Paper: CDGraph: Dual Conditional Social Graph Synthesizing via Diffusion Model (ICLR-2024)

Reviewer comment (Writing):
The author's real motivation is confusing. Why the dual conditional model is important and why not more (than two) should be clearly pointed out before explaining the challenges (maybe not just giving examples).

Author response:
**Response**:  The generation of dual conditional social graphs is highly valuable for social applications, as these applications often require the consideration of users' diverse social contexts and connections, which typically satisfy more than one condition in real-world applications. Moreover, they should also retain the unique characteristics of social graphs, i.e., social homophily and social contagion, to ensure that the phenomena inherent in social applications can be captured. Social homophily [6], which involves the tendency to form social connections with people who share similar profiles, plays a significant role as a primary factor in link formation [1]. On the other hand, social contagion [7,8], where behavior spontaneously spreads through a social graph, is largely influenced by network topology [2]. Since homophily and contagion are generally intertwined [3], many studies often simultaneously consider both to facilitate social applications. For instance, Li et al. [4] explore the effects of social homophily and contagion on users' behavior for social recommendation. Sankar et al. [5] leverage social homophily and contagion to accurately predict the spread of information.

In downstream applications like social marketing, it is common to select individuals with strong influence within their social circles for word-of-mouth promotion. These social circles are parts of users' ego social graphs, preserving the essential characteristics of social homophily and social contagion to accurately capture social influence propagation. Specifically, for luxury golf club brands targeting golf enthusiasts and high-income individuals, analyzing users' influence within social circles that include friends sharing a passion for golf and possessing high incomes can help identify individuals well-suited for promoting the brands. Similarly, in link prediction, recommending friends with similar social circles is fundamental. Therefore, focusing on users' social circles characterized by a shared interest in birdwatching and frequent visits to locations in Japan ensures precise recommendations of friends involved in birdwatching within Japan.

Additionally, we agree that a multi-conditional model is expected to be more general and applicable to a broader range of social applications. We provide the method of extending CDGraph to support additional conditions as follows; however, due to time constraints, the implementation of such extensions is deferred to future work.

To support more than two conditions, we need to modify the probability distribution of the reverse process by augmenting the node embeddings $\mathbf{X_{c_i}}$ to indicate the satisfaction of the conditions. And the probability distribution of the classification guidance model can be modified by additionally considering the dependency between the conditions.
We noticed that Eq. (6) can be extended by adding more feature matrices of node conditions, which can be written as follows.
$$
p_{\theta} ( \lbrace \mathbf{X}_{c_i}^{(t-1)} \rbrace _{i=1}^{K}, \mathbf{E}^{(t-1)} | \lbrace \mathbf{X} _{c_i}^{(t-1)} \rbrace _{i=1}^{K}, \mathbf{E}^{(t)}) = \prod _{i=1}^{K} p _{\theta} (\mathbf{X} _{c_i}^{(t-1)} | \mathbf{X} _{c_i}^{(t)}, \mathbf{E}^{(t)}) p _{\theta} (\mathbf{E}^{(t-1)} | \mathbf{E}^{(t)}, \lbrace \mathbf{X} _{c_i}^{(t-1)} \rbrace _{i=1}^{K}),
$$

where $K$ is the number of specified conditions.
On the other hand, the dual-conditional classification guidance model can also be extended by specifying more conditions as follows:
$$
q (G_C^{(t-1)} | G_C^{(t)}, \lbrace c_i \rbrace _{i=1}^{K}) \propto q(c_i | G_C^{(t-1)}, \lbrace c_j \rbrace _{j \neq i}) \prod _{j \neq i} q (c_j | G_C^{(t-1)}) q (G_C^{(t-1)} | G_C^{(t)})
$$

where $K$ is the number of specified conditions, and $\lbrace c_j \rbrace _{j \neq i}$ are conditions dependent with $c_i$.

Reviewer comment (Theory):
Both social homophily and social contagion are based on the assumption of network homogeneity. However, more and more works point out the heterogeneity of real-world networks, especially in social networks (e.g., Facebook and Twitter). Therefore, it is concerning whether the assumptions on which the method is based are good for social graphs.

