Paper: Attributed Graph Clustering via Coarsening with Modularity (ICLR-2024)

Reviewer comment (Novelty):
Given that the proposed optimization objective in Equation 5 comes from simply adding a modularity-based regularization term $\mbox{tr}(C^TBC)$ to the original objective function of FGC (Kumar et al, 2023), the originality of this work is very limited, both methodology-wise and technical-wise. It looks like this paper simply (1) adds a regularization term to an existing work and (2) conducts some experiments on a few selected datasets. If there is something very novel, I would recommend the authors emphasize on those aspects.

Author response:
The modularity term can not be regarded as a regularization term as can be seen from the ablation studies Fig 4b that it "guides" the model a majority of the way not to mention the significant increase in performance from baseline methods FGC (**+153 %**) and DMoN (**+40 %**), which shows that our contribution is to **allow potentially any coarsening method to work in a clustering setting**, as there are lots of **theoretical benefits** in common coarsening algorithms that have not been studied in clustering. Moreover, that is why we have included very **comprehensive analyses** surrounding the new objective function, with theoretical guarantees as well. We include a heavy load of supporting proofs ranging from **proofs of convexity, proofs of KKT optimality, proofs of convergence, complexity analysis, ablation studies on the behavior of the different loss terms, and how drastically different it is from FGC, and even recovery on a degree-corrected stochastic block model**. This is not incremental. \
Also, our experiments are not on a few selected datasets but rather range from the **benchmark attributed datasets** used in graph clustering to **non-attributed datasets**, and **even to very large datasets** that most literary works skip out on. For example, CoauthorCS, CoauthorPhysics, AmazonPhoto, AmazonPC, and ogbn-arxiv, even though our work is not specialized for very large datasets. \
We have also extensively worked on **integrating our methods** natively with **various GNN architectures**, which was not done in FGC, and also performing ablation studies, including **contributions of the loss terms** and how they **work together**, the **differences in evolution of latent space** for different models, comparison of runtime including **complexities**, **comparison of modularity** to make it as comprehensive as possible, both empirically and theoretically.

Reviewer comment (Writing):
The overall quality/clarity of this paper can be significantly improved. First of all, there are a couple of ambiguous statements and overstatements. Here are some examples:  
  - On page 1, in the second paragraph, the authors say that "... the Fiedler vector (the eigenvector of the second smallest eigenvalue of the Laplacian) produces a graph cut minimal in the weights of the edges (Fiedler, 1973)." As a reader I find it difficult to understand what "a graph cut minimal in the weights of the edges" means in this context. If the authors mean minimum cut, then I don't think this statement is correct. The relationship between the Fiedler vector and the minimum cut is not mentioned in the original paper (Fiedler, 1973), and in general it does not give rise to a minimum cut.  
  - On page 1, in the second paragraph, the authors say that "... they assume each node gets mapped to one cluster, which is not consistent with real graph data." This is a clear overstatement. As far as I know, most of the node classification benchmarks used by the GNN community has non-overlapping clusters/classes. These include the 3 citations networks in Table 1, and several other attributed datasets from Table 3 in the appendix.

Author response:
A graph cut minimal in the weights of the edges means that a graph cut such that the sum of edges it "cuts" is minimum over all such possible cuts. This can also be written as the number of edges in the cut. We referenced the Fiedler paper for the Fiedler vector. \
	It is a well known result that the eigenvector related to the second smallest eigenvalue gives us a partition such that the number of edges being cut is minimal, and this result is part of Newman's landmark 2006 work on modularity, also referenced in other places in our paper. We have added this citation there.

Reviewer comment (Writing):
The presentation can be greatly improved if proper definitions/citations are provided in the right place. Here are some examples:  
  - The term Feature Graph Coarsening (FGC) first appears at the end of page 1 without a citation. The authors should cite Kumar et al (2023) here.  
  - Similarly, the abbreviation GNN appears at the end of page 1, but the full definition "Graph Neural Network (GNN)" only appears much later in Section 2, at the end of page 2.  
  - In the first paragraph of Section 3.3, it is unclear what "the volume of inter-cluster edges" means. The authors should define it.  
  - In Equation 1, the expression $\delta(c_i,c_j)$ is not defined.  
  - In the first paragraph of Section 4.1, it is unclear what "the original graph is smooth" means. If the authors mean that the graph has smooth signals, they should just say "the original graph has smooth signals".

