Topological Data Analysis on Graphs: Euler characteristics, Persistent Homology, or Spectrum?

We are grateful to Reviewer icUB for their time and detailed comments. We reply to your comments/questions below.

The expressivity of persistent homology (PH) or topological descriptors (TD) is difficult to define without specifying the space of filtration functions. Without this specification, one could theoretically achieve any level of distinguishing power between graphs by selecting suitable filtration functions (or colorings).

Thank you for the opportunity to clarify aspects of our work. In fact, we did clarify what the space of filtration functions are in Definition 1 (starting at Line 147) of the original submission. This is now Line 149 of the new submission. The color-based vertex and edge filtration functions on the graph $G$ are given by $F_v$ and $F_e$, which are induced from the functions $f_v: X \to \mathbb{R}$ and $f_e: X \times X \to \mathbb{R}_{>0}$.

Stability: The stability result presented here is incremental, as it mainly builds upon foundational work by Cohen-Steiner (2006). There is limited novel contribution to the understanding of stability in this context.

Thank you for the comment! There are three difficulties in extending the result to RePHINE. The first was that the original definition of Bottleneck distance did not generalize to RePHINE diagrams as it did not take into account of the $\alpha$ and $\gamma$ parameters. The second is that the original proof for Bottleneck stability is not applicable to our setting because it does not account for the $\alpha$ and $\gamma$ parameters. While the $\alpha$ values can be interpreted as the birth time of the vertices, there was not a clear interpretation of what the $\gamma$ values are in terms of the birth time / death time of a simplex. Furthermore, it was unclear how the setting of Bottleneck stability would change since the original metric on persistence pairs in Cohen-Steiner et al. (2006) were given by the $\ell_{\infty}$-norms, but in our paper the norm we adopted is the $\ell_1$-norm.

In the new version of the manuscript, we have also investigated and added in a discussion on the stability of $\operatorname{RePHINE}^{Spec}$. Specifically, we have proven that $\operatorname{RePHINE}^{Spec}$ is locally stable under the assumption that the functions $f_v: X \to \mathbb{R}$ (which assigns each vertex color type with a real value) and $f_e: X \times X/\sim \to \mathbb{R}_{>0}$ (which assigns each edge volor type with a positivie real value) are both injective.

By locally stable, we mean that tiny variations in the filtration functions used will not drastically change the output of $\operatorname{RePHINE}^{Spec}$. In our original submission, what we proved for $\operatorname{RePHINE}$ is actually somewhat stronger - we showed that the bottleneck distance of any two RePHINE diagrams on the same graph are bounded in terms of some variant of the $\ell^{\infty}$ metric on the filtration functions that created them. This is a global statement of stability, whereas it suffices for local stability to hold in applications.

While $\operatorname{RePHINE}$ was shown to be globally stable by us, we also showed that $\operatorname{RePHINE}^{Spec}$ is not globally stable. Furthermore, when injectivity does not hold, it may not be locally stable either (but on a measure zero subset of the space of filtration functions). We have provided counter-examples for this at the end of Appendix A.3 (Example 1).

To discuss the word local, we needed a notion of topology on $(f_v, f_e)$. The convenient choice for us turned out to be thinking of $(f_v, f_e)$ as an element of $\mathbb{R}^{n} \times \mathbb{R}^m$ (where $n$ is the size of the vertex coloring set $X$ and $m$ is the size of the correspondent edge coloring set). In this case, $f_v$ being injective is the same as saying its coordinates have no repeated entries. The main idea behind proving the local stability of $\operatorname{RePHINE}^{Spec}$ was that the proof for the stability $\operatorname{RePHINE}$ may be adapted in a convex open neighborhood of $f_v$ in $\mathbb{R}^n$ and $f_e$ in $\mathbb{R}^m$.

Product of Graphs: The purpose of this section is unclear, as it does not seem connected to the main theme of the paper. The motivation for ML and relevance to the other sections should be clarified.

