Transfer Learning on Edge Connecting Probability Estimation Under Graphon Model

We thank the reviewer for these constructive comments. We will improve figure quality, and add the following clarifications and discussions in the revision.

Comment 1: Assumption on similarity between source and target

Response: We agree that the effectiveness of GTRANS depends on the assumption that the target graph exhibits structural similarity to a larger source graph, as is standard in transfer learning.

However, we would like to highlight the fact that our method is robust even in the absence of strong source-target similarity. When GW distance is large, our debiasing step ensures GTRANS performs on par with target-only NS, as shown in Table 1 for dissimilar pairs like 8 and 9. This robustness means that, while structural similarity enhances transfer effectiveness, GTRANS remains a safe and broadly applicable tool even when ideal source-target matching is not available. We will further clarify in the revised manuscript.

Comment 2: Theorem 4.1 is for Entropic GW or GW?

Response: Theorem 4.1 is for the Entropic Gromov-Wasserstein (EGW) distance instead of GW distance. The $\epsilon$ in the statement refers to the entropic regularization parameter. We will state it more clearly in the revised version of the manuscript.

Comment 3: Validating assumptions of Theorem 4.1

Response: Theorem 4.1 relies on two key conditions: (i) $\epsilon > C_1|\pi^\star|_\infty |\mathbf{P}_s \otimes \mathbf{P}_t|{\rm op}$ for some $C_1 > 2$, and (ii) the estimation errors for $\mathbf{P}_s$ and $\mathbf{P}_t$ must be sufficiently small.

While the first condition involves unobservable population-level quantities and cannot be directly verified, our simulations suggest it holds for a broad range of $\epsilon$. Table E (please find in our response to Reviewer aKUK) reports computed lower bounds on feasible $\epsilon$ values (with $C_1 = 2.01$) for Graphons 1–3 (defined in Section E.1), each scaled to have operator norm 1, with $n_s = 100$ and $n_t = 50$. Any $\epsilon$ above the reported thresholds satisfies the condition, including $\epsilon = 0.01$ used in our real data analysis.

The second condition is well-supported by established graphon estimation literature, as it requires $\hat{\mathbf{P}}_s^{\text{ini}}$ and $\hat{\mathbf{P}}_t^{\text{ini}}$ to be consistent. For example, Ref. [62] shows that NS-based estimators achieve rates of $\sqrt{\log n_s / n_s}$ and $\sqrt{\log n_t / n_t}$ for the source and target, respectively.

Comment 4. Whether selfloops or not, Why is the MSE divided by n^2:

Response: Self-loops: Our theoretical results can be applied to both networks with and without self-loops. In our simulation experiments, for simplicity, we conducted exepriments on networks without self-loops. Our real-data also does not contain self-loop.

MSE Normalization: The MSE is calculated as the average squared error over all $n^2$ entries in the probability matrix, following conventions established in prior graphon estimation studies [1-3]. While the graphon is symmetric, this normalization is widely adopted for consistency and comparability across the literature.

[1] Zhang, Y., Levina, E., & Zhu, J. (2017). Estimating network edge probabilities by neighbourhood smoothing. Biometrika, 104(4), 771-783.

[2] Chan, S., & Airoldi, E. (2014). A consistent histogram estimator for exchangeable graph models. In International Conference on Machine Learning (pp. 208-216). PMLR.

[3] Qin, Y., Yu, L., & Li, Y. (2021). Iterative connecting probability estimation for networks. Advances in Neural Information Processing Systems, 34, 1155-1166.

Comment 5. GW optimal coupling uniqueness:

Response: In general, the Gromov-Wasserstein (GW) optimization problem is non-convex, so multiple optimal or near-optimal coupling matrices may exist in theory. To clarify our implementation, we use the ot.gromov_wasserstein function from the POT library [1] in Python to compute the GW distance and corresponding coupling. While the GW problem may have multiple optima, the POT algorithm deterministically converges to a single stationary point for fixed inputs and initialization. This ensures reproducibility, though not mathematical uniqueness. We will revise the manuscript to state "an optimal coupling" and clarify.

[1] Flamary, R., Courty, N., Gramfort, A., et al. (2021). POT: Python Optimal Transport. Journal of Machine Learning Research, 22(78), 1–8.

