricci curvature into graph learning. While the paper claims a task-specific reinterpretation for KGC (down-weighting high-curvature edges instead of strengthening them), the conceptual difference from prior curvature-guided weighting or rewiring approaches remains somewhat incremental. The authors should clarify more explicitly what is fundamentally new beyond adapting curvature weighting to the KGC setting.

Second, the empirical evaluation is limited. Experiments are conducted only on FB15k-237 and WN18RR, both relatively standard benchmarks. It would strengthen the paper to evaluate on additional datasets (e.g., YAGO, DBpedia, or larger KGs) to demonstrate scalability and robustness. Moreover, the reported improvements over CompGCN are relatively small (e.g., MRR 0.355 → 0.363 on FB15k-237), so statistical significance or multiple runs should be reported.

Third, the computational overhead of ORC computation is not sufficiently discussed. Since curvature must be computed for all edges prior to training, it would be useful to report preprocessing cost, scalability behavior, and whether approximate ORC methods were used.

Fourth, some design choices appear heuristic and require justification. For example, the curvature-based weighting 1-k, the semantic aggregation relies on cosine dissimilarity without learnable parameters, and self-loop curvature is fixed at +1.0. It would help to include sensitivity analysis or alternatives.

Finally, the writing could be improved in several places. Some sections are longer than necessary and repeat known background (e.g., the detailed ORC explanation). Tightening the presentation and clarifying the main technical contributions would improve readability.

Summary of the paper: ORCA-GCN integrates Ollivier-Ricci Curvature into a GNN framework that down-weights redundant high-curvature intra-cluster edges and amplifies low-curvature bridge edges, achieving improvements on knowledge graph completion benchmarks FB15k-237 and WN18RR.
(1) Originality: The core idea is to integrate Ollivier-Ricci Curvature (ORC) into the message-passing mechanism for knowledge graphs, which is interesting. The model adopts a two-layer architecture, where the first layer performs geometry-aware aggregation using ORC values, while the second layer transitions to semantic-aware aggregation based on cosine dissimilarity. However, it does not introduce fundamentally new methodology.
(2) Significance of results: Experimental results demonstrate improvements on most metrics on both FB15k-237 and WN18RR benchmarks. However, the baseline comparison is incomplete and outdated. For example, the classic GNN method SE-GNN[1] is absent, whose performance on certain metrics surpasses that of ORCA-GCN.
[1] How Does Knowledge Graph Embedding Extrapolate to Unseen Data: A Semantic Evidence View (AAAI2022)
(3) Quality of writing: The paper is well organized and easy to be understood.