Thank you for giving us the opportunity to clarify our work! A key motivation for us to study graph products is that, when dealing with large or complex graphs that have the structural property of being some product, it is often easier to work with the components of the graph product rather than the graph as a whole. This decomposition can be particularly useful in how they relate to persistent homology methods and spectral methods. In particular, we were interested in how much topological information about the whole graph we can recover by just analyzing the components. This is for example the content of Proposition 3, 4, and 5 in the original manuscript. In Proposition 3, we show that the product of vertex filtrations is the max of vertex colors. In Proposition 4, the product of edge filtrations is the natural edge coloring on the product. These two propositions give a natural way to decompose two color-based filtrations of graph products in terms of its components. Finally, Proposition 5 gives a way to compute the 0-dimensional persistence diagram of the graph product based on just its components.

We have also added a description of these motivation for graph products at the beginning of the section.

Death-time Filtration: The relevance of this section to the other sections is also unclear. Its connection to the overall analysis needs to be better justified. The authors derive several new notions and make comparisons between them. However, the motivation is again unclear.

Thank you for the opportunity to clarify aspects of our work. In fact, we did we explain what the motivation for death-time filtrations are and how they relate back to the overall analysis on line 454 - 459 of the original submission. As explained in the paper, in the construction of the max versions of $\operatorname{RePHINE}^{Spec}, \operatorname{RePHINE},$ and $\operatorname{EC}$, for an edge $(w_1, w_2)$, the edge value $f_e(v, w)$ is defined as the minimum of $f_v(w_1)$ and $f_v(w_2)$. On the other hand, $f_v(w_1)$ and $f_v(w_2)$ are the birth times of the vertices $w_1$ and $w_2$ in the vertex filtration. Thus, a motivating question to ask is - what if we replace the birth time in the definition of the max versions with the death time?

Based on your input, we have moved the discussion on the death-time filtrations to the appendix.

Molecular Graphs: Since this is an ML venue, I would expect experimental validation. Some experiments demonstrating the utility of the discussed approaches on molecular graphs would be valuable to illustrate practical implications.

We greatly appreciate your feedback. We focus here on several key theoretical underpinnings from multiple perspectives to bridge prominent gaps in our understanding of various TDA methods from a machine learning perspective.

We fully agree with you that a comprehensive empirical investigation would be useful, and hope that the theoretical insights presented in this work pave way for such detailed empirical investigations in future studies.

I recommend narrowing the scope to one or two of these topics, accompanied by a much deeper analysis, with clearly defined goals and statements.

We greatly appreciate your perspective and feedback. The key reason we focus on all the three axes – expressivity, computation, and stability – is that they together provide a rather complete and more realistic picture of the applicability of the different methods than if considered in isolation. For example, methods such as those based on Laplace spectrum and PH are more expressive than EC, but in practice, depending on the computational budget, the latter might still be preferable in some settings.

Many thanks for your insightful review and constructive feedback. We hope your concerns have been satisfactorily addressed, and if so, would appreciate if you could revisit your score to reflect the same. We are also committed to engaging further if you have any additional questions, concerns, or suggestions.

Thank you for your responses. While I acknowledge that this work presents several interesting findings, I believe it would benefit from a major revision to narrow its scope and provide a deeper analysis of the key results. As such, I will keep my original score.

We are grateful to Reviewer pi7v for their time and detailed comments. We reply to your comments/questions below.

Some of the main conclusions and findings of the paper are not that surprising, such as the Euler characteristic being strictly less expressive than persistent homology. This is in quite fact obvious, given that the (persistent) Euler characteristic is an alternating sum of Betti numbers, while persistent homology by construction contains more information.

We agree with the reviewer. It is only for completeness that we concluded it in the statement of Theorem 1. The purpose of Theorem 1 (and its proof) is to give an organized summary of comparisons and a concrete record for a collection of (counter)examples to rigorously justify the distinctions. We believe that the most interesting parts of Theorem 1 come in the incomparability of $\operatorname{RePHINE}^{mSpec}$ and $\operatorname{RePHINE}$, and the equivalence of $\operatorname{EC}$ and $\operatorname{EC}^m$, which are non-trivial and require subtle arguments and insights. Furthermore, Theorem 1 is only one of the items on the table of contributions (Figure 1).