Comment 6. \nu and \mu:

Response: In our formulation, $\mu$ and $\nu$ denote the uniform measures on the source and target nodes, respectively; that is, $\mu = (\underbrace{n\_S^{-1}, \dots, n\_S^{-1}}\_{:= n\_S \text{ times}})$ and $\nu = (\underbrace{n\_T^{-1}, \dots, n\_T^{-1}}\_{:= n\_T \text{ times}})$. Their product, $\mu \otimes \nu$, is the product measure, which assigns a uniform weight of $(n_S n_T)^{-1}$ to each possible pair consisting of a source and a target node.

Comment 7. How to obtain initial estimates?:

Response: Initial Estimates: We obtain the initial edge probability matrices using the neighborhood smoothing (NS) method [1] (see lines 60–63 in our paper).

Dependence in the Graphon Model: You are correct that, in the graphon model, edge indicators such as $A_{ij}$ and $A_{ik}$ are dependent because they share the same latent variable $u_i$. Rather than ignoring or explicitly modeling these dependencies, NS actually leverages them. For each node $i$, NS defines a neighborhood $\mathcal{N}_i$ consisting of nodes $j$ whose latent positions $u_j$ are close to $u_i$, which means their edge probabilities pattern are similar. Since the latent variables $u_j$ are unobserved, in practice, NS approximates $\mathcal{N}_i$ by identifying nodes $j$ whose observed connectivity patterns (i.e., rows of the adjacency matrix $A$) are close to that of node $i$ under the $L_2$ norm. NS averages edge values within empirically defined neighborhoods, pooling information from correlated edges to consistently estimate edge probabilities—much like kernel smoothing in nonparametric regression.

[1] Zhang, Y., Levina, E., & Zhu, J. (2017). Estimating network edge probabilities by neighborhood smoothing. Biometrika, 104(4), 771–783.

Comment 8. Theorem 4.1: Consider adding the cutoff and strengthening it to a high-probability statement:

Response: Indeed as the estimation error $||\hat{\mathbf{P}}\_s^{ini} - \mathbf{P}\_s||\_\infty$ and $||\hat{\mathbf{P}}\_t^{ini} - \mathbf{P}\_t||\_\infty$ decays at the order $\sqrt{\log{n\_s}/n\_s}$ and $\sqrt{\log{n\_t}/n\_t}$ respectively under certain assumptions on the underlying graphon (e.g., see Theorem 2 of \cite{}), we can leverage this type of result to argue that the conclusion of our theorem holds with probablity $\ge 1 - C(n\_s \wedge n\_t)^{-c}$ for some constant $C , c > 0$. We will update the description of the theorem in the revised version of the manuscript as per your suggestion.

Comment 9. symbol after "source data."

Response: We are not entirely certain what is meant by “symbol combination after source data.” We assume you are referring to the Kronecker product notation (e.g., $\mathbf{P}_s \otimes \mathbf{P}_t$) that appears in Theorem 4.1. In particular, the Kronecker product between two matrices is a way of combining them to create a larger matrix. This operation expands each entry of the first matrix by multiplying it with the entire second matrix, capturing all pairwise interactions between elements—such as nodes in source and target graphs. In our context, it enables comprehensive comparison within the Gromov-Wasserstein framework. If this doesn't fully address your question, we’d appreciate further clarification.

Comment 10. Comparing graphlet-based methods on graph classifcation:

Response: In response, we have included a comparison between our method and a representative graphlet-based graph classification approach. Specifically, we implemented the Graphlet Kernel method. We use the Graphlet Kernel from the GraKeL library in Python to compute pairwise graph similarities, followed by SVM classification. As shown in Table R6, our transfer-based graphon method significantly outperforms Graphlet on two target data (IMDB-BINARY and IMDB-MULTI), while achieving comparable performance as Graphlet on PROTEINS target data.

Table R6: Graph classification accuracy (%) across three target datasets

Comment 11: quantitative bound for how “small” graph means:

Response: Defining universal bounds on "small" $n$ is challenging as it depends on graphon complexity and source-target similarity rather than fixed thresholds. Complex graphons require larger $n_t$. Existing works provide asymptotic error rates (e.g., $O(\sqrt{\log n / n})$ for NS method) but lack specific "small" cutoffs.

