We sincerely thank the reviewer for the time and effort in reviewing our paper. We take all comments seriously and try our best to address every raised concerns. Please feel free to ask any follow-up questions.

---

S1

The typos on line 652 will be corrected in the next version, and we will carefully proofread the entire text again.

---

Q1

The calculation of local hyperbolicity is not related to the number of nodes but depends on the links to determine the hierarchical properties of the graph. When we compute the hyperbolicity of local subgraphs, we observe the link structure between nodes to decide whether to adopt a hyperbolic embedding. For example, in a graph containing 100 nodes, if the connections between nodes form a grid-like structure, the graph is more suited to Euclidean space. On the other hand, if the connections resemble a full tree, the graph is better suited for hyperbolic space.

Local subgraphs, defined as k-hop subgraphs centered around a node, vary in size depending on k. We analyzed the hyperbolicity distribution in 100 randomly selected local subgraphs from the HepPh dataset.

| | # Nodes $0-3000$ | # Nodes (3000-6000] | # Nodes (6000-9746] | | ----------------- | :----------: | :-------------: |:-------------: | | Hyperbolicity δ=0 | 6 | 10 | 18 | | Hyperbolicity δ≠0 | 9 | 31 | 26 |

The results show that hyperbolicity varies independently of the node count.

---

Q2

Reasons for using the tangent space:

• The tangent space serves as a local linear approximation of the hyperbolic space. It enables the use of Euclidean space operators while preserving the hyperbolic characteristics, ensuring that feature vectors remain within the manifold. This approach reduces the embedding loss caused by geometric heterogeneity, thereby enhancing the model's accuracy.
• Our framework combines Euclidean and hyperbolic spaces, with the tangent space serving as an intermediate space that allows for flexible transformation, facilitating alignment operations within the framework.

As you pointed out, if the feature vectors do not lie in the tangent space, the exponentiation might require an additional isomorphism operation. However, in the Poincaré model, since the feature vectors naturally reside in the tangent space, the exponential map can be directly applied without the need for an extra isomorphism operation.

---

Q3

When the curvature of each layer in the HGCN is fixed to a constant value, its forward propagation process may indeed degrade into the situation you mentioned.

Our framework uses adaptive curvature, so the exponential and logarithmic mappings between layers cannot be canceled. These operations, based on the geometric properties of hyperbolic space, ensure node embeddings stay within the tangent space of the Lorentz model, helping the model capture hierarchical features and reduce modeling losses caused by geometric heterogeneity.

If operations were conducted in Euclidean space with the exponential mapping applied only at the final layer, the model would reduce to a single-layer hyperbolic network. While simpler, this approach overlooks the unique properties of hyperbolic space, particularly its negative curvature, which is beneficial for modeling hierarchical data. This would likely reduce the model’s ability to capture complex structures.

---

Q4

Yes, theoretically, a graph can exhibit positive curvature locally.

When considering embedding into hyperbolic space, we rely on the consensus in the community that hyperbolic space is better suited for graph data with hierarchical structures. Similarly, when considering embedding a graph into spherical space, it is crucial to assess whether the graph’s structural characteristics align with those of spherical space. However, there is currently no widely accepted standard in the community to determine if a graph is suitable for embedding in spherical space.

To validate whether the spherical space is superior, we completely embed the graph in a spherical (curvature=1) or Euclidean (curvature=0) space and re-conducted experiments. The results show that spherical space method did not outperform other method on HepPh and LFB datasets. This may be due to the fact that spherical space typically exhibits periodic or closed structures, and when the graph data lacks periodicity or global symmetry, the spherical embedding may not effectively capture the relationships between nodes.

We sincerely thank the reviewer for the time and effort in reviewing our paper. We hope that our response can resolve your concerns. Please feel free to ask any follow-up questions.

---

W1

We fully agree that MRR is an important metric for evaluating model performance in dynamic link prediction tasks. Since the original papers of our baseline methods report AUC and AP, we initially adopted these metrics for direct comparison. Following your suggestion, we computed MRR by ranking each positive example within its candidate set (i.e., counting the number of negative examples with higher scores plus one) and averaging the reciprocal of these ranks, integrating it into our evaluation framework. The table below presents the experimental results using MRR (%), where our method achieves the highest scores across all datasets.

---

C1

Our method focuses on discrete-time dynamic graphs (DTDGs), which differ from continuous-time dynamic graphs (CTDGs) in data storage and processing. CTDGs have diverse representations $1$ , leading to various models. We explore extending our method to CTDG link prediction in two ways:

• CTDGs are a special case of DTDGs, with fine-grained timestamps approximating a CTDG. For example, in $1$

$2$ , the UCI and Contact datasets have 58,911 and 8,065 timestamps, respectively. A sliding window approach segments event streams, adapting CTDGs to a snapshot-based framework. Our method, applied to these datasets with fixed timestamps (e.g., daily/monthly), consistently outperforms baselines in AUC, AP, and MRR.

• If the CTDG format remains unchanged, the model must incorporate temporal information, such as decay functions or Neural ODEs. This is a key area for improvement.

