TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
Authors: Mathilde Papillon
Guillermo Bernardez
Claudio Battiloro
Nina Miolane
Venue: ICML 2025 poster (2025)
Keywords: Topological Deep Learning
Graph Neural Network
Graph Expansion
Combinatorial Complex
Cellular Complex
Primary area: deep_learning->graph_neural_networks
TL;DR: TopoTune generalizes any architecture (Graph Neural Network, Transformer, etc.) into a Topological Neural Network that can process higher order structures on relational data.

Abstract: Graph Neural Networks (GNNs) effectively learn from relational data by leveraging graph symmetries. However, many real-world systems---such as biological or social networks---feature multi-way interactions that GNNs fail to capture. Topological Deep Learning (TDL) addresses this by modeling and leveraging higher-order structures, with Combinatorial Complex Neural Networks (CCNNs) offering a general and expressive approach that has been shown to outperform GNNs. However, TDL lacks the principled and standardized frameworks that underpin GNN development, restricting its accessibility and applicability. To address this issue, we introduce Generalized CCNNs (GCCNs), a simple yet powerful family of TDL models that can be used to systematically transform any (graph) neural network into its TDL counterpart. We prove that GCCNs generalize and subsume CCNNs, while extensive experiments on a diverse class of GCCNs show that these architectures consistently match or outperform CCNNs, often with less model complexity. In an effort to accelerate and democratize TDL, we introduce TopoTune, a lightweight software for defining, building, and training GCCNs with unprecedented flexibility and ease.

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
Mathilde Papillon' Guillermo Bernárdez' Claudio Battiloro 2 Nina Miolane *
Abstract
to capture and model higher-order interactions- crucial in complex systems like particle physics, social interactions, or biological networks (Lambiotte et al., 2019). Topologi- cal Deep Learning (TDL) (Bodnar, 2023; Battiloro, 2024) precisely emerged as a framework that naturally encom- passes multi-way relationships, leveraging beyond-gras -gapph combinatorial topological domains such simplicial and cell complexes, or hypergraphs (Papillon et al., 2023).' !
Graph Neural Networks (GNNs) effectively learn from relational data by leveraging graph symme- tries. However, many real-world systems-suchh as biological or social networks feature multi- way interactions that GNNs fail capture. Topo- logical Deep Learning (TDL) addresses this by modeling and leveraging higher-order structures, with Combinatorial Complex Neural Networks (CCNNs) offering a general and expressive ap- proach that has been shown to outperform GNNs. However, TDL lacks the principled and standard- ized frameworks that underpin GNN development, restricting its accessibility and applicability. To address this issue, we introduce Generalized CC- NNs (GCCNs), a simple yet powerful family of TDL models that can be used to systematically transform any (graph) neural network into its TDL counterpart. We prove that GCCNs generalize and subsume CCNNs, while extensive experiments on diverse class of GCCNs show that these architec- tures consistently match or outperform CCNNs. often with less model complexity. In an effort to accelerate and democratize TDL, we introduce Tune,Tune, a lightweight software for defining. building, and training GCCNs with unprecedented flexibility and ease.
In this context, Hajij et al. (2023; 2024a) have recently in- troduced combinatorial complexes, fairly general objects that are able to model arbitrary higher-order interactions along with a hierarchical organization among them-hence generalizing (for learning purposes) most of the combina- torial topological domains within TDL, including graphs. The elements of combinatorial complex are cells, being nodes or groups of nodes, which are categorized by ranks. The simplest cell, a single node, has rank zero Cells of higher ranks define relationships between nodes: rank one cells are edges, rank two cells are faces, and so on. Hajij et al. (2023) also proposes Combinatorial Complex Neural Networks (CCNNs), deep learning architectures that lever- age the versatility of combinatorial complexes to naturally model higher-order interactions. For instance, consider the task of predicting the solubility of a molecule from its struc- ture. GNNs model molecules as graphs, thus considering atoms (nodes) and bonds (edges) (Gilmer et al., 2017). By contrast, CCNNs model molecules as combinatorial com- plexes, hence considering atoms (nodes, i.e., cells of rank zero), bonds (edges, i.c., cells of rank one), and also im- portant higher-order structures such as rings or functional groups (i.e., cells of rank two) (Battiloro et al., 2025).