Author response:
Thank you for these. We have moved the references and abbreviations earlier. We have gone over the paper with a spellchecker and fixed a few typos. If you find any more, please tell us the location. For clarification:
	- Volume of inter-cluster edges means the total number of edges in-between the clusters and not inside them.
	- $\delta(c_i,c_j)$ is the common Kronecker delta which is 1 only when $c_i=c_j$ and 0 otherwise

Reviewer comment (Experiments):
The proposed method has a lot of parameters, i.e. $\alpha,\beta,\gamma,\lambda$ in Equation 5. It is unclear how to select these parameters and how robust are the results with respect to the choice of these parameters.

Author response:
In the paragraph just above Section 6, we mention the observed sensitivity of hyperparameters (The order of sensitivity is $\alpha>\gamma>\beta>\lambda$). Selection is done starting from previous values from FGC and then adjusting accordingly for the modularity term (i.e., general hyperparameter optimization, we can use any method such as a grid search or Bayesian optimization, etc.).

Reviewer comment (Experiments):
In the experiments, how did you pick the parameters $\alpha,\beta,\gamma,\lambda$? How robust are the clustering results with respect to the choice of the parameters?

Author response:
In the paragraph just above Section 6, we mention the observed sensitivity of hyperparameters (The order of sensitivity is $\alpha>\gamma>\beta>\lambda$). Selection is done starting from previous values from FGC and then adjusting accordingly for the modularity term (i.e., general hyperparameter optimization, we can use any method such as a grid search or Bayesian optimization, etc.).

Reviewer comment (Presentation):
The submitted title in the system (Attributed Graph Clustering via Coarsening with Modularity) is different from the title in the paper (ATTRIBUTED GRAPH CLUSTERING VIA MODULARITY AIDED COARSENING).

Author response:
It looks like this was an error.

Reviewer comment (Presentation):
The research motivation is not sufficiently novel or clearly expressed. Additionally, there is an inconsistency between the motivation presented in the introduction and the abstract. For example, Dirichlet energies are stated in the Abstract, but they are not mentioned in Section Introduction. Besides, the reasons for using them are not explained. Reorganizing the abstract and introduction is recommended, particularly in the section discussing the motivation.

Author response:
For novelty concerns, refer to next answer. We do mention Dirichlet energy and all the other terms right under the motivations paragraph inside Introduction section. Also, we have explained the purpose of each term in detail in Section 4.1. We have tried to reorganize a bit.

Reviewer comment (Novelty):
The novelty of this paper is not strong. The proposed method for improving the performance of graph clustering relies on modifying the existing FGC method. Furthermore, the paper fails to explain the shortcomings of the current FGC method and how incorporating modularity would enhance its performance. Overall, the impact of this paper on the field is not significant.

Author response:
In the first paragraph of Section 4.1, we have written that the FGC method is not able to perform on clustering, because the coarsening ratio is too low in clustering tasks. We have multiple paragraphs throughout the paper showcasing the benefits of modularity, starting from the Section 1 Introduction (3rd paragraph) to Section 3.3 in Background to the whole of Section 4 Proposed Method. Basically, this is what our whole paper is about. \
	The impact of the paper on the field is significant because:
	- There is a significant increase in performance from baseline methods FGC (**+153 %**) and DMoN (**+40 %**), which shows that our contribution is to **allow potentially any coarsening method to work in a clustering setting**, as there are lots of **theoretical benefits** in common coarsening algorithms that have not been studied in clustering. Moreover, that is why we have included very **comprehensive analyses** surrounding the new objective function, with theoretical guarantees as well. We include a heavy load of supporting proofs ranging from **proofs of convexity, proofs of KKT optimality, proofs of convergence, complexity analysis, ablation studies on the behavior of the different loss terms, and how drastically different it is from FGC, and even recovery on a degree-corrected stochastic block model**. This is not incremental. \
	- Also, our experiments are not on a few selected datasets but rather range from the **benchmark attributed datasets** used in graph clustering, to **non-attributed datasets**, and **even to very large datasets** that most literary works skip out on. For example, CoauthorCS, CoauthorPhysics, AmazonPhoto, AmazonPC and ogbn-arxiv, even though our work is not specialized for very large datasets. \
	- We have also extensively worked on **integrating our methods** natively with **various GNN architectures**, which was not done in FGC and also performing ablation studies including **contributions of the loss terms** and how they **work together**, the **differences in evolution of latent space** for different models, comparison of runtime including **complexities**, **comparison of modularity** to make it as comprehensive as possible, both empirically and theoretically.

Reviewer comment (Novelty):
The related work section is not comprehensive enough. For instance, the paper does not cite important references such as Kumar M, Sharma A, Saxena S, et al. "Featured Graph Coarsening with Similarity Guarantees" presented at ICML 2023.