Author response:
**Response**: Network homogeneity (or heterogeneity) generally refers to the degree of similarity (or dissimilarity) among users in a social graph. Social homophily and social contagion are widespread phenomena on social graphs, irrespective of whether the graph exhibits homogeneity or heterogeneity. Nevertheless, we agree with the reviewer that network heterogeneity is observed in real-world social networks, indicating that users can also become friends with others with diverse profiles. It is important to note that CDGraph is currently in its early stages and is primarily focusing on fundamental social homophily and social contagion. In the future, we can aim to support a more diverse range of social phenomena by modifying the co-evolution dependency module and introducing an additional distribution to capture the heterogeneity.

Reviewer comment (Presentation):
Figure 2 should be improved for more clarity with some explanations either in the figure or in the caption.

Author response:
**Response**:  The workflow of CDGraph is updated in Figure 2 of the revised paper (previously Figure 3 in the original paper), and we provide the detailed explanation as follows: The forward noising processes of $\mathbf{X_{c_{1}}}^{(t)}$, $\mathbf{E}^{(t)}$, and $\mathbf{X_{c_{2}}}^{(t)}$ are illustrated from above to below, and each $\mathbf{X_{c_{i}}}^{(t)} (i=1,2)$ has dependency with $\mathbf{E}^{(t)}$. The red bending arrow represents the conditional probability distribution $q(\mathbf{X_{c_{i}}}^{(t)}|\mathbf{E}^{(t)})$ of $\mathbf{X_{c_{i}}}^{(t)}$ given $\mathbf{E}^{(t)}$, and the black bending arrow represents the conditional probability distribution $q(\mathbf{E}^{(t)}|\mathbf{X_{c_{i}}}^{(t)})$ of $\mathbf{E}^{(t)}$ given $\mathbf{X_{c_{i}}}^{(t)}$.

On the other hand, the reverse denoising processes of $\mathbf{X_{c_{1}}}^{(t)}$, $\mathbf{E}^{(t)}$, and $\mathbf{X_{c_{2}}}^{(t)}$ are illustrated from above to below, and the blue arrow represents the co-evolution based on social contagion; the red arrow represents the co-evolution based on social homophily.

The illustrations of social homophily and social contagion are updated in Figure 3 of the revised paper (previously Figure 2 in the original paper), and we provide a detailed explanation as follows: The social homophily-based coevolution aims to denoise edge embeddings from given node embeddings of each condition (as stated in Eq. (7)); the social contagion-based co-evolution aims to denoise node embeddings of each condition from given edge embeddings (as stated in Eq. (9)).

Reviewer comment (Experiments):
The ablation studies of each condition and loss are missing.

Author response:
**Response**: We apologize that the results of the ablation study could not be produced within the time limit because the interdependence among modules is complex, and removing the implementation of each module requires a bit more time. We plan to introduce three variants of CDGraph: CDGraph-coevolve, CDGraph-class, and CDGraph-plain. CDGraph-coevolve removes the co-evolution dependency module in the reverse denoising process, CDGraph-class discards the dual conditional classifier module in the reverse denoising process, while CDGraph-plain omits both modules. The ablation study will be included in the final manuscript.

Reviewer comment (Novelty):
Contributions only focus on the design of the model, and its motivation is insufficient, and the motivation for designing the model is not clearly explained to the reader.

Author response:
**W1**: Motivation for the model design.

**Response**: The generation of dual conditional social graphs is highly valuable for social applications, as these applications often require the consideration of users' diverse social contexts and connections, which typically satisfy more than one condition in real-world applications. Moreover, they should also retain the unique characteristics of social graphs, i.e., social homophily and social contagion, to ensure that the phenomena inherent in social applications can be captured. Social homophily [9], which involves the tendency to form social connections with people who share similar profiles, plays a significant role as a primary factor in link formation [4]. On the other hand, social contagion [10, 11], where behavior spontaneously spreads through a social graph, is largely influenced by network topology [5]. Since homophily and contagion are generally intertwined [6], many studies often simultaneously consider both to facilitate social applications. For instance, Li et al. [7] explore the effects of social homophily and contagion on users' behavior for social recommendation. Sankar et al. [8] leverage social homophily and contagion to accurately predict the spread of information.

In social applications like social marketing, it is common to select individuals with strong influence within their social circles for word-of-mouth promotion. These social circles are parts of users' ego social graphs, preserving the essential characteristics of social homophily and social contagion to accurately capture social influence propagation. Specifically, for luxury golf club brands targeting golf enthusiasts and high-income individuals, analyzing users' influence within social circles that include friends sharing a passion for golf and possessing high incomes can help identify individuals well-suited for promoting the brands. Similarly, in link prediction, recommending friends with similar social circles is fundamental. Therefore, focusing on users' social circles characterized by a shared interest in birdwatching and frequent visits to locations in Japan ensures precise recommendations of friends involved in birdwatching within Japan.