1. Introduction
Graph Neural Networks (GNNs) (Scarselli et al., 2008; Corso et al., 2024) have demonstrated remarkable perfor- mance in several relational learning tasks by incorporating prior knowledge through graph structures (Kipf& & Welling, 2017; Zhang& & Chen, 2018). 8). However, constrained by the pairwise nature of graphs, GNNs are limited in their ability
TDL Research Trend. To date, research in TDL has largely progressed by taking existing GNNs architectures (convolutional, attentional, message-passing, etc.) and gen- eralizing them one-by-one to a specific TDL counterpart, whether that be on hypergraphs (Feng et al., 2019; Chen et al., 2020a; Yadati, 2020), on simplicial complexes (Rod- denberry et al., 2021; Yang & Isufi, 2023; Ebli et al., 2020; Giusti et al., 2022a: Battiloro et al., 2024; Bodnar al.,
"Equal contribution Universiity Califomia Santa Barbara, USA "Harvard University, USA. Correspondence to: Mathilde Papillon <papillon @ ucsb.edu>..
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
'iimliccaal and cell complexes model specific higher-order interactions organized hierarchically, while hypergraphs model arbitrary higher-order interactions but without any hierarchy.

---

Contributions. This works seeks to accelcrate TDL re- search and increase its accessibility and standardization for TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
Layer 1
Layer + 1
H
GNi nodes nodes
WN1
M
WN1
Rank-Level Aggregation N E Nc
H N
H
Ng edges -- edges
WN2
N₂
WN2
Readout
H
9 Ng faces edges
N₂
N₂
WN3
W WN3
A.
B.
C.
B.
C.
Figure 1. Generalized Combinatorial Complex Network (GCCN), The input complex C has neighborhoods Nec (N,,,N2,Nss A. The complex is expanded into three augmented Hasse graphs 9x., i = (1, 2, 3), each with features Hv, represented as a colored disc. B. A GCCN layer dedicates one base architecture wx; (GNN, Transformer, MIP, etc.) each neighborhood. C. The output of all the architectures wN; is aggregated rank-wir wise, then updated. In this example, only the complex's 's edge features (originally pink) are aggregated across multiple neighborhoods (N2 and N3).
2021b; Maggs et al., 2024; Lecha et al., 2025), on cell com-
outside practitioners. To that end, we introduce novel joint methodological and software framework that easily enables the development of new TDL architectures in principled way-overcoming the limitations of existing works. We outline our main contributions and specify which of the field's open problems (OPs), as defined in Papamarkou et al. (2024), they help answer:
plexes (Hajij al., 2020; Giusti et al., 2022b; Bodnar al., 2021a), or combinatorial complexes (Battiloro et al., 2025; Eitan et al., 2024). Although overall valuable and insightful, such a fragmented research trend is slowing the development of standardized methodologies and software for TDL, as well as limiting the analysis of its cost-benefits trade-offs (Papamarkou al., 2024). We argue these two challenges hindering the use and application of TDL be- yond the community of experts. This is partieularly relevant as practitioners are beginning to turn to TDL for tackling rpplication-specifii scenarios, such as computer network modelling (Bernárdez et al., 2025).
Systematic Generalization. We propose the first method to systematically generalize any neural network to its topological counterpart with minimal adaptation. Specifi- cally, we define a novel expansion mechanism that trans- forms a combinatorial complex into a collection of graphs, enabling the training of TDL models as an ensemble of synchronized models. To our knowledge, this is the first method designed to accommodate many topological do- mains. (OPs 6. 11: foundational, cross-domain TDL.)
Current Efforts and Gaps for TDL Standardization. TopoX (Hajij et al., 2024b) and TopoBench(Telyattike et al., 2024) have become the reference Python libraries for developing and benchmarking TDL. models, respectively. However, despite their potential in defining and implement- ing novel standardized methodologies in the field, the cur- rent focus of these packages is replicating and analyzing existing pesssag-ppassngg CCNNs. Works like Jogl et al. (2022b;a) have instead focused on making TDL accessi- ble by porting models to the graph domain. They do so via principled transformations from combinatorial topolog- ical domains to graphs. However, although these architec- tures are as expressive as their TDL counterparts (using the Weisfeiler-Lehmen criterion (Xu et al., 2019b)), they are neither formally equivalent to generalization of their TDL counterparts. Due to collapse of rank information dur- ing the graph expansion, the GNNs on the resulting graph do not preserve the same topological information.