Author response:
The related work section includes papers on Graph Coarsening/Pooling, Modularity Optimization and Deep Graph Clustering. The work you have referenced is itself an extension of FGC (Featured Graph Coarsening), JMLR'23, by the same authors, which our method inherits from and references multiple times.

Reviewer comment (Experiments):
The experimental section lacks sufficient detail in its description. For example, the experimental setup is missing information. More specifically, the authors did not provide detailed experimental settings for the baselines. Additionally, the experimental section merely presents the experimental results without providing an explanation for the superior performance of the proposed algorithm in this paper.

Author response:
We refer the reviewer to the paragraph after Section 5.2, which is about Training Details and our experimental setup is explained in detail there (Also refer to answer of Question#2). We explain the performance benefits of modularity all throughout the paper, as written in the answer to #3. We also analyze our results and methods in Section 5.5 Ablation Studies.

Reviewer comment (Writing):
The writing of the paper needs to be improved. There are also some typos in this paper.

Author response:
Thank you for all these, we have moved the references and abbreviations first and fixed typos. And we have added semicolons in the paragraph you requested. Also, we have added references in Table 1. All of them already have experimental results, so we are not sure what you mean by "do not provide experimental results". \
	Figure 2 and 4 are divided into 2a), 2b) and 4a),4b), all four of which have captions. We have grouped them because the table and graphs are small and would not fit in the space constraints otherwise. \
	We use the latex command `\ref` to refer to tables, figures and equations and that generates the output 4a and not 4(a). We are following the ICLR custom latex style and format provided on the conference website. This is important to keep references linked (i.e. they will take you to the related figure/table/eqn when clicked on). \
	As visible from the architecture, Q-GCN has 3 GCN layers. And just like any other GCN or even MLP model, the dimensions change as: `input_size`--(GCN1)-->`hidden_size 1`--(GCN2)-->`hidden_size 2`--(GCN3)-->`output_size` \
	For the x-axis in 4b: We tried to improve this before submission but it is rendered as latex text within the generated plot pdf (using the plotly package), and still somehow increasing the size was making it being rendered as blurry. If you zoom in it is better. We have tried to make this better by changing the latex to `$\large{ ... }$`

Reviewer comment (Novelty):
The algorithm (Feature Graph Coarsening (FGC)) is first given, but no references are given.

Author response:
Thank you for all these, we have moved the references and abbreviations first and fixed typos. And we have added semicolons in the paragraph you requested. Also, we have added references in Table 1. All of them already have experimental results, so we are not sure what you mean by "do not provide experimental results". \
	Figure 2 and 4 are divided into 2a), 2b) and 4a),4b), all four of which have captions. We have grouped them because the table and graphs are small and would not fit in the space constraints otherwise. \
	We use the latex command `\ref` to refer to tables, figures and equations and that generates the output 4a and not 4(a). We are following the ICLR custom latex style and format provided on the conference website. This is important to keep references linked (i.e. they will take you to the related figure/table/eqn when clicked on). \
	As visible from the architecture, Q-GCN has 3 GCN layers. And just like any other GCN or even MLP model, the dimensions change as: `input_size`--(GCN1)-->`hidden_size 1`--(GCN2)-->`hidden_size 2`--(GCN3)-->`output_size` \
	For the x-axis in 4b: We tried to improve this before submission but it is rendered as latex text within the generated plot pdf (using the plotly package), and still somehow increasing the size was making it being rendered as blurry. If you zoom in it is better. We have tried to make this better by changing the latex to `$\large{ ... }$`

Reviewer comment (Presentation):
Several algorithms in Table 1 are missing references, and some do not provide experimental results.

Author response:
Thank you for all these, we have moved the references and abbreviations first and fixed typos. And we have added semicolons in the paragraph you requested. Also, we have added references in Table 1. All of them already have experimental results, so we are not sure what you mean by "do not provide experimental results". \
	Figure 2 and 4 are divided into 2a), 2b) and 4a),4b), all four of which have captions. We have grouped them because the table and graphs are small and would not fit in the space constraints otherwise. \
	We use the latex command `\ref` to refer to tables, figures and equations and that generates the output 4a and not 4(a). We are following the ICLR custom latex style and format provided on the conference website. This is important to keep references linked (i.e. they will take you to the related figure/table/eqn when clicked on). \
	As visible from the architecture, Q-GCN has 3 GCN layers. And just like any other GCN or even MLP model, the dimensions change as: `input_size`--(GCN1)-->`hidden_size 1`--(GCN2)-->`hidden_size 2`--(GCN3)-->`output_size` \
	For the x-axis in 4b: We tried to improve this before submission but it is rendered as latex text within the generated plot pdf (using the plotly package), and still somehow increasing the size was making it being rendered as blurry. If you zoom in it is better. We have tried to make this better by changing the latex to `$\large{ ... }$`

Reviewer comment (Presentation):
Figure 2 and Figure 4 do not have a caption (it is suggested to separate the tables in Figure 2(a) and Figure 4(a) from the figure itself).

