Title: On topological descriptors for graph products
Venue: NeurIPS 2025
Accepted: yes
Keywords: topological data analysis

graph neural networks

graph machine learning
Abstract: Topological descriptors have been increasingly utilized for capturing multiscale structural information in relational data. In this work, we consider various filtrations on the (box) product of graphs and the effect on their outputs on the topological descriptors - the Euler characteristic (EC) and persistent homology (PH). In particular, we establish a complete characterization of the expressive power of EC on general color-based filtrations. We also show that the PH descriptors of (virtual) graph products contain strictly more information than the computation on individual graphs, whereas EC does not. Additionally, we provide algorithms to compute the PH diagrams of the product of vertex- and edge-level filtrations on the graph product. We also substantiate our theoretical analysis with empirical investigations on runtime analysis, expressivity, and graph classification performance. Overall, this work paves way for powerful graph persistent descriptors via product filtrations. Code is available at https://github.com/Aalto-QuML/tda_graph_product.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies persistent homology (PH) and Euler characteristics (EC) of graph products and present several new theoretical results. The authors introduce a computationally cheaper variant of EC, called max-EC, and proves that it has the same expressive power for graphs and fore higher-dimensional simplicial complexes. It is shown that PH on graph products captures strictly more structural information than PH on individual graphs, while EC does not capture more information on graph products. The authors present new algorithms for computing PH persistence pairs (0 and 1 dimensional) for vertex/edge-level product filtrations, and new characterizations of product filtrations via colorings on graph products. As application, the authors propose to use graph product filtrations as enhanced topological descriptors in GNN.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n- To my knowledge, max-EC and the virtual node technique for boosting PH expressivity in graph products are new.  max-EC gives a lightweight, theoretically grounded alterative to EC. Virtual nodes have been used to improve expressivity of GNN, but its use in TDA seems new.\n- The combinatorial characterization of EC expressivity for graphs and simplicial complexes, as well as the study of PH/EC on graph products are highly valuable to topological data analysis. max-EC and PH might be useful for exploiting graph product structures in GNN.\n- The proofs (edge and vertex color counting arguments, filtration refinements, Betti number calculations) look sound to me. The authors acknowledge that their algorithms for persistence pairs only apply to product filtrations, not general color-based filtrations.\n\n Weaknesses:\n- It lacks discussions on computational complexity and scalability of the proposed algorithms for PH on product filtrations.\n  \n- It lacks experiments or concrete use cases where graph product aware PH improves GNN performance.\n\n- It lacks discussion on the effects of virtual nodes on additional graph features and the appropriate choice of f_v and f_e.\n\nQUESTIONS:\n- The proposed algorithms for PH on product filtrations track births-deaths inductively, which seems expensive to scale. Do the authors has any estimation of their computational complexity for large graphs?\n  \n- The authors suggest that graph products can better capture relational data structures. Could the authors give some experiments or discuss some use cases where graph product aware PH improves GNN performance? Will the improvement outweigh the cost of product filtration computations?\n\n- While using virtual nodes can boost PH expressivity, does it change or add some artificial features (like circles or cliques) to the graphs? If it does, how do the additional features affect the descriptors? How to choose f_v and f_e?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates topological descriptors on graph products,  EC and PH. Specifically, the authors prove the expressive power of EC and PH on graph products and propose a PH computation algorithm tailored for filtrations on such structures.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n1. The study of topological data analysis methods on graph products is both novel and insightful. It has the potential to inspire future work in the machine learning community.\n\n**Weaknesses**\n\n1. As a submission to a machine learning conference, the absence of experiments is a major shortcoming. Although the authors emphasize the potential of PH in neural network applications, no experimental results are provided to support this claim.\n\n2. The lack of an experimental section also affects the overall structure of the paper. In particular, Section 3 feels disconnected and could be moved to the appendix to allow the main text to focus more clearly on topological descriptors for graph products. Accordingly, the introduction should also be revised to reflect this structural shift.