Do you mean Euler characteristic or persistent Euler characteristic? Definition 5 is the static version, but a persistent version is cited with Turner et al. being the reference.

We thank the reviewer for raising this question. Definition 5 is the persistent version of Euler characteristic we had in mind, where we did a simplification from the original setting of the Euler characteristic transform (ECT) in Turner et al. The ECT in Turner et al. originally is an integeral transform that takes in a shape $S$ embedded in Euclidean space $\mathbb{R}^n$ and produces a function $\mathbb{S}^{n-1} \times \mathbb{R} \to \mathbb{Z}, (\nu, t) \mapsto \chi(\{x \in S\ |\ x \cdot \nu \leq t\}).$ When we adapt the definition to our setting, there are two changes we had to make. First, our graph is not embedded in some particular Euclidean space, so there is no clear notion of directional transform. Therefore, instead of considering the directions given by $\mathbb{S}^{n-1}$, our directions are the two color filtrations given by vertices and edges.

Since there are only two filtrations in this case, and the Euler characteristic can only change whenever $f_v$ (resp. $f_e$) changes. Our ECT can be simplified as just two lists of integers, which is currently in Definition 5.

We also omitted the time variable from the output because for two different graphs with the same $f_v$ and $f_e$ (ie. in expressivity comparisons), both graphs will experience possible changes in their EC at the exact same time-stamp. The inclusion of the time variable only becomes important in stability analysis (ie. different color filtrations on the same graph.)

Overall, I find the presentation of the topological descriptors studied confusing; it is not very clear to me whether the persistent version is being studied for the three types or the static ones, although the title of the paper and the introduction seems to emphasize persistent versions of all descriptors. Why are persistent versions studied in particular and not static ones?

We thank the reviewer for raising this question. As a general question, we want to ask if the reviewer could clarify what a static vs. persistent descriptor is in their mind. To us, whenever we are looking at how a topological invariant changes over a given filtration, we would consider that as ``persistent". Whereas to us, a static descriptor is defined on a graph without the usage of a filtration. Under our interpretation, all of our descriptors that show up in Section 2 are persistent versions.

The reason why we want to study persistent versions instead of static versions is because, under our interpretation, capturing how topological data changes over time contains more information than looking at a graph without filtrations. For example, consider two graphs $G$ and $H$ where $G$ is the circle graph on $3$ vertices, all labeled red, and $H$ is the circle graph on $3$ vertices, with two labeled yellow and one labeled red. If we only look at the static Euler characteristic of both graphs, then under our interpretation they would both output $0$, so there is no difference. However, if we look at how the Euler characteristic changes with an induced color filtrations on $G$ and $H$, then their EC diagrams would be different

We have added a clarification of that we mean persistent in Section 2 of the manuscript.

Dear Reviewer pi7v,

As the discussion period is ending soon, we wanted to check-in again. We have addressed most of your concerns in the rebuttal, notably:

• Missing the list of all contributions: Figure 1 of the current submission lists a full table summarizing our contributions in the paper.

• Static vs. Persistent: We have clarified our interpretation of what persistent and static methods are, explained that all methods we consider are persistent in the document, and explained why persistent method is more advantageous than static.

• We also wanted to bring your attention that we also analyzed and prove the (local) stability of $\operatorname{RePHINE}^{Spec}$ in the updated manuscript, which we believe adds more novelty to our submission. See our global response for more details.

We hope your concerns have been satisfactorily addressed, and if so, would appreciate if you could revisit your score to reflect the same. If you have any specifc questions or concerns, we would be happy to address them. Thank you again for your feedback and help to improve the paper.

In Definition 8, what does $h_1 \sim h_2$ mean?

We thank the reviewer for raising this question. By $h_1 \sim h_2$, we mean that $h_1$ and $h_2$ are related by an edge in the graph $H$. We have added a clarification of this notation in the definition now.

Line 491: Bond Coloring Representation -> Bond Counting Representation?

We thank the reviewer for point out this typo. We have corrected the typo in the updated manuscript now.