General Architectures. Our method induces novel wide class of TDL. architectures, Generalized Combina- torial Complex Networks (GCCNs), portrayed in Fig. 1. GCCNs (i) formally generalize CCNNs, (ii) are cell per- mutation equivariant, and (íii) are as expressive as CCNNs. (OP consolidating TDL advantages a unified theory.)
Implementation. We provide TopoTune, a lightweight PyTorch module for developing GCCNs fully integrated into TopoBench (Telyatnikov et al., 2024). (OP 4: need for software). Using TopoTune, practitioners can, for the first time, casily define and iterate upon TDL models, making TDL a much more practical tool for real-world datasets (OP I: need for accessible TDL).
Benchmarking. Using TopoTune, we create a broad class of GCCNs using four base GNNs and one base Trans-
2

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
former over two combinatorial topological spaces (simpli- cial and cell complexes). Unlike prior works that compare models under heterogeneous conditions, our systematic benchmarking provides a controlled evaluation of GCCNs across diverse architectures, datasets, and topological do- mains. A wide range of experiments on graph-level and node-level benchmark datasets shows GCCNs generally outperform existing CCNNs, often with smaller model sizes. Some of these results are obtained with GCCNs that cannot be reduced to standard CCNNs, further underlin- ing our methodological contribution. A guide to the code is available at ccometri-iintlligenne.. wwwpllliennnggggb..oooooo.o github.io/topotume (OP 3: need for standardized benchmarking.)
Ni,t and Nr, defined
N1,r(0) { C rk(r) - (I) Nr, = {T E c rk(r)
Therefore, a k 11+-1-cet 1-cell is neighbor of k-cell a w.r.t to Ni.t if a is contained in т; analogously, a k: 1-cell t is a neighbor of a k-cell a w.r.t. to NN r is contained in o. These incidences induce up/down adjacencies NA,1 and NA.1 as
NA.((o) fr ( rk(o),
(2)
C :k())) (k)))
NA. = (7
Outline. Section provides necessary background. Sec- tion 3 motivates and positions our work in the current TDL literature. Section 4 introduces and discusses GCCNs. Sec- tion 5 describes TopoTune. Finally, Section 6 showcases extensive numerical experiments and comparisons.
E rk()) -
- -,, 1,,
Therefore, k-cell is a neighbor of k-cell a w.r.t. to.V if they are both contained in a k 1-cell à; analogously, k-cell is neighbor of k-cell w.r.t. to Vo, if they both contain a k 1-cell 6. Other neighborhood functions can be defined for specific applications (Battiloro et al., 2025).
2. Background
To properly contextualize our work, we revisit the funda- mentals of combinatorial complexes and CCNNS- closely following the works of Hajij et al. (2023) and Battiloro al. (2025)aas as well as the notion of augmented Hasse graphs. Appendix A briefly introduces all topological domains used in TDL, such as simplicial and cell complexes.
Combinatorial Complex Passsae--Passing Neural Net- works. Let be a combinatorial complex, and Ne a collee- tion of neighborhood functions. The I-th layer of a CCNN updates the embedding h' E R of cell o as
h
h NENe TEN((r
(aaa(()) (b', e R
Combinatorial Complex. A combinatorial complex is triple (V, rk) consisting of a set V, subset C of the powerset P(V) ( and rank function rk CC Zzo with the following properties:
(4) where h = h, are the initial features, + is an intra- neighborhood aggregator, X is an inter-ncighborhood ag- gregator. The functions v.N.((()) RFF + F' and the update function & are learnable functions, which are typi- cally homogeneous across all neighborhoods and ranks. In other words, the embedding of cell is updated in a learn- able fashion by first aggregating messages with neighboring cells per each neighborhood, and then by further aggregat- ing across neighborhoods. We remark that by this definition, all CCNNs they can only leverage neighborhood functions that consider all ranks in the complex.
1. for all v E (v E C and rk((u)) = =0;
2. the function rk is order-preserving, i.e., if af, E C satisfy a C r, then rk(a) S rk(r).