Reviewer comment (Presentation):
Some of the figures in the paper are difficult to read, due to the fact that too many models are included in the same figure.

Author response:
**W2**: Detailed explanation of the workflow of CDGraph. 

**Response**:  The workflow of CDGraph is updated in Figure 2 of the revised paper (previously Figure 3 in the original paper), and we provide the detailed explanation as follows: The forward noising processes of $\mathbf{X_{c_{1}}}^{(t)}$, $\mathbf{E}^{(t)}$, and $\mathbf{X_{c_{2}}}^{(t)}$ are illustrated from above to below, and each $\mathbf{X_{c_{i}}}^{(t)} (i=1,2)$ has dependency with $\mathbf{E}^{(t)}$. The red bending arrow represents the conditional probability distribution $q(\mathbf{X_{c_{i}}}^{(t)}|\mathbf{E}^{(t)})$ of $\mathbf{X_{c_{i}}}^{(t)}$ given $\mathbf{E}^{(t)}$, and the black bending arrow represents the conditional probability distribution $q(\mathbf{E}^{(t)}|\mathbf{X_{c_{i}}}^{(t)})$ of $\mathbf{E}^{(t)}$ given $\mathbf{X_{c_{i}}}^{(t)}$.

On the other hand, the reverse denoising processes of $\mathbf{X_{c_{1}}}^{(t)}$, $\mathbf{E}^{(t)}$, and $\mathbf{X_{c_{2}}}^{(t)}$ are illustrated from above to below, and the blue arrow represents the co-evolution based on social contagion; the red arrow represents the co-evolution based on social homophily.

The illustrations of social homophily and social contagion are updated in Figure 3 of the revised paper (previously Figure 2 in the original paper), and we provide a detailed explanation as follows: The social homophily-based coevolution aims to denoise edge embeddings from given node embeddings of each condition (as stated in Eq. (7)); the social contagion-based co-evolution aims to denoise node embeddings of each condition from given edge embeddings (as stated in Eq. (9)).

Reviewer comment (Experiments):
In the experimental part, the author focuses on the lack of sufficient comparison algorithms on the performance of different data sets in the designed method, which is not convincing enough. In particular, the comparison algorithm lacks the classical algorithm.

Author response:
**W3**: More sufficient comparisons (with the classical algorithm).

**Response**: We add a new baseline RW [1], which is a random-walk-based graph generation algorithm. In addition to the original metrics, including validity and discrepancy with the input graphs (i.e., relative error ratios of #node, #edges, and density, as well as MMD of clustering coefficients), we further consider two homophily metrics, including assortativity and EI homophily index, to evaluate discrepancy with the input graphs. Note that both assortativity and the EI homophily index are presented using relative error ratios. Assume that there are two attributes associated with each node, and each node has a binary value (0/1) for each attribute. A homo-edge is an edge where the end nodes share the same value for every attribute, whereas a hetero-edge is an edge where the end nodes differ in values for at least one attribute.
* Assortativity [2]: The ratio of the number of homo-edges $|E_{homo}|$ to the number of edges $|E|$ in the whole graph:  $$r=\frac{|E_{homo}|}{|E|}.$$

* EI homophily index [3]: The ratio of the difference between the number of hetero-edges $|E_{hetero}|$ and the number of homo-edges $|E_{homo}|$ to to the number of edges $|E|$ in the whole graph: 
\begin{equation*}
EI=\frac{|E_{hetero}|-|E_{homo}|}{|E|}.
\end{equation*}