To provide practical guidance, we ran simulations with Graphon 1, fixing $n_s = 1000$ and varying $n_t$ from 100 to 700 (Table R4, in our response to Reviewer JN9a). The relative gain from transfer learning decreases as $n_t$ increases but remains positive; even at $n_t = 700$, GTRANS still outperforms target-only NS, showing that transfer learning provides substantial benefit even for moderately large target graphs.

Comment 1: The success of the small target graph estimation seems to rely a bit on whether there is a large and similar graph that can serve as the source graphon estimation. Maybe, it could be better if the authors provide more real life examples to suggest that it is very common and easy to find the similar source graphs to work on.

Response: Thank you for raising this important point. In many real-world settings, it is actually quite common to find large graphs that are structurally similar to smaller ones, especially within the same application domain. For example, in our experiments (Table 2), we successfully transferred from large protein networks in D&D to smaller ones in PROTEINS-Full, and from large social or collaboration graphs (Reddit-B, COLLAB) to smaller IMDB-B or IMDB-M co-actor networks. Another example is in our link prediction task (Section D.2 in Appendix), we successfully transferred from large Wiki-Vote to smaller four networks, including Dolphins, Firm, Football and Karate. For your covenience, we put the link prediction results in Table A.

Beyond our main experiments, here are some potential examples: (1) In systems biology, large gene regulatory or protein interaction networks (e.g., from STRING or BioGRID) can be used as source for smaller subnetworks in specific tissues or cell types. (2) In online social platforms, massive public graphs (from Facebook, Twitter, or Reddit) can be leveraged to inform smaller networks. (3) In neuroscience, large-scale brain connectomes can be sources for smaller regional networks. (4) In transportation research, traffic network of a large city (e.g., Chicago, NY) can be used to improve the traffic system of small urban areas. Due to space limitations, further empirical studies on these additional datasets are beyond the scope of this work. We believe our results provide a strong foundation, and we hope to systematically explore these broader applications in future work.

We will revise our manuscript to include discusssions about these potential application examples.

Table A: Link prediction AUC (Mean ± SD, multiplied by 100) using Wiki-Vote as the source graph ($n_s = 889$).

Comment 2: It is good that we have the real dataset application being enhancing the graphon estimation used in the G-Mixup method. I'm wondering that is there any other applications and usage to showcase the benefit from more accurate small graph graphon estimation.

Response: Thank you for your encouraging comment and interest in broader applications of our method. Beyond the G-Mixup graph classification task, we have evaluated GTRANS on the link prediction problem, as detailed in Appendix D.2. Our results in Table A show that enhanced graphon estimation significantly improves link prediction performance compared to baselines, highlighting the method’s versatility across practical tasks.

In essence, GTRANS can enhance any application requiring graphon estimation. For instance, researchers have utilized estimated graphons for community detection [1] and graph neural network training [2]. Due to page constraints, we defer empirical exploration of these areas to future work.

[1] Klimm, F., Jones, N. S., & Schaub, M. T. (2022). Modularity maximization for graphons. SIAM Journal on Applied Mathematics, 82(6), 1930-1952.

[2] Hu, Z., Fang, Y., & Lin, L. (2021, December). Training graph neural networks by graphon estimation. In 2021 IEEE International Conference on Big Data (Big Data) (pp. 5153-5162). IEEE.

Comment 3: Are there any criterions or measurements that suggest a good choice of source graph or have you encountered the case where the transferability is nearly impossible even with the designed debiasing process?

Response: Thank you for the valuable comment.

Criterion for Source Graph Selection: As mentioned in Section 3.2.2 (line 194), we measure the source-target domain difference using the Gromov-Wasserstein (GW) distance between the initial graphon estimators $\hat{\mathbf{P}}^{\text{ini}}_s$ and $\hat{\mathbf{P}}^{\text{ini}}_t$, each obtained using the neighborhood smoothing method. When multiple candidate source graphs are available, we use the estimated GW distance between inital graphon estimator as the criterion for selecting the most appropriate source graph for transfer (see Section D.1 in the appendix, lines 982-992). Our new theoretical analysis (see detailed response for Comment 2 of Reviewer JN9a) shows that $GW_2^2(\hat{\mathbf{P}}^{ini}_s, \hat{\mathbf{P}}^{ini}_t)$ approximates $GW_2^2(\mathbf{P}_s, \mathbf{P}_t)$, thereby justifying its use as a criterion for source selection with theoretical guarantees.

