TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks

Thank you for your thoughtful comments. We believe they strengthened our work. We are happy to read you found the work accessible for someone not coming from TDL.

Note on Reviewer's Summary: GCCNs and TopoTune extend not only GNNs but also non-GNN neural networks. Our framework is also not limited to simplicial and cellular complexes—it generalizes to combinatorial complexes and other higher-order relational structures. While our experiments focus on simplicial and cellular domains, the methodology itself is broadly applicable.

Responding to performance concerns: The values in Table 1 serve a benchmarking purpose rather than direct comparison to end-to-end optimized GNNs. TopoBenchmark ensures controlled comparisons by standardizing key components—encoder/decoder design, readout, dataset splits, omission of edge features, and so on. While this impacts absolute performance, it enables a controlled evaluation of architectural differences, where we see GCCNs outperform prior TDL works and standard GNNs when subjected to these constraints.

Explanation of proofs: We agree that it is important to briefly describe the idea of the proof. In the updated manuscript, we will add: “Proof of Prop. 4.1 relies on setting the $\omega_\mathcal{N}$ of a GCCN to a simple, single-layer convolution. Proof of Prop 4.2 hinges on the node-wise permutation equivariance of the $\omega_\mathcal{N}$ and the permutation invariance of the inter-neighborhood aggregation. Proof of Prop. 4.3 shows that GCCNs surpass CCNNs in expressivity by relating CCNNs to WL and GCCNs to $k$-WL on augmented Hasse graphs.”

Weaknesses

• Beyond section 4, the general construction (GCCNs) are introduced in the introduction as part of the contributions both conceptually and empirically. Introducing them before necessary TDL background is tricky–we would appreciate any input on this. A GCCN is pictured in Fig. 1, showing how it is built from $\omega_\mathcal{N}$ (ex.: GCN, see caption line 70).
• Experiments: We have uploaded an anonymized version of the repository here: https://anonymous.4open.science/r/TopoBench-1F1C/topobench/nn/backbones/combinatorial/gccn.py. Due to the sheer amount of dataset/domains considered, it would be too expensive to run hyperparameter tuning for each of the roughly 50 GCCNs being tested. While we agree tuning would certainly lead to better performance, we argue these results are sufficient to show the superior performance of GCCNs over CCNNs.

Questions

1. Addressing in two parts:

• Datasets: The datasets, albeit not recent, were chosen because they are used in TopoBenchmark and thus represent the current norms for benchmarking TDL models. In the future, we will continue testing GCCNs as new datasets inevitably appear in TopoBenchmark. See Q2 for added larger datasets.
• Models: we chose the base architectures largely based on the vanilla GNNs that are often used as inspiration for TDL models. For example, GAT is the inspiration for CAN (Giusti et al.) and GSAN (Battiloro et al. arxiv.org/abs/2309.02138), GCN for SCNN (Maosheng et al. arxiv.org/abs/2110.02585), and GIN for CWN (Bodnar et al., arxiv.org/abs/2106.12575). We aim to show the TDL community how simple it can be to leverage pre-existing infrastructure from the well-established GNN community. We now also include two more recent GNNs (see Q2).

2. We improve the strengths of our empirical results.

• Dataset-wise, we now include 3 larger node-level benchmark datasets (Amazon Ratings, Roman Empire, Minesweeper) that our machine can support memory-wise and that CCNNs have previously been benchmarked on (due to strict word limit here, table can only be in paper). Summary: GCCNs achieve similar performance to regular CCNNs, outperforming them by a significant margin on Minesweeper. We note that all TDL models are constrained by available liftings, as large graph-based datasets significantly increase in size (see arxiv.org/abs/2409.05211 for active research efforts here).
• Model-wise, we now include experiments with GATv2 and PNA in the cellular domain. Results (which we will include in the updated paper) show how the GCCNs built with these models perform consistently well across node-level and graph-level tasks on the cell domain, often <1$\sigma$ of best standard-GNN GCCN, but only outperform them on MUTAG.

3. Lifting generally does improve performance, as seen in comparisons between CCNNs and vanilla GNNs (TopoBenchmark Table 1). We will add a row in our Table 1 showing that standard GNNs match GCCNs/CCNNs in 3 out of 8 datasets, but never outperform them. We also mention very recent work (e.g., Battiloro et al. arxiv.org/abs/2405.15429) showing that simple, general-purpose TDL models outperform GNNs heavily tailored for specific tasks (e.g. molecules).

To conclude, your suggestions helped us summarize theoretical results, contextualize contributions w.r.t. GNNs, and expand our experiments. Please let us know if any concerns remain.