The results are presented in the following table. For validity, compared to diffusion-based and deep generative models, RW cannot simultaneously meet the conditions since it only focuses on structural information. Regarding the discrepancy with the input graphs, those generated by RW typically exhibit the largest disparity because RW lacks global awareness and is highly influenced by biases in local structure. Concerning homophily, RW yields high relative error ratios of assortativity and EI homophily index since it neglects the attributes of nodes when generating graphs.
| Metric   | Validity | Node     | Edge   | Density | Clust. coeff. | Assortativity | EI homophily index |
|----------|----------|----------|--------|---------|---------------|---------------|--------------------|
| RW       | 0        | 5.170    | 8.225  | 0.698   | 0.242         | 0.351         | 1.437             |
| SPECTRE  | 0.276    | _0.110_| 1.113  | 0.439   | 0.288         | 0.042         | 0.172             |
| GSM      | 0.274    | 0.147    | _0.677_ | 0.457 | 0.031         | 0.043         | 0.175             |
| EDGE     | 0.307    | 0.897    | 2.424  | 0.926   | _0.03_      |  **0.006**     | **0.025**             |
| DiGress  | _0.375_| __0.002__| 0.725  | _0.297_| 0.259         | 0.067         | 0.275             |
| CDGraph  | __1__    | 0.149    | __0.186__| __0.002__ | __0.022__   | _0.013_         | _0.052_             |

Reviewer comment (Experiments):
The author proposed three algorithms, which are CDGraph Training for pseudocode of the sampling of the diffusion process, CDGraph Training for sampling procedure and Conditional Sampling for pseudocode of the sampling of the diffusion process. But no ablation test was offered. How do these algorithms perform and how are they represented?

Author response:
**W4**: Ablation study.

**Response**: We apologize that the results of the ablation study could not be produced within the time limit because the interdependence among modules is complex, and removing the implementation of each module requires a bit more time. We plan to introduce three variants of CDGraph: CDGraph-coevolve, CDGraph-class, and CDGraph-plain. CDGraph-coevolve removes the co-evolution dependency module in the reverse denoising process, CDGraph-class discards the dual conditional classifier module in the reverse denoising process, while CDGraph-plain omits both modules. The ablation study will be included in the final manuscript.

Reviewer comment (Presentation):
The table headers of some graphs are confusing, and the reviewer suggests changing the table headers or table lines appropriately.

Author response:
**W5**: The headers of some figures and tables.

**Response**: As suggested, we revise the titles of tables and figures as follows: 
* Table 1 in the revised paper (previously Table 2 in the original paper): Validity and discrepancy with the input graphs of different approaches on all datasets.
* Figure 2 in the revised paper (previously Figure 3 in the original paper): Workflow of CDGraph, where $\mathbf{E}$ denotes the edge embedding, and $\mathbf{X_{c_{1}}}$, and $\mathbf{X_{c_{2}}}$ are embeddings of nodes indicating the satisfaction of conditions $c_1$ and $c_2$.
* Figure 3 in the revised paper (previously Figure 2 in the original paper): Co-evolution dependency incorporating social homophily and social contagion.
  - (a) Social Homophily-based Co-evolution.
  - (b) Social Contagion-based Co-evolution.

The other figures and tables are moved to the appendix due to the space constraint.

Reviewer comment (Theory):
How does the conditional diffusion model ensure that the model learns the specified conditions? Please explain in detail.

Author response:
**Q1**: Detailed explanation of learning the specified conditions.

**Response**: The conditional diffusion model exploits a pre-trained classifier to guide the learning process, ensuring that the model learns the specified conditions by classifying graphs according to whether the majority of nodes in a graph satisfy both conditions. Specifically, a pre-trained classifier $f$ learns the distribution $p(c_{i}|G^{(t)})$ for a graph classification problem $c_{i} \approx f(G^{(t)})$ in each step $t$ of the denoising process (as shown by the brown arrow in Fig. 2(b)) so that the denoised graphs can be predicted to have the specified condition $c_{i}$, which can be derived by Bayesian Theorem. As there is one additional condition $c_{j}$ to be considered, we need to consider a conditional probability distribution $p(c_{j}|G^{(t)}, c_{i})$ to further guide the denoising process to meet the second condition given that the first condition is satisfied.

Reviewer comment (Experiments):
Methods such as DDPM are specifically mentioned in related work, but they are not compared in related experiments. If there is any relevant work, please add or if there is none or the result is not satisfactory, please explain the reason.

Author response:
**Q2**: Explanation regarding the comparisons with DDPM.

**Response**: According to [1], there are two types of DDPMs: continuous and discrete. Since continuous DDPMs may hurt the graph sparsity, and the structural information of noisy graphs in the diffusion process may be lost [2], we opt for discrete DDPMs as our baselines (i.e., DiGress and EDGE), rather than continuous DDPMs.

**References**

[1] Wenqi Fan, et al. "Generative diffusion models on graphs: Methods and applications." IJCAI, 2023.