Author response:
Thank you for all these, we have moved the references and abbreviations first and fixed typos. And we have added semicolons in the paragraph you requested. Also, we have added references in Table 1. All of them already have experimental results, so we are not sure what you mean by "do not provide experimental results". \
	Figure 2 and 4 are divided into 2a), 2b) and 4a),4b), all four of which have captions. We have grouped them because the table and graphs are small and would not fit in the space constraints otherwise. \
	We use the latex command `\ref` to refer to tables, figures and equations and that generates the output 4a and not 4(a). We are following the ICLR custom latex style and format provided on the conference website. This is important to keep references linked (i.e. they will take you to the related figure/table/eqn when clicked on). \
	As visible from the architecture, Q-GCN has 3 GCN layers. And just like any other GCN or even MLP model, the dimensions change as: `input_size`--(GCN1)-->`hidden_size 1`--(GCN2)-->`hidden_size 2`--(GCN3)-->`output_size` \
	For the x-axis in 4b: We tried to improve this before submission but it is rendered as latex text within the generated plot pdf (using the plotly package), and still somehow increasing the size was making it being rendered as blurry. If you zoom in it is better. We have tried to make this better by changing the latex to `$\large{ ... }$`

Reviewer comment (Writing):
Equation 8 and Eqn. 8 => Eq. (8).

Author response:
Thank you for all these, we have moved the references and abbreviations first and fixed typos. And we have added semicolons in the paragraph you requested. Also, we have added references in Table 1. All of them already have experimental results, so we are not sure what you mean by "do not provide experimental results". \
	Figure 2 and 4 are divided into 2a), 2b) and 4a),4b), all four of which have captions. We have grouped them because the table and graphs are small and would not fit in the space constraints otherwise. \
	We use the latex command `\ref` to refer to tables, figures and equations and that generates the output 4a and not 4(a). We are following the ICLR custom latex style and format provided on the conference website. This is important to keep references linked (i.e. they will take you to the related figure/table/eqn when clicked on). \
	As visible from the architecture, Q-GCN has 3 GCN layers. And just like any other GCN or even MLP model, the dimensions change as: `input_size`--(GCN1)-->`hidden_size 1`--(GCN2)-->`hidden_size 2`--(GCN3)-->`output_size` \
	For the x-axis in 4b: We tried to improve this before submission but it is rendered as latex text within the generated plot pdf (using the plotly package), and still somehow increasing the size was making it being rendered as blurry. If you zoom in it is better. We have tried to make this better by changing the latex to `$\large{ ... }$`

Reviewer comment (Presentation):
Table 2a  => Table 2(a).

Author response:
Thank you for all these, we have moved the references and abbreviations first and fixed typos. And we have added semicolons in the paragraph you requested. Also, we have added references in Table 1. All of them already have experimental results, so we are not sure what you mean by "do not provide experimental results". \
	Figure 2 and 4 are divided into 2a), 2b) and 4a),4b), all four of which have captions. We have grouped them because the table and graphs are small and would not fit in the space constraints otherwise. \
	We use the latex command `\ref` to refer to tables, figures and equations and that generates the output 4a and not 4(a). We are following the ICLR custom latex style and format provided on the conference website. This is important to keep references linked (i.e. they will take you to the related figure/table/eqn when clicked on). \
	As visible from the architecture, Q-GCN has 3 GCN layers. And just like any other GCN or even MLP model, the dimensions change as: `input_size`--(GCN1)-->`hidden_size 1`--(GCN2)-->`hidden_size 2`--(GCN3)-->`output_size` \
	For the x-axis in 4b: We tried to improve this before submission but it is rendered as latex text within the generated plot pdf (using the plotly package), and still somehow increasing the size was making it being rendered as blurry. If you zoom in it is better. We have tried to make this better by changing the latex to `$\large{ ... }$`

Reviewer comment (Reproducibility):
In Section 5.2, the authors state that “Q-GCN is composed of 3 GCN layers”, but only two hidden sizes of 128 and 64 are provided.

