Higher-Order Graphon Neural Networks: Approximation and Cut Distance

Authors: Daniel Herbst, Stefanie Jegelka
Primary area: learning on graphs and other geometries & topologies
Keywords: graph neural networks, invariant graph networks, universal approximation, graph limits, graphons, transferability, homomorphism densities, machine learning theory.

Abstract:
Graph limit models, like *graphons* for limits of dense graphs, have recently been used to study size transferability of graph neural networks (GNNs). While most literature focuses on message passing GNNs (MPNNs), in this work we attend to the more powerful *higher-order* GNNs. First, we extend the $k$-WL test for graphons (Böker, 2023) to the graphon-signal space and introduce *signal-weighted homomorphism densities* as a key tool. As an exemplary focus, we generalize *Invariant Graph Networks* (IGNs) to graphons, proposing *Invariant Graphon Networks* (IWNs) defined via a subset of the IGN basis corresponding to bounded linear operators. Even with this restricted basis, we show that IWNs of order $k$ are at least as powerful as the $k$-WL test, and we establish universal approximation results for graphon-signals in $L^p$ distances. This significantly extends the prior work of Cai & Wang (2022), showing that IWNs—a subset of their *IGN-small*—retain effectively the same expressivity as the full IGN basis in the limit. In contrast to their approach, our blueprint of IWNs also aligns better with the geometry of graphon space, for example facilitating comparability to MPNNs. We highlight that, while typical higher-order GNNs are discontinuous w.r.t.\ cut distance—which causes their lack of convergence and is inherently tied to the definition of $k$-WL—transferability remains achievable.

Decision: accept (spotlight)
Reviewer scores: 8, 8, 8, 8, 8

Reviews and discussion:
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creation date:
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creator: reviewer_p53n

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summary:
This paper proposes the Invariant Graphon Network (IWN) architecture for graphons as an extension of the Invariant Graph Network GNN architecture for discrete graphs. The paper's results are framed as a more natural extension and generalization of the related work of Cai & Wong 2022. The authors provide an overview of the relevant graphon theory and an in-depth introduction to a novel signal-weighted homomorphism density concept, providing a comprehensive characterization of weak isomorphism of graphon-signals using this definition, as well as showing a convergence equivalence to cut distance convergence. A generalization of linear equivariant layers is then introduced, through measure-preserving functions rather than permutations in the discrete case. Several central theorems are shown about this function space, including invariance under discretization, construction of a canonical basis, and analysis of dimension. The IWN architecture is then built and proven to be an instance of an IGN-small network. Continuity with regard to $\delta_p$ is then shown, as well as lack of continuity for $\delta_{\square}$ unless the activation function is linear and thus expressivity is lost. Finally, expressivity of IWNs are shown in the sense that order $k+1$ IWNs are able to distinguish with the strength of $k$-WL for graphons.

strengths:
- There is very thorough analysis of relation to previous work throughout this paper. This work's placement in the literature is clear.
- Original contributions are outlined clearly and presented in a cohesive framework.
- The ideas and results are very original, with several novel concepts defined. The definitions are shown to make sense through the strength of the theory, providing good evidence that these ideas are pointing toward a very useful framework.
- The proof sketch narration is generally clear, and the full proofs in the appendices are nicely written. It is clear there was a good deal of care taken toward ensuring the quality of presentation for these main results.
- In relation to the cited related works, the results shown in this paper are a significant contribution to the community.

weaknesses:
- The discussion of the $\delta_{\square}$ discontinuity limiting transferability seems like it could be more complete. Especially as the definitions leading to this point are novel, it seems important to justify this not being a product of a definitional choice that could be reworked and rather a facet of the true underlying problem. Reading the appendix helped clear up my confusions here, but it would be ideal for the section itself to contain just a bit more of that insight.
- Due to the heavy theoretical nature, this paper requires a very high degree of engagement to fully understand the stated results. It would be much easier to digest the main ideas on an initial reading with more time, even a short section, dedicated to showcasing some of the concepts you develop, perhaps in some experiments. While convergence experiments such as those in Cai & Wong 2022 are not relevant here, I think there are great possibilities for a showcase of the continuity result and the expressivity result which would really elevate the overall impact of the paper.