[2] C. Vignac, I. Krawczuk, A. Siraudin, B. Wang, V. Cevher, and P. Frossard, “Digress: Discrete denoising diffusion for graph generation.” ICLR, 2023.

Reviewer comment (Presentation):
Figure 3 lacks relevant figure captions, making it slightly difficult to read.

Author response:
The workflow of CDGraph is updated in Figure 2 of the revised paper (previously Figure 3 in the original paper), and we provide the detailed explanation as follows: The forward noising processes of $\mathbf{X_{c_{1}}}^{(t)}$, $\mathbf{E}^{(t)}$, and $\mathbf{X_{c_{2}}}^{(t)}$ are illustrated from above to below, and each $\mathbf{X_{c_{i}}}^{(t)} (i=1,2)$ has dependency with $\mathbf{E}^{(t)}$. The red bending arrow represents the conditional probability distribution $q(\mathbf{X_{c_{i}}}^{(t)}|\mathbf{E}^{(t)})$ of $\mathbf{X_{c_{i}}}^{(t)}$ given $\mathbf{E}^{(t)}$, and the black bending arrow represents the conditional probability distribution $q(\mathbf{E}^{(t)}|\mathbf{X_{c_{i}}}^{(t)})$ of $\mathbf{E}^{(t)}$ given $\mathbf{X_{c_{i}}}^{(t)}$.

On the other hand, the reverse denoising processes of $\mathbf{X_{c_{1}}}^{(t)}$, $\mathbf{E}^{(t)}$, and $\mathbf{X_{c_{2}}}^{(t)}$ are illustrated from above to below, and the blue arrow represents the co-evolution based on social contagion; the red arrow represents the co-evolution based on social homophily.

The illustrations of social homophily and social contagion are updated in Figure 3 of the revised paper (previously Figure 2 in the original paper), and we provide a detailed explanation as follows: The social homophily-based coevolution aims to denoise edge embeddings from given node embeddings of each condition (as stated in Eq. (7)); the social contagion-based co-evolution aims to denoise node embeddings of each condition from given edge embeddings (as stated in Eq. (9)).

Reviewer comment (Evaluation):
The reasons for some experimental results are not explained in depth. For example, there is a lack of detailed reasons why Relative error ratios of #nodes on the BlogCatalog data set, and Clust. coeff reaches suboptimality.

Author response:
Though the number of nodes in generated graphs is sampled by empirical distribution in CDGraph and GSM, which may result in slight suboptimality of CDGraph’s relative error ratio of #nodes. Nevertheless, density refers to the ratio of the number of edges to the maximum possible number of edges (which is related to the square of the number of nodes) in a graph. Hence, it is more comprehensive to simultaneously examine the relative error ratios of nodes, edges, and density. In CDGraph, the relative error ratios for both edges and density are much lower than GSM since GSM does not aim to optimize the structure discrepancy. Overall, the structure of the graphs generated by CDGraph remains very close to that of the input graphs.

On the other hand, the MMD of clustering coefficients of EDGE slightly outperforms CDGraph on BlogCatalog since EDGE focuses on the guidance of degree changes, leading to smaller differences in the clustering coefficient between input graphs and generated graphs.

Reviewer comment (Evaluation):
It seems that the comparison algorithms only consider one condition. Only the proposed CDGraph considers two conditions, and the evaluation indicator "Validity" evaluates the proportion of nodes that meet the two specified conditions. Is this evaluation indicator unreasonable enough? (in addition to the restructured DiGress)

Author response:
We clarify that validity refers to the portion of generated graphs in which the majority of nodes meet the two specified conditions, rather than the proportion of nodes meeting these conditions. Although the baselines are not designed to control the generation process to meet the two conditions, we extend them to support graphs with nodes having two attributes as input and capable of generating graphs with nodes having two attributes. Through the evaluation of validity, we can conclude that, for baselines other than DiGress, as they do not consider meeting both conditions during generation, even if they learn from graphs with nodes having two attributes, they cannot guarantee the generation of graphs meeting both conditions. While DiGress is also extended to support two conditions, it cannot achieve high validity, indicating the need to design new methods to generate graphs that can meet both conditions.

Reviewer comment (Novelty):
I think the reason why a dual conditional social graph can be stated more clearly in the paper. The scarcity of data is not a sufficient reason to justify the need for a dual conditional social graph. Is the **dual conditional** social graph applicable to any representative downstream scenarios?

Author response:
**W1**: The justification of the need of dual-conditional social graphs (and the downstream scenarios).