Cases Where Transferability is Challenging: We have indeed encountered scenarios where transferability is nearly impossible due to high dissimilarity (large $d$). In our simulations (Table 1 and Table C2), cross-graphon transfers between highly dissimilar pairs (e.g., between graphon 8 and 9, with distinct patterns like gradient vs. striped structures; Figure 3) achieves similar performance as target-only neighborhood smoothing method. However, we would like to emphasize that the debiasing step in our method effectively mitigates the influence of the source graph when it is substantially dissimilar to the target domain.

Thus, while GTRANS excels with similar sources, it remains safe and non-degrading even when transfer is challenging, making it a reliable tool for practitioners.

We thank the reviewer for these constructive comments. We will correct typos, add the following clarifications and discussions in the revision. Due to space constraints, only mean values are shown in the following experiments.

Comment 1: How small should the target data be?

Response: Defining universal bounds on "small" $n$ is challenging as it depends on graphon complexity and source-target similarity rather than fixed thresholds. Complex graphons require larger $n_t$. Existing works provide asymptotic error rates (e.g., $O(\sqrt{\log n / n})$ for NS method) but lack specific "small" cutoffs.

To provide practical guidance, we ran simulations with Graphon 1, fixing $n_s = 1000$ and varying $n_t$ from 100 to 700 (Table R4, 50 runs each). The relative gain, calculated as $\frac{MSE_{NS} - MSE_{GTRANS}}{MSE_{NS}}$ (where $MSE_{GTRANS}$ is the minimum of GTRANS-EGW and GTRANS-GW). The relative gain from transfer decreases as $n_t$ increases but remains positive; even at $n_t = 700$, GTRANS still outperforms target-only NS, showing that transfer learning provides substantial benefit even for moderately large target graphs.

Table R4. Mean MSE value (multiplied by 100) over 50 runs for Graphon 1 with $n_s = 1000$.

Comment 2: lack of theoretical conditions on the target and source graphons under which the proposed method is effective.

Response: We derived new theoretical results which can help us understand how close the graphons should be for effective transfer.

(1) Theoretical Justification for GW Distance as Similarity Measure: We developed new theory showing that GW distance between initial estimators $\hat{\mathbf{P}}^{ini}_s$ and $\hat{\mathbf{P}}^{ini}_t$ is a theoretically justified similarity criterion. Our results establish: (i) $GW_2^2(\mathbf{P}_s, \mathbf{P}_t) \lesssim GW_2^2(\hat{\mathbf{P}}^{ini}_s, \hat{\mathbf{P}}^{ini}_t) + \delta_n$ and (ii) $GW_2^2(\hat{\mathbf{P}}^{ini}_s, \hat{\mathbf{P}}^{ini}_t) \lesssim GW_2^2(\mathbf{P}_s, \mathbf{P}_t) + \delta_n$, where $\delta_n$ represents estimation error of order $\sqrt{\log(n)/n}$ for Lipschitz smooth graphons [1]. Together, these results establish that GW distance between initial estimators reliably proxies true graphon similarity, justifying its use for guiding effective transfer learning.

(2) Theoretical Result of Transferring Step: Our theoretical analysis shows that at the population level, the distance between the transferred estimator and true target probability matrix is bounded by the source-target GW distance: $|{\mathbf{P}}_t^{\rm trans} - \mathbf{P}_t| \leq \mathrm{GW}_2^2(\mathbf{P}_s, \mathbf{P}_t)$. This demonstrates that the transferring step effectively transfers structural information, particularly when source and target graphons are sufficiently similar.

Comment 3: Why called "debiasing step"?

Response: When source and target probability matrices ($\mathbf{P}_s$ and $\mathbf{P}_t$) differ, the transferred estimator may systematically misestimate target probabilities due to structural differences, creating bias that requires correction. "Debiasing" is standard terminology in transfer learning literature [1].

Why Debiasing helps: The residual matrix $\mathbf{R}_t = \hat{\mathbf{P}}^{ini}_t - \hat{\mathbf{P}}^{trans2}_t$ captures target-specific structural patterns and noise. Neighborhood smoothing retains meaningful target patterns while suppressing random fluctuations. The final estimate $\hat{\mathbf{P}}_t = \hat{\mathbf{P}}^{trans2}_t + \hat{\mathbf{P}}^{res}_t$ effectively combines transferred knowledge (shared patterns) with target-specific information.