Thank you for your review. We are happy to read that you believe there is potential for significant impact and acceleration of TDL.

We answer your main question here. Yes, any neural network (not even necessarily a GNN) can be easily incorporated into TopoTune. Practically speaking, as long as the neural network can be imported as a PyTorch module, it can be selected as the chosen base architecture ($\omega_\mathcal{N}$) when defining the GCCN. As such, selecting a model like ChebNet would simply mean choosing a spectral graph network as the $\omega_\mathcal{N}$ function. In this case, features inside each neighborhood would first be spectrally updated (step B of Fig. 1) and then neighborhood-level features would be spatially aggregated (step C of Fig. 1).

We will better emphasize this fact in the paper at line 324 col 1: “Differently from the work in Hajij et al., 2023, the fact that GCCNs can have arbitrary neighborhood message functions implies that non message-passing TDL models can be readily defined. For example, one could choose $\omega_\mathcal{N}$ to be a spectral graph neural network such as Defferrard et al., 2016.”

Please let us know if you have any other questions.

Thank you for your helpful feedback. We address comments and questions below.

(Weakness 1) Hodge Theory and Spectral Filtering: Thank you for your thoughtful comment. While the cited works integrate information across simplices, they are restricted to simplicial complexes due to their reliance on cohomology for defining Hodge Laplacians (not possible for hypergraphs) and the PSD property of Flower-Petals Laplacians (not possible for cell complexes or hypergraphs). Our approach, in contrast, generalizes to diverse higher-order domains without requiring specific spectral notions (e.g., no cohomology theory nor Flower-Petals Laplacian have been developed yet for general combinatorial complexes). However, any spectral model—whether graph-, simplicial-, cellular-, or hypergraph-based—can be used as $\omega_\mathcal{N}$​ (step B, Fig. 1) and combined with spatial aggregation (step C). This highlights the flexibility of GCCNs and TopoTune in enabling new topological architectures. We will clarify that spectral networks can serve as base architectures for non-message-passing models.

(Weakness 2) Experiments in simplicial and cellular domains: You are correct that we limit our experiments to these two domains in part because of computational cost and in part because of the lack of available “standard” liftings for hypergraphs and combinatorial complexes. However, we do in fact compare GCCNs to existing models by comparing the models to the “Best available model in TopoBenchmark”. These models include CWN and MPSN. In the revised version, we will specify which model is the best available model in TopoBenchmark, and provide a complete list of all models included in TopoBenchmark (and thus that we compare against). We emphasize that the purpose of Table 1 is exactly to evaluate the method’s efficacy across different topological domains, with different choices of base architecture ($\omega_\mathcal{N}$).

(Weakness 3) Open problems:

• (OP 1) While we agree the datasets used in these experiments are nothing new, we argue that the unprecedented ease TopoTune provides in defining and training new architectures makes TDL a much more practical option for real-world applications, especially those for which custom GNNs have already been developed. In the updated manuscript, we will rephrase the claim to better emphasize this (line 106, col 2): “Using TopoTune, practitioners can, for the first time, easily define and iterate upon TDL models, making TDL a much more practical tool for real-world datasets (OP 1: need for accessible TDL).”
• (OP 3: need for standardized benchmarking) Our work provides a standardized benchmarking of GCCNs. Unlike traditional studies that tune a single model across datasets, we train ~50 models (varying base architectures and neighborhoods) and compare their performance across multiple datasets and topological domains. This systematic approach is a notable contribution, addressing the field’s reliance on comparisons under heterogeneous training conditions and marking a first step toward solving OP 3. We will clarify this in the contributions: “Unlike prior works that compare models under heterogeneous conditions, our systematic benchmarking provides a controlled evaluation of GCCNs across diverse architectures, datasets, and topological domains.”

Questions

1. Beyond the sweep of possible base architectures and neighborhoods, we did not do a traditional “optimization” of hyperparameters for the GCCNs in the sense that we only considered one set of training hyperparameters for each combination of GCCN and task. This set of hyperparameters was selected from the defaults proposed by TopoBenchmark. If there was no default available, we picked the lowest value considered in TopoBenchmark reported grid search. To answer your question, the CCNNs in TopoBenchmark's hyperparameters were obtained with a wide grid search, as described in Appendix C.2 of their work (https://arxiv.org/pdf/2406.06642). We will clarify this in the “Experimental Setup” subsection: “While CCNN results reflect extensive hyperparameter tuning by Telyatnikov et. al, 2024 (see that work's Appendix C.2 for details),...”