**Response**: The generation of dual conditional social graphs is highly valuable for social applications, as these applications often require the consideration of users' diverse social contexts and connections, which typically satisfy more than one condition in real-world applications. Moreover, they should also retain the unique characteristics of social graphs, i.e., social homophily and social contagion, to ensure that the phenomena inherent in social applications can be captured. Social homophily [6], which involves the tendency to form social connections with people who share similar profiles, plays a significant role as a primary factor in link formation [1]. On the other hand, social contagion [7,8], where behavior spontaneously spreads through a social graph, is largely influenced by network topology [2]. Since homophily and contagion are generally intertwined [3], many studies often simultaneously consider both to facilitate social applications. For instance, Li et al. [4] explore the effects of social homophily and contagion on users' behavior for social recommendation. Sankar et al. [5] leverage social homophily and contagion to accurately predict the spread of information.

For downstream scenarios like social marketing, it is common to select individuals with strong influence within their social circles for word-of-mouth promotion. These social circles are parts of users' ego social graphs, preserving the essential characteristics of social homophily and social contagion to accurately capture social influence propagation. Specifically, for luxury golf club brands targeting golf enthusiasts and high-income individuals, analyzing users' influence within social circles that include friends sharing a passion for golf and possessing high incomes can help identify individuals well-suited for promoting the brands. Similarly, in link prediction, recommending friends with similar social circles is fundamental. Therefore, focusing on users' social circles characterized by a shared interest in birdwatching and frequent visits to locations in Japan ensures precise recommendations of friends involved in birdwatching within Japan.

Reviewer comment (Evaluation):
The crucial phenomena of social homophily and social contagion of social graphs are stated in the paper. However, I am wondering if the error ratios and the MMD metrics are sufficient to evaluate the performance of the proposed model in capturing the social homophily and social contagion is not included in the paper, and I think more metrics and experiment forms may be added, as the nature of network co-evolution is the primary motivation of the proposed model.

Author response:
**W2**: More metrics and evaluations.

**Response**: We conduct additional experiments for evaluation with assortativity [9] and EI homophily index [10], detailed as follows. Note that to demonstrate the discrepancy with the input graphs, these metrics are presented using relative error ratios.  Assume that there are two attributes associated with each node, and each node has a binary value (0/1) for each attribute. A homo-edge is an edge where the end nodes share the same value for every attribute, whereas a hetero-edge is an edge where the end nodes differ in values for at least one attribute.
* Assortativity [9]: The ratio of the number of homo-edges $|E_{homo}|$ to the number of edges $|E|$ in the whole graph:  $$r=\frac{|E_{homo}|}{|E|}.$$
* EI homophily index [10]: The ratio of the difference between the number of hetero-edges $|E_{hetero}|$ and the number of homo-edges $|E_{homo}|$ to to the number of edges $|E|$ in the whole graph: 
\begin{equation*}
EI=\frac{|E_{hetero}|-|E_{homo}|}{|E|}.
\end{equation*}

The results are presented in the table below. For both assortativity and the EI homophily index, CDGraph obtains the second-lowest relative error ratios, indicating the effectiveness of the design based on social homophily and social contagion co-evolution. It's worth noting that EDGE achieves the lowest relative error ratios, as its generation process is guided by nodes' degrees, which may implicitly learn associations with node attributes from the input graphs. Despite EDGE performing well in homophily discrepancy, its validity is significantly lower than that of CDGraph, indicating that it still cannot generate appropriate dual conditional social graphs. Other baselines, lacking specific consideration for the relationship between structure and attributes, exhibit higher relative error ratios.

|            | Assortativity | EI homophily index |
|------------|---------------|--------------------|
| RW         | 0.351         | 1.437              |
| SPECTRE    | 0.042         | 0.172              |
| GSM        | 0.043         | 0.175              |
| EDGE       | **0.006**     | **0.025**          |
| DiGress    | 0.067         | 0.275              |
| CDGraph    | _0.013_         | _0.052_              |

Reviewer comment (Reproducibility):
The network parameterization (denoiser architecture) should be provided in the paper (could be in the appendix). Is there any specified design of the network structure in light of this problem?

Author response:
**W3**: Denoising network parameterization.