The stability section just talks about the stability of $RePHINE$ diagrams, which is a straightforward extension of the stability of persistence diagrams. What about the stability of $RePHINE^{spec}$? Can that be characterized as well? What about the stability of EC?

Thanks to this question raised by the reviewer (and also reviewer ZQBM), we have looked into the stability of $\operatorname{RePHINE}^{Spec}$ and $\operatorname{EC}$. In the new updated manuscript, we have proven that $\operatorname{RePHINE}^{Spec}$ is locally stable under the assumption that the functions $f_v: X \to \mathbb{R}$ (which assigns each vertex color type with a real value) and $f_e: X \times X/\sim \to \mathbb{R}_{>0}$ (which assigns each edge volor type with a positivie real value) are both injective.

By locally stable, we mean that tiny variations in the filtration functions used will not drastically change the output of $\operatorname{RePHINE}^{Spec}$. In our original submission, what we proved for $\operatorname{RePHINE}$ is actually somewhat stronger - we showed that the bottleneck distance of any two RePHINE diagrams on the same graph are bounded in terms of some variant of the $\ell^{\infty}$ metric on the filtration functions that created them. This is a global statement of stability, whereas it suffices for local stability to hold in applications.

While $\operatorname{RePHINE}$ was shown to be globally stable by us, we also showed that $\operatorname{RePHINE}^{Spec}$ is not globally stable. Furthermore, when injectivity does not hold, it may not be locally stable either (but on a measure zero subset of the space of filtration functions). We have provided counter-examples for this at the end of Appendix A.3 (Example 1).

To discuss the word local, we needed a notion of topology on $(f_v, f_e)$. The convenient choice for us turned out to be thinking of $(f_v, f_e)$ as an element of $\mathbb{R}^{n} \times \mathbb{R}^m$ (where $n$ is the size of the vertex coloring set $X$ and $m$ is the size of the correspondent edge coloring set). In this case, $f_v$ (resp. $f_e$) being injective is the same as saying its coordinates have no repeated entries. The main idea behind proving the local stability of $\operatorname{RePHINE}^{Spec}$ was that the proof for the stability of $\operatorname{RePHINE}$ may be adapted in a convex open neighborhood of $f_v$ in $\mathbb{R}^n$ and $f_e$ in $\mathbb{R}^m$.

Finally, for EC, We have also found that the authors of [1] have proven the stability for EC already in their Proposition 3.2.

Many thanks for your insightful review and constructive feedback. We hope your concerns have been satisfactorily addressed, and if so, would appreciate if you could revisit your score to reflect the same. We are also committed to engaging further if you have any additional questions, concerns, or suggestions.

[1] Dłotko, Paweł and Gurnari, Davide, Euler characteristic curves and profiles: a stable shape invariant for big data problems

We are grateful to Reviewer NVLY for their time and detailed comments. We reply to your comments/questions below.

Last line of the first paragraph “the theoretical foundations of TDs remain rather elusive.” This statement, as it reads, is incorrect. All the topological descriptors have a rich theory backing them. What the authors, perhaps, want to say is the expressivity analysis of TDs is rather elusive.

We thank the reviewer for giving this suggestion. We have changed the wording of the sentence in the edited draft from theoretical foundations to expressivity analysis.

Definition 1: $f_e(c,c’) = f_e(c,c’) -> f_e(c,c’) = f_e(c’,c)$

We thank the reviewer for pointing out this typo, and we have corrected it in the edited draft.

Most of the comparisons in Theorem 1 seem fairly obvious from the definitions.

We thank the reviewer for raising this comment. We agree that most of the comparisons in Theorem 1 are straight-forward. The purpose of Theorem 1 (and its proof) is to give an organized summary of comparisons and a concrete record for a collection of (counter)examples to rigorously justify the distinctions. We believe that the most interesting parts of Theorem 1 come in the incomparability of $\operatorname{RePHINE}^{mSpec}$ and $\operatorname{RePHINE}$, and the equivalence of $\operatorname{EC}$ and $\operatorname{EC}^m$, which are non-trivial and require subtle arguments and insights.