Author response:
Thank you for all these, we have moved the references and abbreviations first and fixed typos. And we have added semicolons in the paragraph you requested. Also, we have added references in Table 1. All of them already have experimental results, so we are not sure what you mean by "do not provide experimental results". \
	Figure 2 and 4 are divided into 2a), 2b) and 4a),4b), all four of which have captions. We have grouped them because the table and graphs are small and would not fit in the space constraints otherwise. \
	We use the latex command `\ref` to refer to tables, figures and equations and that generates the output 4a and not 4(a). We are following the ICLR custom latex style and format provided on the conference website. This is important to keep references linked (i.e. they will take you to the related figure/table/eqn when clicked on). \
	As visible from the architecture, Q-GCN has 3 GCN layers. And just like any other GCN or even MLP model, the dimensions change as: `input_size`--(GCN1)-->`hidden_size 1`--(GCN2)-->`hidden_size 2`--(GCN3)-->`output_size` \
	For the x-axis in 4b: We tried to improve this before submission but it is rendered as latex text within the generated plot pdf (using the plotly package), and still somehow increasing the size was making it being rendered as blurry. If you zoom in it is better. We have tried to make this better by changing the latex to `$\large{ ... }$`

Reviewer comment (Writing):
In section 5.2: “We surpass all existing methods…”  => “Our proposed model surpasses all existing methods...”.

Author response:
Thank you for all these, we have moved the references and abbreviations first and fixed typos. And we have added semicolons in the paragraph you requested. Also, we have added references in Table 1. All of them already have experimental results, so we are not sure what you mean by "do not provide experimental results". \
	Figure 2 and 4 are divided into 2a), 2b) and 4a),4b), all four of which have captions. We have grouped them because the table and graphs are small and would not fit in the space constraints otherwise. \
	We use the latex command `\ref` to refer to tables, figures and equations and that generates the output 4a and not 4(a). We are following the ICLR custom latex style and format provided on the conference website. This is important to keep references linked (i.e. they will take you to the related figure/table/eqn when clicked on). \
	As visible from the architecture, Q-GCN has 3 GCN layers. And just like any other GCN or even MLP model, the dimensions change as: `input_size`--(GCN1)-->`hidden_size 1`--(GCN2)-->`hidden_size 2`--(GCN3)-->`output_size` \
	For the x-axis in 4b: We tried to improve this before submission but it is rendered as latex text within the generated plot pdf (using the plotly package), and still somehow increasing the size was making it being rendered as blurry. If you zoom in it is better. We have tried to make this better by changing the latex to `$\large{ ... }$`

Reviewer comment (Presentation):
The font size of the x-axis values in Figure 4(b) is too small.

Author response:
Thank you for all these, we have moved the references and abbreviations first and fixed typos. And we have added semicolons in the paragraph you requested. Also, we have added references in Table 1. All of them already have experimental results, so we are not sure what you mean by "do not provide experimental results". \
	Figure 2 and 4 are divided into 2a), 2b) and 4a),4b), all four of which have captions. We have grouped them because the table and graphs are small and would not fit in the space constraints otherwise. \
	We use the latex command `\ref` to refer to tables, figures and equations and that generates the output 4a and not 4(a). We are following the ICLR custom latex style and format provided on the conference website. This is important to keep references linked (i.e. they will take you to the related figure/table/eqn when clicked on). \
	As visible from the architecture, Q-GCN has 3 GCN layers. And just like any other GCN or even MLP model, the dimensions change as: `input_size`--(GCN1)-->`hidden_size 1`--(GCN2)-->`hidden_size 2`--(GCN3)-->`output_size` \
	For the x-axis in 4b: We tried to improve this before submission but it is rendered as latex text within the generated plot pdf (using the plotly package), and still somehow increasing the size was making it being rendered as blurry. If you zoom in it is better. We have tried to make this better by changing the latex to `$\large{ ... }$`

Reviewer comment (Writing):
In section 5.1, “GCN-based architecures”   => “GCN-based architectures”.

Author response:
Thank you for all these, we have moved the references and abbreviations first and fixed typos. And we have added semicolons in the paragraph you requested. Also, we have added references in Table 1. All of them already have experimental results, so we are not sure what you mean by "do not provide experimental results". \
	Figure 2 and 4 are divided into 2a), 2b) and 4a),4b), all four of which have captions. We have grouped them because the table and graphs are small and would not fit in the space constraints otherwise. \
	We use the latex command `\ref` to refer to tables, figures and equations and that generates the output 4a and not 4(a). We are following the ICLR custom latex style and format provided on the conference website. This is important to keep references linked (i.e. they will take you to the related figure/table/eqn when clicked on). \
	As visible from the architecture, Q-GCN has 3 GCN layers. And just like any other GCN or even MLP model, the dimensions change as: `input_size`--(GCN1)-->`hidden_size 1`--(GCN2)-->`hidden_size 2`--(GCN3)-->`output_size` \
	For the x-axis in 4b: We tried to improve this before submission but it is rendered as latex text within the generated plot pdf (using the plotly package), and still somehow increasing the size was making it being rendered as blurry. If you zoom in it is better. We have tried to make this better by changing the latex to `$\large{ ... }$`

Reviewer comment (Evaluation):
In the Motivation of the introduction section, when stating that "We aim to utilize the feature graph coarsening framework (which does not perform well on clustering, as seen in the results) for graph clustering.", does the phrase "the results" refer to the experimental results in the experimental section? It is recommended to provide specific details on which results are being referred. Also, why does the feature graph coarsening framework not perform well on clustering?