**Response**: We provide the architecture of the denoising network and elaborate the details as follows. Following DiGress [11], we construct a Graph Transformer by first extracting graph-theoretic (structural and spectral) features from node and edge embeddings and then updating the embeddings of nodes and edges, as well as graph-level features through graph transformer layers, each of which consists of a self-attention module. We extend the architecture of Graph Transformer employed in DiGress by augmenting the node embeddings since nodes in the condition indication graph are associated with two conditions (there is a 1-hot encoding for each condition). The graph-theoretic features are extracted by following the way in DiGress.

Reviewer comment (Experiments):
The effect of the classifier guidance is not sufficiently demonstrated in the experiments. An ablation study may be needed, by comparing the performance of the proposed model with and without the classifier.

Author response:
**W4**: Ablation study.

**Response**: We apologize that the results of the ablation study could not be produced within the time limit because the interdependence among modules is complex, and removing the implementation of each module requires a bit more time. We plan to introduce three variants of CDGraph: CDGraph-coevolve, CDGraph-class, and CDGraph-plain. CDGraph-coevolve removes the co-evolution dependency module in the reverse denoising process, CDGraph-class discards the dual conditional classifier module in the reverse denoising process, while CDGraph-plain omits both modules. The ablation study will be included in the final manuscript.

Reviewer comment (Evaluation):
The details of the calculations of the metrics should be provided in the appendix.

Author response:
**W5**: Calculations of the evaluation metrics.

**Response**: The definitions of the metrics are stated as follows: 
* Validity (the higher, the better): Suppose $D= \lbrace G_1,...,G_n \rbrace$ is the set of graphs generated by a graph generation algorithm. Let $D_{val}= \lbrace G_i \in D|r_{c_1,c_2}(G_{i}) \geq 0.5 \rbrace$ denote the set of valid graphs in $D$, where $r_{c_1,c_2}(G_{i})$ is the ratio of the nodes in $G_{i}$ satisfying the specified conditions $c_1$ and $c_2$. The validity is calculated as follows:
$$Validity = |D_{val}|/|D|$$
* Relative error ratios (the lower, the better): Suppose $D= \lbrace G_1,...,G_n \rbrace$ is the set of original input graphs and $\hat{D}= \lbrace \hat{G}_1,...,\hat{G}_n \rbrace$ is the set of graphs generated by a graph generator based on $D$. The relative error ratio regarding a graph metric $m$ (e.g., average #nodes and #edges, average density) is calculated by: $m(\hat{G})/m(G)$.
* MMD (maximum mean discrepancy; the lower, the better): Suppose $G$ is the original input graph and $\hat{G}$ is the generated graph. The MMD between $G$ and $\hat{G}$ is calculated by the difference between their graph metric distribution $f(G)$ and $f(\hat{G})$, where $f$ can be the degree distribution, local clustering coefficient distribution, etc.

Reviewer comment (Writing):
The dimensions of the notations in Sec 3.1 are not clearly stated, nor the exact size of input of the diffusion process. What's more, there seems to be a mistake in mixing up the $E_c$ and $E$, and the definition of $E$ is not clear. What is the meaning of "E is a one-hot encoding vector representing whether **an edge** between v_n and v_m in GC **satisfies condition c**"?

Author response:
**Q1 and Q3**: Dimensions of the input in the diffusion process.

**Response**: The dimensions of the notations are stated as follows. (Suppose $N$ is the number of nodes in a graph $G$.)
* The one-hot encoding of nodes indicating the satisfaction of condition $c_{i}$: $\mathbf{X} _{c_i} \in \mathbb{R}^{N \times 2}$.
* The one-hot encoding of edges: $\mathbf{E} \in \mathbb{R}^{N \times N \times 2}$.
* The transition matrix for $\mathbf{X_{c_i}}$: $\mathbf{Q}_{X _{c_i}}^{(t)} \in \mathbb{R}^{2 \times 2}$
* The transition matrix for $\mathbf{E}$: $\mathbf{Q}_{E}^{(t)} \in \mathbb{R}^{2 \times 2}$

Besides, we fix the typo and clarify that $\mathbf{E}$ is a one-hot encoding vector representing whether there is an edge between $v_n$ and $v_m$ in the definition of the conditional indication graph. 

However, according to previous works [1, 2], the number of nodes in the generated graph is drawn from an empirical distribution rather than specified by user input, thus becoming an indicator of whether the generated graph can effectively mimic the original graph.

Reviewer comment (Theory):
The formulation in equation 5 does not seem to contribute to the method's design. What is the motivation for introducing the formulation?