Empirical support of the effectiveness of the debiasing step: We conducted an ablation study comparing our method with its variant without the debiasing step. As shown in Figure C2 in the Appendix, GTRANS consistently outperforms its ablated variants GTRANS-NonDebias. This highlights the importance of the debiasing step.

[1] Li, S., Cai, T. T., & Li, H. (2022). Transfer learning for high-dimensional linear regression: Prediction, estimation and minimax optimality. Journal of the Royal Statistical Society Series B: Statistical Methodology, 84(1), 149-173.

Comment 4: What makes the debiasing step adaptive?

Response: The debiasing step is adaptive because it's applied only when GW distance $d > \delta$, where threshold $\delta$ is chosen via cross-validation to minimize prediction error, making it data-driven based on measured source-target similarity.

Comment 5: What's `source-target domain shift'?

Response: Following transfer learning literature [1], "domain shift" refers to differences in data distribution between source and target domains. In our work, we define source-target domain shift as the GW distance between true latent edge probabilities $\mathbf{P}_s$ and $\mathbf{P}_t$, estimated using GW distance between $\hat{\mathbf{P}}_s^{\text{ini}}$ and $\hat{\mathbf{P}}_t^{\text{ini}}$ (Section 3.2.2, lines 194).

[1] Wilson, G., & Cook, D. J. (2020). A survey of unsupervised deep domain adaptation. ACM Transactions on Intelligent Systems and Technology (TIST), 11(5), 1-46.

Comment 6: Why called residual matrix'?

Response: We use "residual matrix" by analogy to regression analysis, where residuals represent the portion of observed outcomes not explained by the fitted model. In our setting, $\hat{\mathbf{P}}_t^{ini} - \hat{\mathbf{P}}_t^{trans2}$ captures the component of the target's initial estimator not explained by the transferred source estimator, isolating target-specific structural patterns and sampling noise not captured by transfer.

Comment 7: Lines 189-190: why additional smoothing?

Response: The additional neighborhood smoothing step after the projection is implemented because the transfer step, while structurally aligning the source and target, may introduce local discontinuities or irregularities, especially when the alignment matrix aggregates nodes with different local structures. As a result, the transferred estimator might inherit sharp transitions or noise not present in the original smooth source graphon. Appendix C.3 provides empirical justification: our ablation study shows GTRANS-NonSmooth (omitting additional smoothing) achieves higher MSE than the full GTRANS method in most cases, demonstrating that additional smoothing improves estimation accuracy.

Comment 8: Line 229- where is the lower bound on \epsilon?

Response: The lower bound on $\epsilon$ is specified in Theorem 4.1, where we require that the penalty parameter satisfies $|\pi^*|_\infty \leq \frac{\epsilon}{C_1 | \mathbf{P}_s \otimes \mathbf{P}_t |{\mathrm{op}}}$ for some $C_1 > 2$. This condition ensures the necessary local convexity for the analysis to hold. For completeness, we will revise line 229 to explicitly reference Theorem 4.1 and clarify that the lower bound on $\epsilon$ is given there.

Comment 9: Theorem 4.1: Does this cover the specific case where the source graph is completely uninformative about the target graph?

Response: Theorem 4.1 establishes that the estimated alignment matrix approximates the population counterpart, i.e., minimizer of GW distance between $\mathbf{P}_s$ and $\mathbf{P}_t$, with deviation governed by the estimation error of initial estimation $\hat{\mathbf{P}}^{ini}_s$ and $\hat{\mathbf{P}}^{ini}_t$. As long as both the source and target graphons are estimated well (e.g., via the neighborhood smoothing method, which we adopt in our work), and the parameter $\epsilon$ exceeds a certain threshold, this theorem holds, regardless of whether the source is informative or not.

Comment 10: Experiments when n_t =100, n_s=150, 200 and when n_t=150 and n_s=200, 250.

Response: Following your suggestion, we have conducted additional experiments. In Table R5, we compared our method (GTrans-EGW and GTrans-GW) with the Target-only method NS. The results show that GTRANS consistently outperforms NS across all cases. Additionally, as the target sample size $n_t$ increases, the relative gain from transfer learning diminishes, which is expected since more target data naturally reduces the advantage of borrowing strength from a large source.