Author response:
Yes, we are talking about the primary results of the paper. We mention in Section 4.1 that FGC alone does not perform well in clustering because the coarsening ratio in clustering (~ < 0.001) is much lower than the ones it is designed for (0.1 - 0.01).

Reviewer comment (Novelty):
The proposed clustering model (problem (5)) is a trivial combination of different previous works, none of (5) is designed by the authors, so the technical contribution of this work is marginal.

Author response:
We note that the modularity term has existed in literature since [Newman, 2006]. However, our work here was to recognize what was lacking in FGC for making it level with the current state of the art in clustering. We worked on proving theoretical guarantees, for example, the Lipschitz continuity of the spectral approximation to the modularity term. There is a significant increase in performance from baseline methods FGC (**+153 %**) and DMoN (**+40 %**), which shows that our contribution is to **allow potentially any coarsening method to work in a clustering setting**, as there are lots of **theoretical benefits** in common coarsening algorithms that have not been studied in clustering. Moreover, that is why we have included very **comprehensive analyses** surrounding the new objective function, with theoretical guarantees as well. We include a heavy load of supporting proofs ranging from **proofs of convexity, proofs of KKT optimality, proofs of convergence, complexity analysis, ablation studies on the behavior of the different loss terms, and how drastically different it is from FGC, and even recovery on a degree-corrected stochastic block model**. This is not incremental. \
	Also, our experiments are not on a few selected datasets but rather range from the **benchmark attributed datasets** used in graph clustering, to **non-attributed datasets**, and **even to very large datasets** that most literary works skip out on. For example, CoauthorCS, CoauthorPhysics, AmazonPhoto, AmazonPC and ogbn-arxiv, even though our work is not specialized for very large datasets. \
	We have also extensively worked on **integrating our methods** natively with **various GNN architectures**, which was not done in FGC and also performing ablation studies including **contributions of the loss terms** and how they **work together**, the **differences in evolution of latent space** for different models, comparison of runtime including **complexities**, **comparison of modularity** to make it as comprehensive as possible, both empirically and theoretically.

Reviewer comment (Experiments):
The ablation studies part is trivial and not informative at all.

Author response:
- `Visualization and Comparison of running times are generally not regarded as ablation studies.` \
	We refer the reviewer to papers DCRN[AAAI'22], HSAN[AAAI'23], SDCN[WWW'20], AGE[KDD'20], SCGC[IEEE TNNLS'23]. Going by these papers in highly regarded graph clustering literature, ablation studies and analyses are usually combined into one, and analyses do include visualization analysis and complexity analysis in all these papers. Note that some papers just have an ablation study on parameters. That is why we have included **both** kinds of ablation studies. \
	- `Modularity Metric Comparison is interesting but its conclusion is pretty trivial:` \
	We wanted to show that our modularity maximization term does contribute to the performance and actually increases the modularity too, since that is a crucial part of our method. \
	- `No free lunch theorem, Ablation studies for special characteristics, rather than trivial observations` \
	The reviewer might think this is a trivial and/or direct conclusion from the formulation because of their experience, however, it is very important to explain the downfalls of modularity in the method, _as pointed out by other reviewers_. For a more *flavorful* observation, observe how in Fig 4b), $\alpha$ + $\beta$ or $\beta$ + $\lambda$ are worse as compared to $\beta$ alone - we hypothesize this is because these terms want to optimize the model in different directions in the high-dimensional parameter space - either all 3 are in different directions or $\alpha$ and $\lambda$ are in the same direction different to $\beta$. Only when all three are together, these effects combine and push the model in a shared direction (which can be noted by seeing the increase in performance of $\alpha + \beta + \lambda$ by around 23%(Cora)/50%(CiteSeer) over just $\beta$). This statement comes from intuition when we think theoretically about the differing outcomes if these terms were optimized independently. \
	Moreover, observe how the plots of latent space evolution pan out so differently by just changing a GCN layer to a GMM, while the whole VGAE architecture including the losses remains the same.

Reviewer comment (Experiments):
Visualization and Comparison of running times are generally not regarded as ablation studies.

