How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees

We sincerely thank for the positive evaluation regarding its theoretical foundation for constructing GFMs. Response:

Q1: Why work well in zero-/one-shot but not in two-/three-shot.

A1: Thank you for the insightful question.

• 0-shot limited for text-attributed graphs: Knowledge transfer based solely on structural information is inherently challenging in zero-shot scenarios. BRIDGE primarily targets text-free graphs, with experiments (RQ5) integrating an LLM as an enhancer to show effectiveness when textual semantics are available.
• Why excels at 1-shot but not 2-/3-shot: BRIDGE’s MoE-based selective knowledge assembly effectively utilizes precise domain-invariant knowledge, providing a principled advantage at extremely sparse supervision (1-shot). At 2-/3-shot, the MoE routing faces increased uncertainty, while simpler alignment strategies (e.g., ProNoG, GCOPE, MDGPT) temporarily gain an advantage with limited additional supervision. When supervision further increases (≥5-shot), BRIDGE’s spectral regularizer dominates by constraining generalization error, restoring its superior performance.

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Q2: The training and inference time.

A2: Thanks for bringing up the important issue.

• Pre-training Phase: We adopt an early-stopping strategy based on loss, so the maximum training time depends on a preset maximum of 1000 epochs. No explicit inference is conducted during pre-training. Results (s):

• Fine-tuning Phase: We report time per epoch, total fine-tuning time, and inference time separately. The spectral regularizer (highlighted as potentially time-consuming) is only computed during fine-tuning loss, particularly at small shot numbers ($m$). To reduce computational costs in eigenvalue decomposition, we implemented approximate techniques (lines 969–989). Results (5-shot Node Cls., s):

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Q3: Including larger datasets (e.g., ogbn-arxiv) for both source and target domains.

A3: Thanks for this valuable suggestion.

• In source domain: Following your advice, we have now included ogbn-arXiv as an additional dataset for pre-training. Results (Acc. %):

• In target domain: We have already validated BRIDGE on Reddit. Notably, Reddit has 232,965 nodes and 114,615,892 edges (Table D.1), substantially larger than ogbn-arXiv (169,343 nodes and 1,157,799 edges). Therefore, the strong performance on Reddit sufficiently demonstrates BRIDGE’s capability for knowledge transfer on large-scale graphs.

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Q4: The ablation studies regarding the prompt.

A4: Thank you for the insightful suggestion. We have conducted an ablation concerning the graph prompts for fine-tuning. Results (Acc. %):

Note that, prompts $\boldsymbol{\mathcal{P}}$ and routing weights $\mathbf{S}$ are the only tunable parameters. Without prompts, only $\mathbf{S}$ is trained, significantly restricting fine-tuning flexibility and degrading performance, highlighting prompts' necessity in BRIDGE.

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Q5: How does 'S' favors same-domain over cross-domain knowledge for selective assembly?

A5: We calculate the learned assignment weights for source experts from the same domain versus cross-domain sources. Results (1-shot, Node Cls.):

Analysis: For Cora (CR), Academic experts (CS, PM) receive significantly higher weights (0.5742, 0.2950) compared to E-Commerce sources (Ph, Com: 0.0486, 0.0822). For Reddit, the structurally similar E-Commerce expert (Computers, 0.2256) and Academic experts (CR, 0.3001; CS, 0.2855) dominate. These results validate that BRIDGE effectively favors knowledge transfer from same-domain or structurally similar-domain experts, supporting our selective assembly design.

Thank you for your thoughtful review and positive feedback. We appreciate your recognition of BRIDGE’s contributions and address your comments and suggestions point by point below.

Q1: The scalability and computational efficiency.

A1: Thank you. For scalability, we address this in Reviewer Efax (Q3) with new results on ogbn-arXiv (source) and analysis on Reddit (target). For efficiency, see Reviewer Efax (Q2), where we provide end-to-end training and fine-tuning time comparisons.

