Disentangled Graph Spectral Domain Adaptation

Q1. The authors fail to clearly attribute the source of performance improvement in their method. Further empirical results are needed to isolate the specific role of disentanglement.

R1. To address your concerns, we have conducted an additional experiment to clearly identify the source of performance improvement. This experiment introduced a variant of DGSDA that uses Bernstein polynomials and directly aligns node representations instead of separately aligning attributes and topology. The variant model's performance is consistently worse than that of our full DGSDA, as shown in the following table. This indicates that the disentanglement play a crucial role in enhancing the performance of our model.

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Q2. Given the lack of labels in the target domain, it is unclear how the model ensures that the target domain parameters accurately capture the topological patterns.

R2. We have conducted two additional experiments to demonstrate that the unsupervised loss can provide effects similar to supervised loss in terms of model parameter optimization. In the experiment, the supervised variant of DGSDA employs the supervised loss from 10% labeled data in the target domain, replacing the unsupervised loss: the spectral alignment loss and the entropy loss. The results are shown below.

The results indicate that the unsupervised DGSDA achieves comparable performance to the supervised version, highlighting the effectiveness of the unsupervised losses. This can be attributed to two key factors. First, by regularizing the coefficients of Bernstein polynomials (in Eq. 6), the method explicitly aligns the spectral filters across different domains. This alignment enables the target filters to inherit topology-aware patterns from the source domain, even without labels. Second, the entropy loss sharpens cluster assignments, which implicitly encourages the model to learn more discriminative topological features.

In addition, we have compared the learned filter curves in both labeled and unlabeled target domains. The results can be found at https://anonymous.4open.science/r/DGSDA/figure/DC.png. In the supervised learning case, the model parameters capture the homophily topological pattern, characterized by increasing low-frequency information and suppressing high-frequency information. Similarly, in the unsupervised learning case, the target domain parameters can capture the same topological pattern.

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Q3. Some equations, such as Eq. (4) and Eq. (9), as well as Theorem 4.4, appear to be slightly misaligned. It is recommended to adjust the line breaks for better formatting.

R3. We will adjust the line breaks in Eq. (4), Eq. (9), and Theorem 4.4 to ensure proper alignment and improve the overall formatting.

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Q4. Expression error on line 182 and spelling mistake of model "PairAlign".

R4. We will perform a thorough review of the manuscript to correct any expression errors and spelling mistakes.

Q1. The title appears to be somewhat ambiguous. The title does not reflect the focus on the unsupervised problem in graph domain adaptation, which is a key aspect of the study.

R1. Thank you for pointing this out. The primary focus of this paper is indeed on the unsupervised problem in graph domain adaptation. While our architecture can also accommodate target labels when available, the unsupervised scenario remains our main emphasis. We will consider revising the title to better reflect this focus.

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Q2. The notation T used in the paper is not clear. For example, X^T represents the node attribute matrix of the target domain, but it can also be interpreted as the transpose of the node attribute matrix.

R2. Thanks for your careful check. We will correct $X^T$ to $X^{\top}$ to denote the transpose of the matrix.

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Q3. The description of the proposed method lacks clarity. While $L_{source}$ and $L_{mmd}$ are described in words, they are not accompanied by clear, formal formulations.

R3. The formal formulations of $L_{source}$ and $L_{mmd}$ are presented as follows:

$\mathcal{L}\_{source} = -\frac{1}{N^{{S}}} \sum\_{i=1}^{N^{{S}}} \sum\_{c=1}^{C} y\_{i,c} \log p\_{i,c}$

$\mathcal{L}\_{mmd} = \frac{1}{\left(N^{{S}}\right)^2} \sum\_{i=1}^{N^{{S}}} \sum\_{j=1}^{N^{{S}}} k\left(H\_i^{{S}}, H\_j^{{S}}\right) + \frac{1}{\left(N^{{T}}\right)^2} \sum\_{i=1}^{N^{T}} \sum\_{j=1}^{N^{T}} k\left(H\_i^{T}, H\_j^{T}\right) - \frac{2}{N^{{S}} N^{T}} \sum\_{i=1}^{N^{{S}}} \sum\_{j=1}^{N^{T}} k\left(H\_i^{{S}}, H\_j^{T}\right)$

where $k(·,·)$ represents the kernel function.

We will add them to the appendix to enhance the clarity of the manuscript.

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Q4. In Section 4.1 Distribution Shift Disentanglement, "Topology alignment in the graph data shift can be simplified from $P^{S}(A,X| Y) \neq P^{T}(A,X| Y)$ to $P^{S}(A| X,Y) = P^{T}(A| X,Y)$" seems to be misstatement. The correct statement should be $P^{S}(A| X,Y) \neq P^{T}(A| X,Y)$.

R4. Thanks for pointing out this important detail. We will correct this formula in the revised manuscript and ensure that all related discussions are consistent with this accurate representation.

Q1. The main weakness is the lack of source code.