I don’t understand the point of stating Theorem 2 separately. Isn’t it already stated in Theorem 1 that $EC$ and $EC^m$ have the same expressive power?

We thank the reviewer for raising this comment. We stated Theorem 2 apart from Theorem 1 to emphasize the importance/surprise that $\operatorname{EC}$ and $\operatorname{EC}^m$ are equivalent. Please let us know if you think it would help to remove it instead.

The flow seems to be lacking from Section 4 onwards. I don’t quite get the motivation behind the product of graphs and death-time filtrations.

We thank the reviewer for raising this comment. On line 454 - 459 of the original submission, we explained what the motivation behind death-time filtrations are. In the construction of the max versions of $\operatorname{RePHINE}^{Spec}, \operatorname{RePHINE},$ and $\operatorname{EC}$, the edge value $f_e(v, w)$ for an edge $(v, w)$ is defined as the minimum of $f_v(w_1)$ and $f_v(w_2)$. On the other hand, $f_v(w_1)$ and $f_v(w_2)$ are the birth times of the vertices $w_1$ and $w_2$ in the vertex filtration. Thus, a motivating question to ask is - ``what if we replace the birth time in the definition of the max versions with the death time?". Based on your input, we have also moved the discussion on death-time filtrations to the appendix in the current submission.

A key motivation for us to study graph products is that, when dealing with large or complex graphs that have the structural property of being some product, it is often easier to work with the components of the graph product rather than the graph as a whole. This decomposition can be particularly useful in how they relate to persistent homology methods and spectral methods. In particular, we were interested in how much topological information about the whole graph we can recover by just analyzing the components. This is for example the content of Proposition 3, 4, and 5 in the original manuscript. In Proposition 3, we show that the product of vertex filtrations is the max of vertex colors. In Proposition 4, the product of edge filtrations is the natural edge coloring on the product. These two propositions give a natural way to decompose two color-based filtrations of graph products in terms of its components. Finally, Proposition 5 gives a way to compute the 0-dimensional persistence diagram of the graph product based on just its components.

We have also added a description of these motivation for graph products at the beginning of the section.

Section 5 seems something new, although I am not sure about its applicability as such.

We thank the reviewer for raising this comment. Please see our explanation for the motivation of graph products in the response to the previous comment.

I didn’t quite understand the overall point being made by Section 7.

We thank the reviewer for raising this question. The overall point of this section is to analyze the persistent homology (PH) and Euler characteristics (EC) of two types of graph representations of a molecule - the Bond Coloring Representation and Bond Counting Representation. The purpose is to analyze under which conditions do the two representations have the same expressivity (ie. Proposition 7 and 8) and when they differ (ie. the remarks near the end of the section). The point we make is that neither subsumes the other, but they are equivalent under specific cases.

We are grateful to Reviewer qz4P for their time and detailed comments. We reply to your comments/questions below.

How does the computational complexity of $RePHINE^{Spec}$ compare to existing methods in practice, and what are the memory requirements for storing the additional spectral information?

We thank the reviewer for raising this question. Compared to RePHINE, $\operatorname{RePHINE}^{Spec}$ needs to store an additional $\gamma$ parameter. While the memory capacity may seem daunting (for example, for a connected tree with 4000 vertices that die at the same time, each $\gamma$ parameter would have a list of 3999 values). In practice, there are various implementation techniques to optimize the memory needed to be used. For example, if multiple vertices die at the same time in the same component, then they really just need to point to the same list. In practice, there is also no need to calculate the entire list of non-zero eigenvalues, often just computing the smallest non-zero eigenvalue (ie. the Fiedler value / algebraic connectivity) can already give a lot of information. There are many optimized methods to do this, such as minimizing the Ralyeigh quotient iteration, etc.

Can the authors provide empirical evidence that the improved expressivity of $RePHINE^{Spec}$ translates to better performance on real-world tasks?