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Q2: Theoretical validation of the generalization error bound tightness.

A2: The bound tightness primarily depends on: (1) how accurately the Wasserstein distance $d_W$ captures true domain divergence, and (2) how precisely the Lipschitz constants ($L_f$, $L_g$) reflect actual smoothness of the learned graph representations. Lower $d_W$, tighter estimates of $L_f$ and $L_g$ yield a tighter bound, more closely approximating empirical generalization errors. We will explicitly evaluate these factors empirically in future revisions.

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Q3: The practical impact of the generalization bound.

A3: The generalization bound indicates how effectively BRIDGE transfers knowledge across domains. A smaller bound means better transfer (e.g., between similar graphs like Cora and PubMed), whereas a larger bound signals difficulty (e.g., between citation networks and social graphs), guiding practical adjustments like additional fine-tuning.

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Q4: How minimizing the bound directly affects transfer performance.

A4: Directly minimizing the bound improves BRIDGE’s transfer by explicitly constraining domain divergence ($d_W$) and smoothness ($L_f$, $L_g$). Practically, this means representations from different domains become closer and smoother, significantly enhancing generalization to unseen graphs. Experiments (Section 5.3, Figure 4) confirm that optimizing this bound via the spectral regularizer consistently leads to higher accuracy on downstream tasks.

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Q5: Whether real-world graphs satisfy optimal transport-based alignment assumption.

A5: The optimal transport-based alignment assumption has been widely adopted in prior studies, and our work follows this common practice. The underlying reason is that real-world graphs often share intrinsic structural similarities (e.g., similar community or connectivity patterns across social or citation graphs), allowing optimal transport methods to meaningfully measure domain divergence.

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Q6: Missing comparisons to general domain adaptation baselines.

A6: Since BRIDGE targets graph-structured data, we compare with strong graph-specific adaptation baselines (e.g., GCOPE, MDGPT). General domain adaptation methods lack structure-aware design, making them less relevant for fair comparison.

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Q7: LLM feature alignment lacks qualitative validation.

A7: Thank you for this valuable comment. We clarify that the LLM-enhanced alignment is not part of BRIDGE's core contributions. Rather, it serves as an auxiliary experiment to demonstrate BRIDGE’s applicability to text-attributed graphs, highlighting BRIDGE’s broad adaptability. We agree qualitative validation would further support this applicability and will consider it in future extensions.

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Q8: Statistical significance tests for ablations.

A8: Thank you for raising this point. Explicit statistical significance tests (e.g., t-tests) for ablation studies are relatively uncommon for papers in AI research communities. Ablations generally aim to intuitively illustrate each component’s individual contribution rather than establish statistical differences rigorously. Consistent with common practice, our ablations provide results averaged over multiple runs (with std), clearly showing reliable performance differences.

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Q9: Compare different MoE architectures.

A9: Our MoE is a lightweight internal module of BRIDGE. Rather than comparing MoE variants in isolation, we focus on end-to-end comparisons with full graph adaptation methods (e.g., MDGPT, GCOPE) to reflect overall effectiveness.

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Q10: Compare against full fine-tuning methods.

A11: BRIDGE focuses on efficient prompt-based tuning, not full model updates. We compare with strong graph prompt baselines (e.g., ProNoG, GraphPrompt), which align better with our design and goals.

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Q11: Negative transfer issues for multi-domain learning.

A11: Negative transfer is a key challenge in multi-domain learning. BRIDGE addresses this via expert assignment weights ($\mathbf{S}$), which down-weight less relevant domains. As shown in our response to Reviewer Efax (Q5), $\mathbf{S}$ effectively favors helpful sources. We acknowledge BRIDGE doesn’t explicitly handle extreme cases where all source domains are harmful, which will be explored in future work.

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Q12: Typos and Grammars.

A12: We will carefully proofread the manuscript.