The elements of V are the nodes, while the elements of C called cells (i.e., group of nodes). The rank of cell C is A ;= rk(a). and we call it a k-cell. C simplifies notation for (V, rk). and its dimension is defined as the maximal rank among its cell: dCm(C)= == max rk(a).
cossage-passingg architectures. Moreover,
Augmented Hasse Graphs. In TDL, a Hasse graph is a graph expansion of a combinatorial complex. Specifically, represents the incidence structure Ni.1 by representing each cell (node, edge, face) as node and drawing edges between cells that are incident to each other. Going beyond just con- sidering Nr,1 given a collection of multiple neighborhood functions, a combinatorial complex C can be expanded into a unique graph representation. We refer to this represen- tation as an augmented Hasse graph (Fig. 2) (Hajij et al., 2023). Formally, let Ne be a collection of neighborhood
Neighborhoods. Combinatorial complexes can be equipped with a notion of neighborhood among cells. In particular, a neighborhood. N: C - P(C) on combinato- rial complex C is a function that assigns to each cell a in C a collection of "neighbor cells" ((ar) c C UUU 0. Examples of neighborhood functions are adjacencies, connecting cells with the same rank, and incidences, connecting cells with different consecutive ranks. Usually, up/down incidences
3

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
OND No 3
OND Nr Nard
C No (A1
{g NNan.
Figure 2. Augmented Hasse graphs. Expansions of combinato- rial complex C (middle) into two augmented Hasse graphs: (left) the Hasse graph induced by Ne = :Ni..): (right) the augmented Hasse graph induced by NC = [Ni....VA.t J- Information on cell rank discarded (we retain rank color for illustrative purposes).
Figure 3. Ensemble of strictly augmented Hasse Graphs. Given complex C with neighborhood structure including both incidence and upper adjacency (left), this graph expansion (right) produces one augmented Hasse graph for each neighborhood.
functions on C: the augmented Hasse graph ONe of C in- duced by .Cc is directed graph 9ie- = (C, ENe) with cells represented as nodes, and edges given by
topological relationships. For instance, in a molecule (cellu- lar complex), two bonds (edges) may simultancously share multiple neighborhoods: lower-adjacent through shared atom (node) and upper-adjacent through a shared ring (face). A GNN on GNa collapses these distinctions, applying the same weights to all connections and losing the structural symmetries encoded in the domain. While this may suffice for preserving expressivity, it is inherently very different computation than that of TDL models.
EN2- = [(T, (T,, EC, E No: N(()))
(5)
The augmented Hasse graph of C is thus obtained by con- sidering the cells as nodes, and inserting directed edges among them if the cells are neighbors in C. Notably, such representation of combinatorial complex discards all information about cell rank.
The Particular Case of Hypergraphs. Hypergraph neu- ral networks have long relied on graph expansions (Telyat- nikov et al., 2023), which has allowed the field to leverage advances in the graph domain and, by extension, much wider breadth of models (Antelmi et al., 2023; Papillon et al., 2023). Most hypergraph models are expanded into graphs using the star (Zhou al., 2006; Solé et al., 1996). the clique (Bolla, 1993; Rodríguez, 2002: Gibson et al., 2000), or the line expansion (Bandyopadhyay et al., 2020). As noted by Agarwal et al. (2006), many hypergraph learning algorithms leverage graph expansions.
3. Motivation and Related Works
As outlined in the introduction, TDL lacks comprehensive framework for easily creating and experimenting with novel topologiaa architectures-unlkk the more established GNN field. This section outlines some previous works that have laid important groundwork in addressing this challenge.
Formalizing CCNNS on graphs. The position paper (Veličković, 2022) proposed that any function over a higher- order domain can be computed via message passing over transformed graph, but without specifying how design GNNs that reproduce CCNNs. Hajij et al. (2023) showed that, given a combinatorial complex C and a collection of neighborhoods Ne. a message-passing GNN that runs over the augmented Hasse graph GNa is equivalent to a specific CCNN as (4) running over C using: i) NC as collection of neighborhoods; ii) same intra- and inter-aggregattios, i.e.,
The success story of hypergraph neural networks motivates further research on new graph-based expansions of CCNNs. These expansions could, at the same time, subsume current CCNNs and exploit progress in the GNN field. Therefore, returning to our core goal of accelerating and democratizing TDL while preserving its theoretical properties, we propose two-part approach: a novel graph-based methodology able to generate general architectures (Section 4), and a lightweight software framework to easily and widely im- plement it (Section 5).
+ X and iii) no rank- and neighborhhodd-deendenn message functions, i... (d-)))) s VN e Ne.