Author response:
- `Visualization and Comparison of running times are generally not regarded as ablation studies.` \
	We refer the reviewer to papers DCRN[AAAI'22], HSAN[AAAI'23], SDCN[WWW'20], AGE[KDD'20], SCGC[IEEE TNNLS'23]. Going by these papers in highly regarded graph clustering literature, ablation studies and analyses are usually combined into one, and analyses do include visualization analysis and complexity analysis in all these papers. Note that some papers just have an ablation study on parameters. That is why we have included **both** kinds of ablation studies.

Reviewer comment (Experiments):
Modularity Metric Comparison is interesting but its conclusion is pretty trivial:  
> Even though modularity is a good metric to optimize for, maximum modularity labelling of a graph does not always correspond to the ground truth labelling. For this reason, it is important to have the other terms in our formulation as well.  
This is a common sense known as the “no free lunch theorem” in machine learning. We generally would like the ablation studies to uncover special and important characteristics of the proposed method, rather than trivial observations.

Author response:
- `Modularity Metric Comparison is interesting but its conclusion is pretty trivial:` \
	We wanted to show that our modularity maximization term does contribute to the performance and actually increases the modularity too, since that is a crucial part of our method. \
	- `No free lunch theorem, Ablation studies for special characteristics, rather than trivial observations` \
	The reviewer might think this is a trivial and/or direct conclusion from the formulation because of their experience, however, it is very important to explain the downfalls of modularity in the method, _as pointed out by other reviewers_. For a more *flavorful* observation, observe how in Fig 4b), $\alpha$ + $\beta$ or $\beta$ + $\lambda$ are worse as compared to $\beta$ alone - we hypothesize this is because these terms want to optimize the model in different directions in the high-dimensional parameter space - either all 3 are in different directions or $\alpha$ and $\lambda$ are in the same direction different to $\beta$. Only when all three are together, these effects combine and push the model in a shared direction (which can be noted by seeing the increase in performance of $\alpha + \beta + \lambda$ by around 23%(Cora)/50%(CiteSeer) over just $\beta$). This statement comes from intuition when we think theoretically about the differing outcomes if these terms were optimized independently. \
	Moreover, observe how the plots of latent space evolution pan out so differently by just changing a GCN layer to a GMM, while the whole VGAE architecture including the losses remains the same.

Reviewer comment (Theory):
In problem (5), why do you propose to optimize both $\tilde{X}$ and $C$ and use $\lVert C\tilde{X}-X\rVert_F^2$ to encourage the consistency, rather than optimizing $C$ only as K-means does?

Author response:
The method is based on graph coarsening, and we typically want to find the output features $\tilde{X}$ in coarsening and not just $C$/loading matrix.

Reviewer comment (Experiments):
Keep the last question in mind, why do you optimize $C$ only in Section 4.2? That makes the experiments inconsistent with your proposal. In addition, what's the difference between the two strategies in terms of clustering performance?

Author response:
We optimize only for $C$ for the reasons mentioned in the 2nd paragraph of section 4.2. No, this does not make the experiments inconsistent as Q-FGC (optimization-based method) still optimizes for both $C$ and $\tilde{X}$. We change this in the GNN architectures because $\tilde{X}$ can't be learnt directly from $X$ using gradient descent and would require manual gradient calculations in PyTorch or to use a separate optimizer just for $\tilde{X}$. 
	We appreciate the author's suggestion and this could be a good point for a future work.

Reviewer comment (Experiments):
In Figure 2(b), what makes the proposed Q-GMM-VGAE faster than its backbone model GMM-VGAE? Are the experimental settings fair?

Author response:
Yes, all the experimental settings were the same, which is essential to any analysis. We modified the implementation of GMM-VGAE to suit our needs and optimized some operations to be vector operations - this is also available in the anonymous code repository. We also made some important normalizations in the implementation to **greatly improve numerical stability** of the GMM (in the original implementation, a lot of the times $C$ would come out to be all zeros because of an unnormalized operation in exponentiating the log probabilities).

Reviewer comment (Presentation):
In Table 1, SCGC and MVGRL have better performance but you marked the proposed method bold, why? Is it a typo?

Author response:
Yes, they should be highlighted in ACC and ARI fields for cora and citeseer. We have fixed this.

Reviewer comment (Evaluation):
The results are a little bit disappointing as, according to Fig 4(b) and the final results, the full loss of eq (6) is not really needed. The modularity term does already a good job by itself, and the others appear to merely modify the results slightly -- even in a weird way as using only 1 term (relaxation of the constraint or the $\ell_{1,2}$ norm regularizer) degrades the performance.