Table R5. MSE (multiplied by 100) of Graphon 1.

Comment 11: Lines 54-57: Are the two problems (alignment and unsupervised learning) linked?

Response: These are distinct challenges. Alignment Challenge: Absence of known node correspondences between graphs of different sizes can cause structural patterns to be incorrectly aligned. Unsupervised Learning Challenge: Unlike conventional transfer learning with labeled data, graphon estimation relies solely on observed adjacency matrices without ground-truth edge probabilities, preventing explicit loss function definition.

Comment 12: update references on graphon estimation.

Response: We have conducted an empirical study comparing our method with GWB, IGNR, and SIGL. Please refer to Table R3 in response to Reviewer aKUK. These references will be added.

We thank the reviewer for these insightful comments. We will correct typos, add the following clarifications, discussions, and experiments in the revised manuscript. Due to space constraints, only mean values are shown in the following experiments.

Comment 1: Similarity between source and target graphs:

Response: We quantify source–target similarity using the Gromov-Wasserstein (GW) distance between their latent probability matrices $\mathbf{P}_s$ and $\mathbf{P}_t$ (Section 2.2). Since these are unobserved in practice, we use the GW distance $d$ between their initial estimators $\hat{\mathbf{P}}^{\text{ini}}_s$ and $\hat{\mathbf{P}}^{\text{ini}}_t$ as a surrogate (Section 3.2.2, line 194; ref. [62] of our method). The entropic regularized GW can also be used for efficiency. To avoid negative transfer, we apply debiasing when $d > \delta$, with the cutoff $\delta$ chosen via network cross-validation (ref. [32], Section 5.1, lines 321–327).

When multiple candidate source graphs are available, we use the estimated GW distance between the initial graphon estimators as the criterion for selecting the most appropriate source graph for transfer (Section D.1 in Appendix, lines 982-992). Our new theoretical analysis (see detailed response for Comment 2 of Reviewer JN9a) shows that $GW_2^2(\hat{\mathbf{P}}^{ini}_s, \hat{\mathbf{P}}^{ini}_t)$ approximates $GW_2^2(\mathbf{P}_s, \mathbf{P}_t)$, thereby justifying its use as a criterion for source selection with theoretical guarantees.

Comment 2: Larger graph from same dataset in G-Mixup framework:

Response: First, we would like to clarify that in the G-Mixup framework, the authors estimate a single graphon per class by aligning and aggregating all graphs within that class, not individual graphons per graph. Specifically, for each class, they: (i) sort nodes in each graph by descending normalized degree, (ii) pad smaller adjacency matrices to match the largest graph, (iii) average the aligned matrices, (iv) estimate a graphon from this averaged matrix. This class-level graphon is meant to capture the common structural pattern of the class, not any individual graph. Our use of GTRANS in this context aims to improve the estimation accuracy of this shared graphon.

Second, when the application demands per-graph graphon estimates (e.g., for individual graph prediction), intra-dataset transfer can be effective. As per your suggestion, we conducted additional experiments on link prediction for single graphs. Taking IMDB-BINARY as target dataset, we select the smallest graph (with $>20$ edges) as the target. For intra-dataset source graph, we considered: (1) the largest graph in IMDB-BINARY, and (2) the graph within IMDB-BINARY that has a small GW distance to the target. For comparison, we also included the largest graph from a different dataset as an inter-dataset source. Graphons were estimated using GTRANS-GW and baselines (NS, ICE, SAS, USVT). Link prediction was performed by the same procedure as described in Appendix D.2.

As we can see from Table R1, the best results are achieved by choosing the IMDB-B source graph with the smallest GW distance to the target, even if it is not the largest. Specifically, transfer from the most structurally similar IMDB-B graph yields the highest link prediction accuracy (AUC 0.96), outperforming both the largest IMDB-B source and all cross-dataset sources. This demonstrates that transfer is most effective when the source and target are structurally similar (small GW distance).

Table R1: Link prediction AUC results on target graphs with various transfer sources.