Retaining expressivity, but not topological symmetry. Jogl et al. (2022a;b) demonstrate that GNNs on augmented Hasse graphs GNa are as expressive as CCNNs on (us- ing the WL criterion), suggesting that some CCNNs can be simulated with standard graph libraries." 2, However, as the authors state, such GNNs do not structurally distinguish be- tween cells of different ranks or neighborhoods, collapsing
4. Generalized Combinatorial Complex Neural Networks
We propose Generalized Combinatorial Complex Neural Networks (GCCNs), novel broad class of TDL architec- tures. GCCNs overcome the limitations of previous graph- based TDL architectures by leveraging the notions of strictly augmented Hasse graphs and per-rank neighborhoods.
"The same authors generalize these ideas to non-standard message-passing GNNs (Jogl et al., 2024).

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
Ensemble of Strietly Augmented Hasse Graphs. This graph expansion method (see Fig- 3) extends from the established definition of an augmented Hasse graph (see Fig. 2). Specifically, given combinatorial complex C and a collection of neighborhood functions Nc. we expand it into |WCll graphs, each of them representing a neighborhood E Nc. In particular, the strictly augmented Hasse graph GN (Cv,Ex) Ex) of a neighborhood N e NN is a directed graph whose nodes Cy and edges Exr are given by:
c
( Na.
(1)) (H)
(1
(iv)
Figure 4. Per-rank neighborhoods. Given a complex (left), we illustrate four examples of per-rank neighhorhoods (right). In each case, they only include rank-specific cells.
CN fo E + 0). Ex ((T, =(T,o))TEN(())) (())).
(6) Following the same arguments from (Hajij et al., 2023), a
method works as follows (see also Fig. 1): (i) C expanded into an ensemble of strictly augmented Hasse graphs one one for each .V' E NC. (ii) Each strictly augmented Hasse graph GN and the features of its cells are independently processed by a base model. (iii) An aggregation module X synchronizes the cell features across the different strictly augmented Hasse graphs (as the same cells can belong to multiple strictly augmented Hasse graphs)
GNN over the strictly augmented Hasse graph GN induced by // is equivalent to a CCNN running over C and using Nc = (A3 up to the (self-) juplat of the cells in C/Cy-
Per-rank Neighborhoods. The standard definition of ad- jacencies and incidences given in Section 2 implies that they applied to each cell regardless of its rank. For instance, consider combinatorial complex of dimension two with nodes (0-cells), edges (I-cells), and faces (2-cells). Employ- ing the down incidence NV., as in (1) means the edges must exchange messages with their endpoint nodes, and faces must exchange messages with the edges on their sides. It is impossible for edges to communicate while faces do not.
This method enables an ensemble of synchronized models per layer- the wys-eachh of them applied strictly augmented Hasse graph.". 2 The rest of this section formalizes the architectures induced by this methodology.
specific
Generalized Combinatorial Complex Networks. We formally introduce a broad class of novel TDL architectures called Generalized Combinatorial Complex Networks (GC- CNs), depicted in Fig. 1. Let be a combinatorial complex containing C| cells and We a collection of neighborhoods on it. Assume an arbitrary labeling of the cells in the complex, and denote the i-th cell with o,. Denote by H R|( the feature matrix collecting some embeddings of the cells on its rows, i... [H], ho,, and by Hv E R/Cw) XF the submatrix containing just the embeddings of the cells be- longing to the strictly augmented Hasse graph GN of N. The 1-th layer of GCCN updates the embeddings of the cells H' E R
This limitation increases the computational cost of standard CCNNs while not always increasing the learning perfor- mance, as experiments will show. For this reason, we in- troduce per-rank neighborhoods, examples of which are depicted in Fig. Formally, per-rank neighborhood func- tion N maps a cell a to the empty set if a cell of rank For example, the up/down r-incidences Ni, and NT.1 are defined as
(TE)) = +10)+11.00 +1, ifrk(o)=1, =r, 0, T C rk(T) rk(o) - -,, rk(o) =
(o)
otherwise,
as
H & H', X wN(H'y Sxr) E gl(xxF NeNo
(o)
0, otherwise.
(8) where HD collects the initial features, and the update function & is a learnable row-wise update function, i.e., (Ø(A, B)], ([A);, BB))) The neighborhoodddependenn sub-module WN RI - RICwIxF'+1 which we re- fer to as the neighborhood message function, a learnable (matrix) function that takes as input the whole strictly aug- mented Hasse graph of the neighborhood, G. and the em- beddings of the cells that are part of it, and gives as output processed version of them. Finally, the inter-neighborhoodd
(7)
and the up/down r-adjacencies NA, and Mi., can be ob- tained analogously. So, it is now straightforward to model setting in which employing only N7. (Fig. 4iii) allows edges to exchange messages with their bounding nodes but not triangles with their bounding edges.