Author response:
It is important to note that even though some of the terms do the heavy lifting, the other regularization terms do contribute to performance and more importantly, change the nature of $C$ : The term $\omega$ corresponds to the smoothness of signals in the graph being transferred to the coarsened graph; you can imagine this would affect $C$ by encouraging local "patches"/groups to belong to the same cluster. The term $\gamma$ ensures that the coarsened graph is connected - i.e. preserving inter-cluster relations, which simple contrastive methods destroy; this affects $C$ by making it so that $\Theta_C$ has minimal multiplicity of 0-eigenvalues (which tells us how many connected components there are, and we want just 1 big connected component). Also refer to answer of Question 1. \
	We feel this is a valuable addition to our paper and will include this explanation there too.

Reviewer comment (Writing):
The article is written in a dense way, possibly too dense, and one has trouble identifying the saillant points. I am not sure that the description of the integration of the method in 3 different deep learning methods for graphs is needed in the main text. The most relevant would be enough and it would leave more space to answer the questions asked underneath.

Author response:
We apologize that you feel this way, we were trying to fit a lot of theory *and* experiments/results in our main text since we felt all of it was important to the paper. However, because of the strict page limit it might feel dense. We have additionally provided theoretical results and more visualizations in the supplementary material. Regarding subsection "4.2 Integration with GNNs", we felt it was important to add an architecture in the maintext since we are comparing our method majorly with other deep learning methods. However, we have tried to reduce it in size.

Reviewer comment (Novelty):
The literature on modularity maximization does not appear to be well quoted. In this context, quoting 1 or 2 of the existing surveys would be expected and useful for the readers. Also, there is a body of literature showing the limitations of modularity, from its intrinsic resolution limit to it being considered 'harmful', and the present article does not say a word on that and on the impact of the weaknesses of modularity on the present work.

Author response:
We mention some excerpts here from the paper that do highlight the weakness of modularity, and why it is essential to have other optimization terms:
    > The usage of these (modularity-maximization) algorithms has plummeted over the years because they rely solely on the topological information of graphs and ignore node features. 

    > Even though modularity is a good metric to optimize for, maximum modularity labelling of a graph does not always correspond to the ground truth labelling. .... By optimizing modularity, we can get close to the optimal model parameters (at which NMI would be 1), but will be slightly off-course. We can think of the other terms as correcting this trajectory

    We have also added a mention of the resolution limit of modularity in the introduction and its weakness to small clusters.

Reviewer comment (Evaluation):
In Fig 4(b): why are the cases $\alpha + \beta$ or $\beta + \lambda$ so much worse as compared to $\beta$ alone?

Author response:
This is the main takeaway from the plot; we hypothesize this is because these terms want to optimize the model in different directions in the high-dimensional parameter space - either all 3 are in different directions or $\alpha$ and $\lambda$ are in the same direction different to $\beta$. Only when all three are together these effects combine and push the model in a shared direction (which can be noted by seeing the increase in performance of $\alpha + \beta + \lambda$ by around 23%(Cora)/50%(CiteSeer) over just $\beta$). This statement comes from intuition when we think theoretically about the differing outcomes if these terms were optimized independently.

Reviewer comment (Theory):
There is always the possibility that the structure, E, is not aligned with the features, X. What would then happen? The method forces the smoothness of X on (V,E); is it always the case? If it's not, is this supposition detrimental?

Author response:
No, this is certainly not always the case - however, it is observed in most graph clustering and coarsening benchmark datasets (including heterogeneous). We believe it would be detrimental if that was the case, but since the smoothness term does not have a big impact on performance, it should not be disastrous. Upon more research, we have found this paper and In it's section 4.2, it shows some datasets with their calculated smoothness values: https://openreview.net/pdf?id=rkeIIkHKvS;

Reviewer comment (Theory):
What would happen if the clusters happen to be affected by the mentioned limits of the modularity?

Author response:
Even if the clusters were small enough to be hit by the resolution limit of modularity, it would still detect that cluster but probably have extra nodes from other clusters because of the regularization terms that enforce connectedness and the balanced $\mathcal{l}_{1,2}$ norm which ensures each cluster has at least 1 node.

Reviewer comment (Novelty):
Modularity has been improved in the past 10 years using the Non-backtracking random walks, then the Bethe-Hessian ansatz, and several variations around that, to detect better clusters or modules. Could these improvements be incorporated?

Author response:
It looks like the question was cut off on openreview, however, we will try to answer it: The Bethe-Hessian definitely has interesting spectral properties and the "deformed Laplacian" view seems like a good concept too. Yes, it looks like these improvements could be integrated into our method as part of a future work.