2. When we mention OP 11, we are specifically addressing the need for cross-domain attentional TDL. Just like in other areas of TDL, existing attentional works (including Ballester et al) are limited to one domain (in this case, cellular complexes). Because of its integration into TopoTune, the GCCN architecture built with a Transformer base architecture is inherently able to accommodate any topological domain.

Thank you again for your time and review. Your comments helped us clarify our discussion on spectral methods and better articulate the scope of experiments. We have also improved the clarity and nature of our contributions. Please let us know if you have further questions.

Thank you for your time as well as your positive and thoughtful review. We are happy to read that the method made sense and the contribution was well justified. We address the raised points about weaknesses and questions below.

(Weaknesses) Comparing to standard architectures: We initially only compared GCCN on topological domains for fairness reasons (comparing topological domains amongst themselves), which is why graphs were not included. However, in practice, we completely agree it is very helpful to know if a simple GNN does better. In the updated manuscript, we will add a line to Table 1 that shows the best-performing GNN tested in TopoBenchmark on each dataset. We can then see that standard GNNs achieve comparable results only in 2 of the 16 dataset/domain combinations GCCNs were tested on (PROTEINS in the cellular domain, PubMed in the simplicial domain). This will now be specified in the Results subsection “GCCNs outperform CCNNs.”

(Weaknesses, Q3) Choosing an optimal base architecture: You are correct that we do observe significant variation in performance between GNNs (see section “Impactfulness of GNN choice is dataset-specific.”, line 411 col 2, as well as Fig. 5). By better leveraging GNN works in TDL, practitioners can build off the extensive benchmarking work performed in the GNN field (see for example the benchmark study https://arxiv.org/abs/2003.00982). For example, it comes as no surprise that the base architecture of GIN is a good choice for the ZINC dataset, as that is how it appears in the GNN world as well. We leave the study of further pre-training optimization of hyperparams such as choice of topological domain and choice of neighborhoods to future work. However, we emphasize that one of the goals of a principled framework like TopoTune is to make such a study, that would be otherwise extremely hard, feasible.

(Comments) Better motivating TDL: TDL architectures perform well in general but are particularly valuable when higher-order multiway interactions matter significantly. Examples include citation networks (Battiloro et al., https://arxiv.org/abs/2309.02138) and human skeleton-based action recognition (Hao et al., https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9329123). A notable SOTA TDL model is TopoRouteNet, used in computer network modeling (Bernardez et al., https://arxiv.org/html/2503.16746v1). Recent work (Battiloro et al., https://arxiv.org/abs/2405.15429) also highlights TDL’s potential in molecular tasks, demonstrating that simple, general-purpose TDL architectures can outperform models heavily tailored for molecules. That said, benchmarking our methods on similar datasets to prior works is a crucial step toward broader application-specific adoption. We will add a note on successful TDL applications at the end of the “TDL Research Trend” section of the introduction.

Choosing tasks (Q1). TopoBenchmark consolidates the most commonly used benchmark tasks in TDL, as well as the highest-performing, most well known models in TDL. By including this variety of tasks, GCCNs benefit from a comprehensive test against the field. Going beyond that, we also specifically use the TopoBenchmark platform to ensure fair comparison amongst architectures, as all the tasks are homogenized in, for example, data split, reported performance metric, and so on (see point below).

SOTA Results (Q2). This performance gap on ZINC primarily arises from the benchmarking framework used in our work. As TopoBenchmark (Telyatnikov et al.) standardizes several components—such as encoder/decoder, readout, dataset splits and the omission of edge features—it ensures a more controlled comparison of model backbones but may come at the cost of absolute SOTA performance. Generally, high-performing GNN models, including Chromatic Self-Attention (Menegaux et al. 2023), are designed as end-to-end architectures with carefully tuned encoders and readouts, whereas TopoBenchmark enforces uniformity in these components to isolate and assess the impact of architectural differences.

In this context, while GCCN may not achieve SOTA performance, the benchmark enables meaningful insights into the role of topological architectures by minimizing confounding factors. This approach aligns with our broader goal: rather than optimizing for peak performance on individual datasets, we aim to establish a fair and standardized evaluation framework that can help guide future architectural improvements. Importantly, TopoBenchmark is flexible enough to accommodate advances in GNN design. This could look like building a GCCN with Chromatic Self-Attention as a base architecture ($\omega_\mathcal{N}$).

Conclusion. Thank you again for your review. Your feedback helped us make (key) comparisons to GNNs, better motivate TDL, and improve our contextualization of results. We appreciate your insights and believe they have made the manuscript stronger.