Generating Graph-based TDL Architectures. We use these notions to define a novel graph- methodology for generating principled TDL architectures. Given a com- binatorial complex C and set Ne of neighborhoods, the
Contrary to past CCNN simulation works that apply a model to the singular, whole augmented Hasse graph.
5

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
aggregation module & synchronizes the possibly multiple neighborhood messages arriving on single cell across mul- tiple strictly augmented Hasse graphs into single message. GCCNs enjoy increased flexibility over CCNS (eq. 4) as their neighhorhoods are allowed to be rank-dependent and their wiy's 's are not necessarily message-passing
Given Proposition 4.1, GCCNs allow us to define general TDL models using any neighhorhood message function wys such any GNN. Not only does this framework avoid having to approximate CCNN computations, as is the case in previous works" (Jogl et al., 2022b;a; 2023), but it also enjoys the same permutation equivariance as regular CCNNs (Proposition 4.2). We show in Appendix that the resulting time complexity of a GCCN is a compromise between a typical GNN and CCNN. 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 WA to be a spectral convolution neural network such as Defferrard et al. (2016). To the best of our knowledge, GCCNs are the only objects in the literature that encompass all the above properties.
Theoretical properties of GCCNs.
1. Generality, GCCNs formally generalize CC- NNs.
Proposition 4.1. LetC be a combinatorial com- plex. Let Ne be a collection of neighborhoods on C. Then, there exists a GCCN that exactly reproduces the computation of CCNN over C using No.
5. Topo
Proof of Prop. 4.1 (Appendix B.1) relies on setting the wix of a GCCN to a simple, single- layer convolution.
Our proposed methodology, together with its resulting GC- CNs architectures, addresses the challenge of systemati- cally generating principled, general TDL models. Here, we introduce Topo'Tune, a software module for defin- ing and benchmarking GCCN architectures the fly- vehicle for accelerating TDL research. A quick start guide to the code and tutorial are provided at geometric- inteligence.github.ioffe github.io/topotun. This section details Topo- Tune's ' main features.
2. Permutation Equivariance. Generalizing CC- NNs, GCCNs layers are equivariant with re- spect to the relabeling of cells in the combina- torial complex.
Proposition 4.2. A GCCN layer is cell permu- tation equivariant if the neighborhood message function is node permutation equivariant and the neighboooho aggregator is cell per- mutation invariant.
Change of Paradigm. TopoTune introduces new per- spective on TDL through the concept of "neighborhoods of interest," enabling unprecedented flexibility in architectural design. Previously fixed components of CCNNs, such as choice of topological domain, become hyperparameters of our framework.
Proof of Prop 4.2 (Appendix B.2) hinges on the node-wise permutation equivariance of wN and the permutation invariance of the inter- neighborhood aggregation.
Accessible TDL. Using TopoTune, a practitioner can in- stantiate customized GCCNs simply by modifying a few lines of a configuration file. In fact, it is sufficient to specify (i) a collection of per-rank neighborhoods Ne. (it) a neigh- borhood message function wx, and optionally (iii) some architectural parameters e-g., the number of GCCN lay- ers. For the neighborhood message function wy, the same configuration file enables direct import of models from Py- Torch libraries such as PyTorch Geometric (Fey & Lenssen, 2019) and Deep Graph Library (Chen et al., 2020b). Topo- Tune's simplicity provides both newcomers and experts with an accessible tool for defining topological architectures.
3. Expressivity. The expressiveness TDL mod- cls is tied to their ability to distinguish non- isomorphic graphs. Variants of the Weisfeiler- Leman (WL) test, like the cellular WL for cell complexes (Bodnar et al., 2021 set upper bounds on their corresponding TDL models' expressiveness, as the WL test does for GNNs (Xu et al., 2019b).
Proposition 4.3. GCCNs are strictly more ex- pressive than CCNNs.
Proof of Prop. 4.3 (Appendix B.3) shows that GCCNs surpass CCNNs in expressivity by re- lating CCNNs to Weisfeiler-Leman (WL) and GCCNs to &-WL on augmented Hasse graphs.
"These models employ GNNs running on one augmented Hasse graph, i.e. GCCN that, given a collection of neighborhoods Ne, uses single neighborhood Ni.. defined, for cell a, Niai(a) UNEN: (ar.