We greatly appreciate your question. We focus here on several key theoretical underpinnings from multiple perspectives to bridge prominent gaps in our understanding of various TDA methods from a machine learning perspective. We fully agree that a comprehensive empirical investigation would be useful, and hope that the theoretical insights presented in this work pave way for such detailed empirical investigations in future studies.

Could the theoretical framework developed in this paper be extended to handle dynamic or temporal graphs where the topology evolves over time?

We thank the reviewer for raising this question. The answer we believe for temporal graphs is yes. The main idea is that $\operatorname{RePHINE}^{Spec}$ is a general filtration method that one can consider augmenting any model that uses color-based PH with, and weaker invariants are general filtration methods that one can consider changing any model that uses color-based PH with to address for computational costs. There are existing methods that have applied persistent homology in temporal network analysis [1]. While the authors used a form of ``zigzag persistence", it shows promise that the zigzag persistence may be combined with color-based filtrations to apply to temporal networks.

Many thanks for your insightful review and constructive feedback. We hope your concerns have been satisfactorily addressed, and if so, would appreciate if you could revisit your score to reflect the same. We are also committed to engaging further if you have any additional questions, concerns, or suggestions.

[1] Audun Myers, David Muñoz, Firas A Khasawneh, and Elizabeth Munch, Temporal network analysis using zigzag persistence

Dear Reviewer qz4P,

As the discussion period is ending soon, we wanted to check-in again. If you have any specifc questions or concerns, we would be happy to address them. Thank you again for your feedback and help to improve the paper.

How sensitive are the proposed descriptors (especially RePHINESpec) to variations in parameters such as filtration type or depth?

We thank the reviewer for raising this question. In our original submission, we already noted that the stability of persistent homology was shown by Cohen-Steiner et al. [3]. Furthermore, we showed in the original submission that $\operatorname{RePHINE}$ is stable under our proposed bottleneck distance, meaning that tiny variations in the filtration functions used will not drastically change the output of $\operatorname{RePHINE}$. What we proved is actually somewhat stronger - we showed that the bottleneck distance of any two RePHINE diagrams on the same graph are bounded in terms of some variant of the $\ell^{\infty}$ metric on the filtration functions that created them. This is a global statement of stability, whereas it suffices for local stability to hold in applications.

Thanks to this question raised by the reviewer (and also reviewer NVLY), we have looked into the stability of $\operatorname{RePHINE}^{Spec}$ and $\operatorname{EC}$. In the new updated manuscript, we have proven that $\operatorname{RePHINE}^{Spec}$ is locally stable under the assumption that the functions $f_v: X \to \mathbb{R}$ (which assigns each vertex color type with a real value) and $f_e: X \times X/\sim \to \mathbb{R}_{>0}$ (which assigns each edge volor type with a positivie real value) are both injective. When injectivity does not hold, we have also provided a counter-example (Appendix A.3, Example 1) to show that local stability fails.

To discuss the word local, we needed a notion of topology on $(f_v, f_e)$. The convenient choice for us turned out to be thinking of $(f_v, f_e)$ as an element of $\mathbb{R}^{n} \times \mathbb{R}^m$ (where $n$ is the size of the vertex coloring set $X$ and $m$ is the size of the correspondent edge coloring set). In this case, $f_v$ being injective is the same as saying its coordinates have no repeated entries. The main idea behind proving the local stability of $\operatorname{RePHINE}^{Spec}$ was that the proof for the stability $\operatorname{RePHINE}$ may be adapted in a convex open neighborhood of $f_v$ in $\mathbb{R}^n$ and $f_e$ in $\mathbb{R}^m$.

We have also found that the authors of [4] have proven the stability for EC already in their Proposition 3.2.

Many thanks for your insightful review and constructive feedback. We hope your concerns have been satisfactorily addressed, and if so, would appreciate if you could revisit your score to reflect the same. We are also committed to engaging further if you have any additional questions, concerns, or suggestions.