One small typo:
- In Section 6, Paragraph 1, the phrase "...that IWNs, as a subset their of IGN-small, retain..." appears. Removing the "their" would fix this.

questions:
- You state that IWNs being only continuous for the weaker $\delta_p$ distances, combined with the lack of compactness, prevent the approach from being applied to the entire space. Is there anything interesting to be said about limited regions of this space under the $\delta_p$ distance? Is there any hope for an interesting bound in the $\delta_{\square}$ distance regardless of discontinuity, or is the behavior fundamentally unworkable?
- Have you attempted to build some toy examples of the model, even without any sort of training?
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creation date:
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creator: reviewer_cdhd

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summary:
This paper studied Invariant Graphon Networks, a generalization of Invariant Graph Networks (IGN) from Maron (2018) to graphons. The authors show that Invariant Graphon Networks can be characterized using a subset of the IGN basis functions, due to additional discretization invariance. The authors proved that Invariant Graphon Networks of order $k+1$ are at least as powerful as the $k$-dimensional Weisfeiler-Leman test for graphons proposed in Böker (2023). These expressivity results extend the convergence results in Cai and Wang (2022), by making use of signal-weighted homomorphism densities as a new tool for the analysis. The authors show that Invariant Graphon Networks are typically discontinuous with respect to cut distance, in contrast to the continuity results of MPNNs defined on graphons (Levie 2023). This implies a negative result on the size transferability of IGNs.

strengths:
1. The paper makes novel contributions on extending IGNs to graphons and studying their expressivity and continuity properties.

2 .The paper is clearly written and well motivated.

3. The notion of signal-weighted homomorphism densitiy is an interesting combination of the graphon-signal space in Levie (2023) and homomorphism densities in Böker (2023).

weaknesses:
Practical consequences: The authors remark that since Invariant Graphon Networks (IWNs) are discontinuous with respect to the cut distance, IWNs can assign vastly different values to similar graph inputs (line 96). It will be nice to substantiate this remark using empirical evidence, either through observation made in existing works using IGNs or original experiment studies.

questions:
1. The definition of signal-weighted homomorphism densities (Defn 3.1) seems standard from the graphon literature, see for example homomorphism density on weighted graphs per equation 5-6 in [1], or various notions of homomorphism densities on multigraphs per chapter 17 in [2]. Can the authors remark the connections to these existing definitions?

2. Invariance under discretization: As stated in Lemma 4.3 and Lemma D.1, invariance under discretization implies the output $L(U)$ is constant almost everywhere. This also leads to a much smaller dimension of the space of linear equivariant layers in Theorem 4.2. Does it make sense to consider relaxing the notion of invariance under discretization (i.e., approximate invariance). Will the authors expect this leads to stronger expressivity results than those stated in Section 5?

3. Nontrivial IWN in Proposition 5.1: Proposition 5.1 (2) states that IWN is continuous with respect to the cut-distance if and only if the pointwise acitivation function is a linear function, where the authors treat it as "trivial" (Line 444). However, such linear IWN may not be trivial if the intermediate layers use higher-order tensors. Can the authors clarify this?

4. The effect of pooling: In the current definition of Invariant/Equivariant Graphon Networks (Defn 4.4), the network consists of linear equivariant maps interleaving with pointwise nonlinearity. In the original works by Maron et al. [3], [4], an invariant pooling operation is often used in practice. Can the authors comment on the role of pooling on the continuity property of IGNs?


References:

[1] Borgs, Christian, et al. "Counting graph homomorphisms." Topics in Discrete Mathematics: Dedicated to Jarik Nešetřil on the Occasion of his 60th Birthday. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006. 315-371.

[2] Lovász, László. Large networks and graph limits. Vol. 60. American Mathematical Soc., 2012.