\n\nQUESTIONS:\nThe authors could at least propose some preliminary experimental designs, which would significantly strengthen the paper. For example, one could verify that graph products enhance the expressive power of persistent homology (PH) on synthetic graphs [1]. Additionally, for a graph classification task, it would be valuable to compare the effectiveness of PH and Euler characteristic (EC) under the graph product structure, observing the performance improvements of classification models [2].\n\n[1] Ballester R, Rieck B. On the expressivity of persistent homology in graph learning[J]. arXiv preprint arXiv:2302.09826, 2023.\n\n[2] Zhao Q, Ye Z, Chen C, et al. Persistence enhanced graph neural network[C]//International Conference on Artificial Intelligence and Statistics. PMLR, 2020: 2896-2906.\n\nLIMITATIONS:\nThe authors have already discussed the key limitations of their work.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors establish a characterization of the expressivity of persistent homology (PH) and Euler Characteristic (EC) of the filtrations over graph products. The authors also establish a characterization of EC on general color-based filtrations. Further, they show that PH of graph products captures strictly more information than PH of single graphs while EC does not. They also give algorithms to compute the PH diagrams of vertex- and edge-level filtrations on the graph product.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nThe paper is well-presented. I like the boxes around theorems.\n\nWeaknesses:\n\nThe choice of notation for $F_v$ seems slightly confusing. I think having the notation depending on $f_v$ makes more sense than $v$. $f_v$ makes sense because it is tied to a vertex $v$. However, $F_v$ is not tied to a vertex, rather it is tied to the function $f_v$.\n\nThe authors have motivated the need for graph products in a few lines towards the beginning of Section 4. However, the motivation seems lacking in concrete intuition and examples for the need of graph products. E.g., \"From an expressivity standpoint, graph products also capture information that the individual components may not have\", this line is vague and doesn't get across the idea that the authors are trying to convey.\n\nRefer to Questions.\n\nTypos:\n\nLine 65: “A graph is a the…” -> “A graph is the data…”?\n\nQUESTIONS:\n1. In line 65, shouldn’t it be $E \\subseteq V \\times V$, instead of $E \\subseteq V \\times X$?\n\n2. What is the larger picture of having defined coloring filtrations? One could have just started with two different filtration functions: first where one starts with a filtration function on the vertices and extends is piecewise linearly onto the edges, and another where all the vertices have zero filtration values and has some non-zero values on edges. I feel defining the coloring based filtrations, especially the way it is defined is rather confusing for the meaning it is conveying.\n\n3. Line 91-92: What do you mean by “filtration of a diagram”?\n\n4. The definition of $f_v$, seems to imply that one has a function $f_v$ for each vertex $v$. Moreover, since the function is not defined as $f_v : V \\to \\mathbb{R}$, but as $f_v : X \\to \\mathbb{R}$, it is understandable that each vertex has a potentially different function $f_v$. However, the writing seems to hint at the fact that $f_v$ is just one function that is defined as a filtration function on the vertices, which doesn’t seem right because the domain is not the set of vertices. Moreover, if it is so, then what is the point of the coloring function $c$? One could just consider $f_v \\circ c$ as a filtration function on the vertices which is the standard setting for PH?\n\n5. Moreover, it seems like this is an analysis of a bifiltration over a graph where $f_v$ and $f_e$ are two different filtrations.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper analyzes topological descriptors (persistent homology (PH) and Euler characteristic (EC)) and their behavior when lifting to (Cartesian) graph products, providing a thorough theoretical framework for understanding their behavior under color-based filtrations.\n\nKey contributions include:\n\n- The introduction of max PH and max EC diagrams, with proofs of bijections between vertex and edge-level filtrations.\n- A complete characterization of EC expressivity for graphs with color filtrations, showing EC and its proposed “max” variant are equally expressive yet the max version is more computationally efficient.\n- A formal study of **graph products**, demonstrating that while EC gains no new expressivity on products, PH of product graphs (with  the addition of “virtual” nodes) is strictly more expressive than for components alone.\n- Algorithms to compute **0- and 1-dimensional persistence diagrams** for product filtrations derived from vertex- and edge-level colorings, showing how the product filtration can be decomposed into simpler component filtrations.