[1] Mathieu Carrière, Frédéric Chazal, Yuichi Ike, Théo Lacombe, Martin Royer, and Yuhei Umeda, PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures

[2] Audun Myers, David Muñoz, Firas A Khasawneh, and Elizabeth Munch, Temporal network analysis using zigzag persistence

[3] David Cohen-Steiner, Herbert Edelsbrunner, and John Harer, Stability of Persistence Diagrams

[4] Paweł Dłotko and Davide Gurnari, Euler characteristic curves and profiles: a stable shape invariant for big data problems

We are grateful to Reviewer ZQBM for their time and detailed comments. We reply to your comments/questions below.

The paper’s applications are primarily molecular. Can this be expanded to other fields that use GNNs, such as social network analysis?

We thank the reviewer for raising the question. While our focus is primarily molecular, there are numerous other models that incorporate persistent homology (PH) diagrams that can be easily adapated to our proposed models in the manuscript. One such example of PersLay [1], which is a simple and flexible layer in neural network to process topological data and has been used in application to analyze data from the social media site Reddit. Another example is the application of persistent homology in temporal network analysis [2]. While the authors used a form of ``zigzag persistence", zigzag persistence may be combined with color-based filtrations to apply to temporal networks. The main idea is that $\operatorname{RePHINE}^{Spec}$ is a general filtration method that one can consider augmenting any model that uses color-based PH with.

How does the use of topological descriptors like RePHINESpec compare with traditional GNN methods, particularly in tasks that do not require high expressivity?

We thank the reviewer for raising this question. What filtration methods like $\operatorname{RePHINE}^{Spec}$ and $\operatorname{RePHINE}$ provide are not just expressibity, but also access to global information. Traditional GNN methods have an inherent inductive bias towards local information and can run into data/network homophily, so augmenting global information is what we wanted to emphasize with this work. For example, for any work where you want to count the number of cycles in a graph, the presence of these kind of TDs are still needed.

The paper proposes several filtrations for molecular graphs. What criteria were used to select these?

We thank the reviewer for raising this question. We considered two types of filtrations - one based on vertex coloring and other based on edge coloring, and several topological descriptors along these two types of filtrations. Our original goal was to look at how topological descriptors interact with color-based filtrations of graphs, which most commonly includes persistent homology (PH). We noted that RePHINE is a more expressive version of PH, so we also included that as well. There is a classical theorem (see Appendix B for a more precise description) that asserts that zeroth homology is the kernel of graph Laplacian, so a natural generalization for us was to consider the non-zero eigenvalues of the Laplacian. Finally, we also wanted to consider weaker invariants to address for computational costs, hence we considered the Euler characteristics (EC) and the max versions of $\operatorname{RePHINE}^{Spec}, \operatorname{RePHINE},$ and $\operatorname{EC}$.

What is the STABILITY OF REPHINE DIAGRAMS? explanation is missing. Can the authors provide practical examples where stability significantly impacts application outcomes?

We thank the reviewer for raising this question. By the ``stability of RePHINE diagram", we mean that RePHINE diagrams are robust to slight perturbations in the input filtrations. This notion is made more precise in the statement of Theorem 4 in the original submission. Specifically, Theorem 4 states that the bottleneck distances (Definition 7) of two RePHINE diagrams on the same graph may be explicitly bounded in terms of the $\ell^{\infty}$-norms of the input functions. We have also added further explanation to this before the statement of Theorem 4 in the edited submission.

There are practical examples where the stability of persistent homology (PH) have benefitted applications outcomes. In the original paper by Cohen-Steiner et al. [3] proving bottleneck stability, the authors applied their stability results to the estimation of the homology of a closed subset of a metric space from a finite point sample, and make concrete statements about the stability of barcodes (a representation of persistence diagrams) with relevance to image processing and morphology.

Dear Reviewer ZQBM,

As the discussion period is ending soon, we wanted to check-in again. We have addressed all your questions in our reply. Notably, we expanded on the stability section, included a new discussion on the (local) stability of $\operatorname{RePHINE}^{Spec}$, and found a reference for the stability of EC.

We hope your concerns have been satisfactorily addressed, and if so, would appreciate if you could revisit your score to reflect the same. If you have any specifc questions or concerns, we would be happy to address them. Thank you again for your feedback and help to improve the paper.