Comment 3: Joint estimation by pooling:

Response: To address your concern, we conducted additional experiments comparing our method to two alternatives: (1) Pooled Estimation: pooling source and target graphs together, estimating a graphon using the G-Mixup procedure with neighborhood smoothing (NS) (see our response of your comment 2), and (2) Pooled-then-Transfer: using the pooled estimator as the initial source estimate $\hat{P}^{\text{ini}}_s$ in our transfer framework, followed by the GTRANS procedure (denoted PGTRANS-GW and PGTRANS-EGW). Following Section 5.2, we evaluated all methods within the G-Mixup framework for graph classification.

As shown in Table R2, GTRANS-GW/EGW consistently outperforms pooled estimators. Naive pooling obscures domain-specific features and introduces negative transfer; While Pooled-then-Transfer partially recovers through post-pooling adaptation, it cannot correct initial information dilution. In contrast, GTRANS maintains domain separation until adaptation, leveraging source information more effectively.

Table R2: Graph classification accuracy (%) across three target datasets.

Comment 4: Comparison with recent methods.

Response: We conducted additional experiments comparing our method with GWB, IGNR, and SIGL in the graph classification task. As shown in Table R3, our proposed transfer learning method outperforms these three non-transfer learning methods.

Table R3: Graph classification accuracy (%).

Comment 5: Why smoothing residual?

Response: The residual matrix $\mathbf{R}_t = \hat{\mathbf{P}}^{\text{ini}}_t - \hat{\mathbf{P}}^{\text{trans2}}_t$ captures (a) target-specific structural patterns not explained by the transferred estimator and (b) random noise due to small sample size of the target graph. To retain meaningful structure while suppressing noise, we apply NS to the residual, without which it simply reduces to the target-only NS estimator. Our results consistently show GTRANS outperforming NS, demonstrated by lower MSE (Figure 4 and Table 1) and higher classification accuracy (Table 2), highlighting the critical role of residual smoothing.

Rationale for adding the smoothed residual in the debiasing step: When GW distance is large ($d > \delta$), transferred estimator $\hat{\mathbf{P}}^{\text{trans2}}_t$ alone may cause negative transfer. However, transfer remains useful as global properties like smoothness are often shared. Debiasing adaptively combines source structure with target-specific corrections via smoothed residuals. For example, graphons 7 & 8 (Figure 3) share smooth diagonal connectivity; the transferred estimator captures this smoothness while debiasing recovers sharper local patterns (Figure 5c).

Comment 6: Source selection in G-Mixup:

Response: In the G-Mixup framework, a single graphon is estimated per class by aggregating and aligning all graphs within that class, rather than estimating graphons for individual graphs (see our response to your Comment 2 for details). As a result, there is no issue of selecting a specific source graph within the source dataset.

Further, transfer is performed at the class level: for each target class, we estimate a class-level graphon. To identify a suitable source for transfer, we estimate initial class-level graphons for all source and target classes using neighborhood smoothing. We then calculate pairwise Gromov-Wasserstein distances between these initial estimators across all class pairs, to select the source class closest to the target (see Section D.1 of Appendix).

Comment 7: Validating theoretical assumptions:

Response: Theorem 4.1 relies on two key conditions: (i) $\epsilon > C_1|\pi^\star|_\infty |\mathbf{P}_s \otimes \mathbf{P}_t|{\rm op}$ for some $C_1 > 2$, and (ii) the estimation errors for $\mathbf{P}_s$ and $\mathbf{P}_t$ must be sufficiently small.

While the first condition involves unobservable population-level quantities and cannot be directly verified, our simulations suggest it holds for a broad range of $\epsilon$. The following Table E reports computed lower bounds on feasible $\epsilon$ values (with $C_1 = 2.01$) for Graphons 1–3 (defined in Section E.1), each scaled to have operator norm 1, with $n_s = 100$ and $n_t = 50$. Any $\epsilon$ above the reported thresholds satisfies the condition, including $\epsilon = 0.01$ used in our real data analysis.

The second condition is well-supported by established graphon estimation literature, as it requires $\hat{\mathbf{P}}_s^{\text{ini}}$ and $\hat{\mathbf{P}}_t^{\text{ini}}$ to be consistent. For example, Ref. [62] shows that NS-based estimators achieve rates of $\sqrt{\log n_s / n_s}$ and $\sqrt{\log n_t / n_t}$ for the source and target, respectively.

Table E: Lower bound of $\epsilon$ for various choices of source and target graphons.