Thank you for your insightful and constructive review, and for recognizing the contributions of BRIDGE. We address your questions briefly below.

Q1: The empirical validation of the generalization bound.

A1: To empirically validate the effectiveness of the bound, we conducted specific ablation studies by removing the spectral regularizer (Section 5.3). Removing the spectral regularizer resulted in a clear performance drop: specifically, accuracy decreased from 42.18% to 41.33% (−0.85%) for 1-shot node classification, and more significantly from 60.20% to 54.51% (−5.69%) for 5-shot node classification on the CiteSeer dataset. This substantial performance degradation empirically demonstrates the effectiveness of our generalization bound in improving transferability. These findings explicitly confirm the bound’s role in guiding practical model design.

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Q2: Empirical verification of Lipschitz behavior reduction should be included.

A2: Thank you for raising this important point. While our experiments clearly show that including the spectral regularizer improves transfer performance (Section 5.3), the original submission did not explicitly measure the Lipschitz constant's empirical evolution. Theoretically, the spectral regularizer is specifically designed to constrain the Lipschitz constant by penalizing large variations in learned representations. Given your valuable suggestion, we plan to explicitly measure and report the evolution of the empirical Lipschitz constant in future version, to more clearly demonstrate how our method reduces the representation sensitivity across domains.

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Q3: Still requires at least one labeled example for good performance.

A3: Indeed, BRIDGE is specifically designed for text-free graphs, where zero-shot knowledge transfer based solely on structural information is inherently limited, which is a fact well-supported by prior works. Pure structural information lacks sufficient semantic cues, leading to poor zero-shot performance, as extensively confirmed in the literature.

To demonstrate that BRIDGE can also effectively leverage textual semantics when available, we conducted additional experiments integrating textual information generated by an LLM as an enhancement strategy (Section 5.6, Table 3). Results clearly showed notable improvements: e.g., node classification accuracy on Reddit improved from 55.72% (structure-only) to 57.16% (structure + LLM) under zero-shot conditions. These experiments indicate that BRIDGE, though originally targeted for text-free scenarios, can effectively incorporate textual semantics, significantly improving zero-shot transfer performance when such information is available.

We will explicitly clarify BRIDGE’s intended scope (text-free graphs) and discuss the potential benefits of textual augmentation thoroughly in the revision.

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Q4: Spectral regularization may be expensive. Runtime analysis.

A4: Thanks for bringing up this important issue. We reply the same concerns for Reviewer Efax (Q2). To explicitly address this, we adopted efficient approximation techniques (lines 969–989) based on truncated eigenvalue decomposition and eigenvalue approximations.

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Q5: Tuning challenging.

A5: Your point regarding the complexity of hyperparameter tuning due to multiple moving parts is valid. Recognizing this issue, we provided a detailed sensitivity analysis of key hyperparameters (Figure 6, Section 5.5), covering: $K$, $\alpha$, $\beta$, and $\gamma$. These clearly presented sensitivity analyses significantly simplify practical hyperparameter tuning by offering explicit recommendations and robust ranges. We also have provided the suggested hyperparameter settings in Appendix F.1 and the configuration files.

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Q6: Difference between training all graphs together versus one by one.

A6: Thank you for this insightful question. In BRIDGE, all graphs from different domains were trained simultaneously in the pre-training phase, explicitly enabling cross-domain alignment and knowledge transfer. This approach encourages the model to learn shared domain-invariant features across multiple domains simultaneously, resulting in enhanced transfer performance. However, we recognize that training all graphs jointly can pose substantial memory overhead.

Training graphs separately (one domain at a time) reduces memory consumption significantly but sacrifices explicit cross-domain feature alignment, potentially weakening transferability. While this sequential training strategy was not explicitly investigated in the original submission, we acknowledge your point and will explicitly clarify our current training strategy and discuss potential trade-offs in the revision. Detailed empirical comparisons of joint vs. sequential training strategies will be included in future work to thoroughly address this important point.