Author response:
**Q2**: Motivation for introducing the formulation in Eq. (5).

**Response**: The two equations formulated in Eq. (5) state the forward process of the nodes and edges respectively, in which the dependencies between nodes and edges are captured by the second term in both equations, and the first term in the first (second) equation models the forward processes of $\mathbf{X_{c_{i}}}^{(t)}$ given $\mathbf{X_{c_i}}^{(t-1)}$ and $\mathbf{E}^{(t-1)}$ ($\mathbf{E}^{(t)}$ given $\mathbf{E}^{(t-1)}$ and $\mathbf{X_{c_i}}^{(t-1)}$). However, due to the Markovian property of $q$, we have: 
$$q(\mathbf{X_{c_{i}}}^{(t)}|\mathbf{X_{c_{i}}}^{(t-1)}, \mathbf{E}^{(t-1)})=q(\mathbf{X_{c_{i}}}^{(t)}|\mathbf{X_{c_{i}}}^{(t-1)}); q(\mathbf{E}^{(t)}|\mathbf{X_{c_{i}}}^{(t-1)}, \mathbf{E}^{(t-1)})=q(\mathbf{E}^{(t)}|\mathbf{E}^{(t-1)})$$

Reviewer comment (Theory):
The proposed method does not seem to be applicable to graphs with more than 2 conditions. However, the profiles of users in a social graph often consist of multiple properties. Is there any future direction to extend the method to graphs with more than 2 conditions?

Author response:
**Q4**: Future direction for CDGraph to support more than two conditions.

**Response**: To support more than two conditions, we need to modify the probability distribution of the reverse process by augmenting the node embeddings $\mathbf{X_{c_i}}$ to indicate the satisfaction of the conditions. And the probability distribution of the classification guidance model can be modified by additionally considering the dependency between the conditions.
We noticed that Eq. (6) can be extended by adding more feature matrices of node conditions, which can be written as follows.
$$
p_{\theta} ( \lbrace \mathbf{X}_{c_i}^{(t-1)} \rbrace _{i=1}^{K}, \mathbf{E}^{(t-1)} | \lbrace \mathbf{X} _{c_i}^{(t-1)} \rbrace _{i=1}^{K}, \mathbf{E}^{(t)}) = \prod _{i=1}^{K} p _{\theta} (\mathbf{X} _{c_i}^{(t-1)} | \mathbf{X} _{c_i}^{(t)}, \mathbf{E}^{(t)}) p _{\theta} (\mathbf{E}^{(t-1)} | \mathbf{E}^{(t)}, \lbrace \mathbf{X} _{c_i}^{(t-1)} \rbrace _{i=1}^{K}),
$$

where $K$ is the number of specified conditions.
On the other hand, the dual-conditional classification guidance model can also be extended by specifying more conditions as follows:
$$
q (G_C^{(t-1)} | G_C^{(t)}, \lbrace c_i \rbrace _{i=1}^{K}) \propto q(c_i | G_C^{(t-1)}, \lbrace c_j \rbrace _{j \neq i}) \prod _{j \neq i} q (c_j | G_C^{(t-1)}) q (G_C^{(t-1)} | G_C^{(t)})
$$

where $K$ is the number of specified conditions, and $\lbrace c_j \rbrace _{j \neq i}$ are conditions dependent with $c_i$.

Reviewer comment (Evaluation):
The density metric of the CDGraph in the Twitter dataset is the worst among all the methods. Is there any explanation for this?

Author response:
**Q5**: Explanation of the performance of CDGraph’s density metric on the Twitter dataset.

**Response**: We are sorry that we got the data wrong. The correct relative error ratio of density in each method is listed as follows:
* GSM: 0.4580
* SPECTRE: 0.4410
* EDGE: 0.0531
* DiGress: 0.0840
* CDGraph: 0.2009

The relative error ratio of density in EDGE and DiGress is lower than in CDGraph since EDGE focuses on guiding degree changes, and DiGress attempts to preserve graph sparsity in the diffusion process. Nevertheless, density refers to the ratio of the number of edges to the maximum possible number of edges (which is related to the square of the number of nodes) in a graph. Hence, it is more comprehensive to simultaneously examine the relative error ratios of nodes, edges, and density. In CDGraph, the relative error ratios for both nodes and edges are less than 0.05. Overall, the structure of the graphs generated by CDGraph remains very close to that of the input graphs.