---

C2

Yes, another common geometric space is the spherical space, which has a constant positive curvature. We replace the hyperbolic space with the spherical space, and the experimental results on two datasets are shown in the table below.

The experimental results show that the spherical space does not provide any gain. This may be because spherical space is suitable for graphs with periodic or closed structures, and if the data lacks periodicity or global symmetry, spherical embeddings may not effectively capture the relationships between nodes.

---

References:

[1] Foundations and modeling of dynamic networks using dynamic graph neural networks A survey

[2] Towards Better Dynamic Graph Learning: NewArchitecture and Unified Library, NeurIPS 2023

[3] Towards Better Evaluation for Dynamic Link Prediction, NeurIPS 2022

We sincerely thank the reviewer for the time and effort in reviewing our paper. We take all comments seriously and try our best to address every raised concerns. Please feel free to ask any follow-up questions.

---

No Supplementary Material

Our appendices include notations, geometric background, experimental details, supplementary experimental results. A zipped code package is available in the Supplementary Material.

---

W4

Our work has several key innovations. Geometrically, we propose a novel cross-geometric dynamic modeling method by analyzing geometric properties from a local graph perspective in snapshots, advancing the theoretical boundaries of cross-geometric dynamic link prediction. Algorithmically, we design a dynamic link prediction framework with a Temporal State Aggregator layer for efficient historical information extraction. Optimizing from a microscopic viewpoint, we designed a novel Link Evolution Loss function. Practically, our framework achieves state-of-the-art results across various types and scales of continuous graph data.

---

W5 & Q3

Since GRU has inherent memory capabilities, the TSA module primarily serves as an auxiliary enhancement. Its design was iteratively refined through extensive experiments. We analyzed and simplified corresponding components in SOTA methods, finding that an attention-based MLP retained the ability to extract key historical information without performance loss while significantly reducing computational complexity, which guided the final design. The table below shows that TSA offers comparable MRR(%) with other modules but with the least time consumption(ms).

---

W6

We discussed the hyperparameter k (line 430, Figure 6), where k ≥ 4 stabilizes model performance. Figure 7 shows that varying embedding dimensions has minimal impact on the results.

Scalability：

• Task: Our method can be extended to link prediction on continuous-time dynamic graphs (CTDGs) (Please refer to our responses to Reviewer yHfU's C1).
• Framework: Each component of our model can be extended and upgraded. The results of component replacement are shown below: ||HepPh||LFB|| |-|-|-|-|-| | |AUC|AP|AUC|AP| |GRU->LSTM|96.82|94.34|92.45|88.96| |GCN->GAT|97.47|94.87|93.02|89.15| |TSA->Transformer|97.93|95.23|93.86|89.89| |Original|97.19|94.68|93.31|89.58|

---

W7

The legends for Figures 1 and 3 are in the dashed box at the bottom. We apologize for any confusion and will adjust the legend size and provide further explanations if revised.

---

Q1

In Figure 6 of the paper, we present the AUC and AP values for different tasks on HepPh and LFB datasets as k varies. The results show that when k ≥ 4, the results stabilize, indicating that the geometric features of the local subgraph can be fully described in most cases when k is greater than or equal to 4, which is consistent with the description in work [1].

---

Q2

We designed experiments to test the model performance under the two scenarios you suggested. The results demonstrate that our method exhibits strong noise resistance and robustness even when some edges are missing. The experimental design is as follows:

• Noisy(N): Randomly add 5%, 10%, and 20% of non-existing edges to the graph.
• Partially Missing(M): Randomly remove 5%, 10%, and 20% of the existing edges. ||DBLP||Ia-Enron| |DBLP(new)||Ia-Enron(new)|| |-|-|-|-|-|-|-|-|-| ||AUC|AP|AUC|AP|AUC|AP|AUC|AP| |Original|93.11|92.32|91.43|89.17|91.23|90.14|89.59|87.16| |5%N|90.60|89.69|86.26|79.91|88.63|86.90|85.33|79.17| |10%N|88.49|89.50|83.24|76.87|86.82|86.89|81.69|74.83| |20%N|82.68|84.86|81.78|77.38|79.65|80.49|79.52|73.77| |5%M|92.16|89.79|87.91|81.17|89.57|86.26|88.26|81.84| |10%M|91.91|89.15|87.61|79.59|89.37|84.66|88.57|81.06| |20%M|92.25|88.17|89.19|80.43|89.36|83.30|89.06|80.80|

---

Q4

Figure 5 in paper shows the average epoch training and inference time for the Euclidean SOTA method GRUGCN, the hyperbolic SOTA method HGWaveNet, and our method. Following your suggestion, we measured runtime on the largest dataset in [2][3] Contact (2,426,280 dynamic edges) and a million-scale UNVote dataset (1,035,742 dynamic edges). The results (in ms) indicate that our method computational complexity scales linearly while maintaining the least time overhead.

---

References:

[1] Tree-like structure in large social and information networks, ICDM 2013

[2] Towards Better Dynamic Graph Learning: NewArchitecture and Unified Library, NeurIPS 2023

[3] Towards Better Evaluation for Dynamic Link Prediction, NeurIPS 2022