[3] Maron, Haggai, et al. "Invariant and Equivariant Graph Networks." International Conference on Learning Representations. 2018.

[4] Maron, Haggai, et al. "Provably powerful graph networks." Advances in neural information processing systems 32, 2019.
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creation date:
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creator: reviewer_1fqg

<content>
summary:
This work proposes Invariant Graphon Networks (IWNs), which is an extension of Invariant Graph Networks (IGNs) defined on discrete graphs, to *graphons*. They study its separation power in relation to the $k$-WL test and its approximation capability for functions defined on graphons with vertex features.

They extend the notion of homomorphism densities (which, in turn, is an extension of *homomorphism counting*) from plain graphons to graphons with vertex features, and show that this notion can be used to determine the $k$-WL separation power of graphon neural networks. They then use it to show that their construction yields architectures at least powerful as the $k$-WL test, which in turn implies that it has universal approximation power of functions defined on vertex-featured fraphons.

Notably, their construction is more economic than that of the discrete IGN counterpart, due to a reduced basis for linear layers, leading to slower asymptotic growth in the number of basis elements required.

Finally, they expose a fundamental limitation of IWNs, and thus of IGNs, in that they are discontinuous with respect to the cut distance, which limits their applicability, and in particular their generalization capability, in certain learning scenarios. This is unlike MPNNs, which are continuous w.r.t. the cut distance.

strengths:
I believe this work offers significant contribution to the community, in that it provides useful theoretical tools to analyze neural networks on vertex-featured graphons, and these tools are already shown useful in this work in their universal approximation and their discontinuity results. The theoretical analysis of IWNs' separation power with the $k$-WL test could guide future architectural innovations or alternative GNN designs on graphons.

weaknesses:
The paper does not provide numerical experiments. Nonetheless, it provides valuable theoretical contribution that can potentially explain empirical results, motivate the design of architectures and also lead to new theoretical insights on GNNs. While numerical experiments could have provided practical insights into the theoretical results, the theoretical results themselves have high potential for practical impact.

questions:
### Minor comments
1. Ln. 140-141: "Note that $\delta_{\square} \leq \delta_p$." - Does it follow from the L1-Lp inequality? This could benefit a short explanation or a citation.
2. Ln. 151: "If simple edges are further sampled..." - unclear definition of G(W,X).
3. Eq. (3), (4): Perhaps missing "as $n \to \infty$".


### Suggestions
1. Ln. 135-141: Some readers may not be familiar with the concept of cut distance. A brief explanation, perhaps as a short note or in a footnote, could benefit readers as well as help highlight your results on discontinuity.
2. An intuitive explanation for the cut-distance discontinuity, possibly with an illustrative example, could help comprehension.
3. Ln. 314 and the subsequent paragraph: To reassure the reader, I recommend stating explicitly that the dimension does not depend on $n$, and that the following theorem shows that this independence carries through to the case of graphons.
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creation date:
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creator: reviewer_fxhj

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summary:
The paper extends IGNs to the graphon-signal space by introducing Invariant Graphon Networks (IWNs). It provides a generalization of linear equivariant layers to measure spaces, specifically the unit interval [0,1]. The authors introduce signal-weighted homomorphism densities to extend the concept of homomorphism densities to graphon-signals. They prove expressivity results for IWNs, showing that IWNs of order $k+1$ are at least as expressive as $k$-WL for graphons and that they are universal approximators on any compact subset of graphon-signals. This extends the standard results from Maron et al. from graphs to graphons. The paper also discusses the continuity properties of IWNs, highlighting that they are discontinuous wrt the cut distance.

strengths:
- The paper seems sound and the theoretical results extend the results from Cai & Wang (2022), Böker et al. (2023), and Levie (2023)
- The authors connect their work to previous studies, such as those by Cai & Wang (2022) and Levie (2023), situating their contributions within the broader context of graph(on) neural networks

weaknesses:
- It's not clear what the consequences for ML on graphs are
- The non-continuity of IWNs wrt to sampling is mentioned and proved, but it is not clear what consequences it has. Also, the authors mention it as a rather negative point, however, this is not clear to me. Do you have an example where this property hinders size generalization? Or is it rather a worst-case property that doesn't affect IGNs in practice?