R1. The source code has been made available at (https://anonymous.4open.science/r/DGSDA) for verification purposes. We promise to make the code public once this paper is accepted.

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Q2. Whether the predicted pseudo-labels on the target domain can be employed?

R2. To answer your valuable question, we have conducted experiments to verify the feasibility of using predicted pseudo-labels on the target domain. This experiment introduces a variant model named DGSDA+PL, which combines pseudo-labels of the target domain. The compared results reveal that pseudo-labels consistently lead to performance degradation in all domain adaptation scenarios, demonstrating the infeasibility of the mentioned scheme.

This is primarily due to the low reliability of the pseudo-labels generated in the early stages of training, which can cause error accumulation in learning processes, and the noise amplification effect in graph neural networks, where erroneous pseudo-labels propagate through message-passing mechanisms. This is also the reason why the proposed method outperforms topology alignment with pseudo-labels.

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Q3. Does the decomposition strategy possess solid theoretical findings?

R3. We acknowledge that the current analysis focuses on demonstrating the feasibility of disentanglement without providing a precise error bound between the entangled and disentangled representations. This is a common limitation in the graph disentanglement field, where theoretical guarantees are still lacking. We will strive to address this in future work.

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Q4. Unsupervised graph domain adaptation is often composed of multiple terms as the objective function, and thus, training it is difficult to balance the impacts of terms. Can these terms be unified to felicitate the training?

R4. We understand your concern for the stability of training. Unfortunately, these terms cannot be unified, as each of the loss terms focuses on different objectives as other GDA methods. To be specific, $L_{source}$ targets minimizing prediction error in the source domain, ensuring effective training on labeled source data. $L_{align}$ focuses on aligning spectral coefficients between the source and target domains. $L_{mmd}$aims to align feature representations to reduce distribution differences. $L_{target}$ promotes model adaptation to the target domain through unsupervised learning. Moreover, the experiments in the hyper-parameter analysis demonstrated that our model is relatively robust to the hyper-parameters used for weighting these terms. We will explore integrating these loss terms in future work to facilitate easier balancing.

Q1. The symbols, especially the theory part, are too complex to read.

R1. Thanks for your feedback. We will thoroughly review and modify all the symbols to make them easier to read.

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Q2. Some explanations should clarify the theoretical differences from [You et al., 2023].

R2. The theoretical differences between this paper and the mentioned work [You et al., 2023] is Polynomial Choice & Lipschitz Properties: The proposed DGSDA adopts Bernstein polynomials, whose Lipschitz constant $C_{\lambda}$ is determined by the ground-truth function (in Theorem 4.3), rather than being restricted by the basic polynomial coefficients as in the work [You et al., 2023]. This allows more flexible and accurate spectral domain adaptation.

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Q3. It is better to provide the source code to enhance reproducibility.

R3. According to your suggestion, the source code has been made available at (https://anonymous.4open.science/r/DGSDA) for verification purposes.

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Q4. Why is the performance of the proposed method lower than that of JHGDA on the traffic network dataset?

R4. The proposed method generally outperforms JHGDA on most tasks in the traffic network dataset and is less effective than JHGDA on the $B → E$ and $E → B$ tasks. The performance weakness may be attributed to overfitting due to limited training data. The Brazil and Europe datasets contain a relatively small number of nodes, which makes the proposed models with multiple constraints more prone to over-capturing the patterns of individual hub nodes. This, in turn, makes it difficult to effectively generalize to the overall structure of the target domain. Nonetheless, the extensive results illustrate the effectiveness of the proposed method.

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Q5. The legend in Figure 2 is not clear. What are the differences between source/target and A/C.

R5. The solid lines represent the filter curves trained on the domain adaptation tasks, while the dashed lines represent the filter curves obtained by training BernNet only on the corresponding datasets. Thus, “source” and “target” in the legend mean the filter curves of source domain encoder and target domain encoder trained on the domain adaptation tasks; “A” and “C” in the legend denote the filter curves of BernNet on the A and C datasets, respectively. We will change them to “Training on A” or “Training on C” in the revised manuscript to enhance readability.

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Q6. Why are the results in Figure 2 continuous curves?

R6. Figure 2 shows the learned polynomials instead of their coefficients, and thus contain continuous curves. The x-axis in Figure 2 denotes the normalized graph signal frequency $\lambda$, and the y-axis represents filter gain $h(\lambda)$. They are independent of the polynomial order $K$ and coefficients $\theta_k$. The Bernstein polynomial is defined as $h(\lambda)=\sum_{k=0}^K\theta_kb^K_k(\frac{\lambda}{2})$, where $b^K_k(t)$ is the Bernstein basis function. For any given $\lambda$, it is first mapped to the [0, 1] interval (i.e., $\frac{\lambda}{2}$), and $h(\lambda)$ is calculated using the learned coefficients $\theta_k$ and the corresponding Bernstein basis functions $b^K_k$. Thus, by sampling sufficiently many $\lambda$ values, we can plot the continuous curves shown in Figure 2.