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n- **Technical depth and rigor:** The paper presents formal, well-structured theorems (e.g., Theorem 2's equivalence between EC and max EC) with detailed, convincing proofs.\n\n- **Novelty:** The approach to EC expressivity via vertex-based filtrations and their “max” counterpart for edge-based filtrations offers a compelling unification, highlighting an instructive example where PH achieves greater expressivity than EC—an interesting trade-off given EC’s computational efficiency.\n\n- **Potential impact:** By formalizing product filtrations and developing a clear calculus for them, the paper provides a solid foundation for improved topological descriptors on graph products. This framework can exploit decomposition properties and clarifies contexts where PH’s expressivity has clear advantages over EC.\n\n**Weaknesses:**\n\n- **Motivation for ML community is underdeveloped:** While the paper mentions subgraph GNNs and multi-way data, the introduction lacks concrete use cases or experiments demonstrating practical relevance. For example, the connection to relational databases (Robinson et al.) is super interesting and could be motivated further to captivate a more general audience.\n- **Justification:** Solid theoretical contribution with meaningful advances in understanding topological descriptors on graph products. It's a borderline accept because its practical ML motivation needs strengthening. I'd be open to raising my score if the authors clarified motivation and improved exposition.\n\nQUESTIONS:\n1. Can you more clearly motivate why product graphs matter for ML tasks? E.g., subgraph GNN architectures, relational data modeling—concrete scenarios or prior work where your theory would plug in? Here I think elaborating in particular on the connection to relational databases modeling via GNNs and how this relates to product graphs could take a front seat in the introduction. I think EC and PH are well-studied and motivated, but the need for the calculus and understanding these descriptors when lifting to products would be better placed here than in Section 4.\n2. Could you separate out and clarify the definition of the **max-induced edge coloring function**? Currently it's embedded in the EC diagram definition and I think it might be more clear (including in the reference in Prop 1) if this got its own definition and explanation and set up the motivation for why this produces the \"max EC diagram\" and collapses the need to deal with both vertex and edge coloring functions.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis submission provides a theoretical analysis of different topological descriptors, i.e., Euler characteristics (EC) and persistent homology (PH). Based on the definition of the EC diagram and max EC diagram for EC, and box product for graph product, it analyzes the expressiveness power of EC and PH and show that PH gains strictly more information on graph products than the individual components. Two algorithms are provided for computing PH diagrams by focusing on the product of vertex-level and edge-level filtrations.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n   - The paper focuses on the theoretical treatment of topological descriptors based on the color-based filtration and provides rigorous proofs and clear characterizations of their properties from the expressiveness perspective; it also provides algorithm frameworks for computing the PH diagrams.\n\nWeakness:\n   - The definition and theoretical formulation are dense, the provided example are not sufficiently explained, which may cause some difficulty for following for people without similar background.\n   - There is no applications for the provided TD and experimental verification for the expressiveness power of defined diagrams and proposed algorithms, which also make the practical value of the proposed TD and algorithms unclear.\n   - It only focuses on the color-based filtration on graph and lacks comprehensive comparison with other types of filtrations and expressiveness aspects;  the analysis of the proposed algorithms in Theorem 4 and 5 are insufficient, e.g., computation complexity and computational cost.\n\nQUESTIONS:\n1. What is the possible application of the proposed TDs, can you provide some experimental results for verification?\n2. Can we use the proposed descriptors to improve the traditional GNN expressiveness? Or how to use them for practical tasks?\n3. Besides the expressiveness, do we need to consider other metrics and aspects to assess / evaluate the topological descriptors? if yes, how about the results on them for the proposed EC and PH descriptors?\n4. The author discuss the persistent descriptors of a graph, what is the counterpart? Nonpersistent? what is the advantage of considering the persistent TD rather than the others?\n\nLIMITATIONS:\nNone", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}