questions:
- Could you discuss your non-continuity results better? What do you think are the consequences?
- Also: how is it possible that the IWNs from Cai & Wang (2022) preserve convergence when sampling graphs from (sufficiently regular) graphon, but yours doesn't? Cai & Wang (2022)  also consider non-linearities, so is the reason that you consider graphon-signal?
- It appears that many results from Böker et al. (2023) and Levie (2023) were reproven using the same techniques for graphon-signal or IWNs. Could the authors discuss their technical contributions. 


## Additional Comments
While the theoretical results appear to be correct (I didn't check all proofs) and generalize some results on graphon neural networks, it is not clear what the broader impact or significance is. I believe answering some of my questions would help to understand this as a reader. I am happy to increase my score if that is done.
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creation date:
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thread_type:
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creator: reviewer_mdwf

<content>
summary:
This paper introduces Invariant Graphon Networks (IWNs), an extension of Invariant Graph Networks (IGNs), to analyze graphons (limit objects of dense graphs) and investigate their expressivity and continuity properties. IWNs leverage a subset of the IGN framework focused on bounded linear operators, allowing them to approximate graphon functions with an expressivity at least as powerful as the k-Weisfeiler-Leman (k-WL) test.

strengths:
This paper introduces Invariant Graphon Networks (IWNs) as an extension of Invariant Graph Networks (IGNs) specifically tailored for graphons, significantly enhancing their expressive power and providing new universal approximation properties for graphon signals. The work advances the theory in graph neural networks by rigorously defining IWNs and aligning them with theoretical results through the introduction of signal-weighted homomorphism densities.
- The paper presents a thorough comparison of IWNs with existing graph neural networks, including Message Passing Neural Networks (MPNNs) and other IGNs, addressing size transferability and continuity aspects. It establishes that IWNs maintain at least the same expressive power as the k-Weisfeiler-Leman (k-WL) test, a known benchmark in GNN expressivity.
- The authors have provided extensive mathematical proofs.

weaknesses:
The absence of empirical experiments and practical application scenarios is the key weakness of this work. Without empirical validation, it is difficult to gauge how IWNs perform on real-world datasets, whether the theoretical advantages translate into meaningful results, and how they compare to established GNN models in practice. IWNs could be applied in scenarios where we need to capture dense graph structures or analyze graphs of varying sizes, which could benefit from their extended expressivity and approximation capabilities.
- It is unclear whether the mathematical assumptions hold across diverse real-world datasets. This could limit the applicability of IWNs to more constrained datasets where these assumptions are valid.
- IWNs are discontinuous with respect to the cut distance. This discontinuity implies that minor structural changes in the discrete graph could lead to significant output changes in the IWN, unlike Message Passing Neural Networks (MPNNs).
- Adapting IWNs to discrete graphs might involve computational overhead, as the model is designed for continuous graphon spaces rather than finite graphs.

questions:
Please see the weaknesses part.
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title:
General Comments by Authors

comment:
We thank all reviewers for their thoughtful and constructive feedback and for **unanimously supporting our work**! We have uploaded a major revision of the script, addressing the reviewers’ feedback. See the highlighted parts for specific updates. Most importantly, we have made the following changes:

1. Throughout the paper, we tried to make the motivation and practical consequences of our work clearer, specifically by **giving a better intuition for the cut distance discontinuity**, as suggested by reviewers fxhj and 1FQG. The cut distance discontinuity is *not* visible when considering finite graphs up to a certain size (as all norms are equivalent there), but only when limits to a graphon are taken. Most prominently, the cut distance discontinuity can be observed when applying an IWN to simple graphs sampled from a graphon-signal. While the simple graphs converge to the underlying graphon-signal in the cut distance (Eq. 8), the function values of the IWN do not converge. This is similar to the result of [1] on non-convergence of IWNs under their “edge probability discrete model”. We have rewritten the section on continuity (now § 5.2) and have dedicated a paragraph (L454-459) to an intuitive explanation.
2. An intriguing point brought up by reviewer p53n is that it is “important to justify [that cut distance discontinuity is not] a **product of a definitional choice that could be reworked** and rather a facet of the true underlying problem”. In the updated version, we have made clearer that the cut distance discontinuity is **inherently tied to the definition of the $k$-WL test**: The $k$-WL test considers multigraph homomorphism densities, which are cut distance discontinuous, and therefore, *any* ML model defined on graphon-signals which is $k$-WL expressive (e.g., also the higher-order models from [3]) would exhibit this discontinuity. This comes from a fundamental difference in the way that 1-WL and $k$-WL consider weighted edges, which are treated as actual weights by 1-WL and only act through the shift operator of the graph(on), while $k$-WL (and IWNs) see them as edge features, capturing their entire *distribution*. We have clarified this in § 5.2 (L460-467). For this discussion, we also extend the $k$-WL test, for which we previously just used the definition for *graphons* from [2], to graphon-signals (see Theorem 3.5 and Appendix C.4).
3. We **significantly refine the discussion of the transferability of IWNs**: As the cut distance discontinuity implies that IWNs do not converge to their graphon limits, this means that the standard approach to prove transferability by invoking the triangle inequality does *not* apply to IWNs. However, it turns out that this **discontinuity can be “fixed”**, in the sense that any IWN can be approximated arbitrarily well by a cut distance continuous function, for which transferability can be obtained. As such, IWNs are just as transferable as MPNNs in the worst case. See Theorem 5.6 for the formal statement, and Appendix E.6 for its proof. We believe that **this observation makes our contribution even stronger**: For practical purposes typically the actual transferability is of greater interest than the convergence to the graphon limit, which is usually just an intermediate step.
4. While we primarily view the work as a theoretical contribution, we agree with the reviewers that **experiments** can improve understanding of the concepts. We added a toy experiment in Figures 1-3, showing the continuity and transferability of IWNs vs. MPNNs, and provide details in Appendix G.
5. To make space for the experiments and discussion of transferability, we had to move some of the proof sketches (Theorem 4.2 and Theorem 5.2) to the appendix.

See below for individual responses to the reviewers.

**References**

[1] Chen Cai and Yusu Wang. Convergence of Invariant Graph Networks. In Proceedings of the 39th International Conference on Machine Learning, pp. 2457–2484. PMLR, June 2022.

[2] Jan Böker. Weisfeiler-Leman Indistinguishability of Graphons. The Electronic Journal of Combinatorics, 30(4):P4.35, December 2023. ISSN 1077-8926. doi: 10.37236/10973.

[3] Christopher Morris, Martin Ritzert, Matthias Fey, William L. Hamilton, Jan Eric Lenssen, Gaurav Rattan, and Martin Grohe. Weisfeiler and leman go neural: Higher-order graph neural networks. In Proceedings of the AAAI conference on artificial intelligence, volume 33, pp. 4602–4609, 2019.
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Metareview:
metareview:
The authors propose Invariant Graphon Networks (IWNs) to generalized Invariant Graph Networks (IGNs) to graphons. The authors prove that IWNs of order (k+1) is at least as powerful as k-dimensional Weisfeiler-Leman (WL) test for graphons. The derived expressivity extend results in Cai and Wang (2022). Additionally, the authors show that IGNs are discontinuous with respect to cut distance, in contrast to the continuity results of MPNNs on graphons (Levie 2023). The Reviewers agree that it is a good submission for ICLR'2025. However, the lack of empirical supporting evidence remains as a concern for the Reviewers. We urge the Authors to incorporate the Reviewers' comments and discussions in the rebuttal to the updated version.