'We provide detailed pseudo-code for TopoTune module in Appendix D.
6

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
Accelerating TDL Research. TopoTune is fully inte- grated into TopoBench (Telyatnikov et al., 2024), a com- prehensive package offering a wide range of standardized methods and tools for TDL. Practitioners can access ready- to-use models, training pipelines, tasks, and evaluation met- rics, including leading open-source models from TopoX (Hajij et al., 2024b). In addition, TopoBench features the largest collection of topological liftings currently avail- able transforatiion that map graph datasets into higher- order topological domains. Together, TopoBench and Topo- Tune organize the vast design space of TDL. into an ac- cessible framework, providing unparalleled versatility and standardization for practitionners.
6.2. Results and Discussion
GCCNs outperform CCNNS. Table compares top- performing CCNNs with our class of GCCNs. GCCNs out- perform CCNNs across all datasets in the simplicial and cel- lular domains and match hypergraph OCNNS-something CCNNs fail to achieve in node-level tasks. Across 16 do- main/dataset combinations, GCCNs exceed the best CCNN by > 1o in 11 cases. In 2 combinations, GCCNs are out- performed by the best GNN included in Telyatnikov et al. (2024). Representing complexes ensembles of augmented Hasse graphs, rather than a single graph, improves results.
Generalizing existing CCNNs to GCCNs improves per- formance. TopoTune makes it easy to iterate upon and improve preexisting CCNNs by replicating their architecture in a GCCN setting. For example, TopoTune can generate a counterpart GCCN by replicating a CCNN's neighborhood structure, aggregation, and training scheme. We show in Table 2 that counterpart GCCNs can achieve comparable better results than SCCN (Yang et al., 2022) and CWN (Bod- nar et al., 2021a) just by sweeping over additional choices of wy. In the single augmented Hasse graph regime, GCCN models are consistently more lightweight, up to half their size (see Table 5).
6. Experiments
We present experiments showcasing broad class of GCCN's constructed with Topo'Tune. These models con- sistently match, outperform, or finetune existing CCNNs, often with smaller model sizes. Topo'Tune's integration into the TopoBench experiment infrastructure ensures a fair com- parison with CCNNs from the literature, as data processing, domain lifting, and training are homogeonized.
6.1. Experimental Setup
We generate our class of GCCNs by considering ten possi- ble choices of neighborhood structure Ne (including both regular and per-rank, see Appendix 1. and five possible choices of w GCN (Kipf& & Welling, 2017), GAT (Velick- ovic et al., 2017), GIN (Xu et al., 2019a), GraphSAGE (Hamilton et al., 2017), and Transformer (Vaswani et al., 2017). We import these models directly from PyTorch Geo- metric (Fey& & Lenssen, 2019) and PyTorch (Paszke et al., 2019). TopoTune enables running GCCNs on both an en- semble of strictly augmented Hasse graphs (eq. 6) and augmented Hasse graph (eq. 5). While CCNN results re- flect extensive hyperparameter tuning by Telyatnikov et al. (2024) (see that work's 's Appendix C.2 for details), we largely fix GCCN training hyperparameters using the TopoBench default configuration.
GCCNs perform competitively to CCNNs with fewer parameters. GCCNs are often more parameter efficient than existing CCNNs in simplicial and cellular domains, and in some instances (MUTAG, NCII, NC109), even in the hypergraph domain. We refer to Table 4. Even as GCCNs become more parameter-intente intensive for large graphs with high-dimensional embeddings-at seen in node-level tasks they remain competitive. (We refer to Appendix H.1 for additional results on larger node-level datasets.) For instance, on the Citeseer dataset, a GCCN (wN = Graph- SAGE) outperforms the best existing CCNN while being 28% smaller. Training times in Appendix G show GCCNs train comparably on smaller datasets but slow down on larger ones, likely due to TopoTune's on-the-fly graph ex- pansion. Preprocessing this expansion in future work could mitigate the lag.
Datasets. We include a wide range of benchmark tasks (see Appendix E.2). MUTAG, PROTEINS, NC101, and NC109 (Morris et al., 2020) are graph-level classification tasks about molecules or proteins. ZINC (Irwin et al., 2012) (subset) is a graph-level regression task about solubility. At the node level, the Cora, CiteSeer, and PubMed tasks (Yang et al., 2016) involve classifying publications within citation networks. We consider two topological domains: simplicial and cellular complexes. We use TopoBench's 's lifting pro- cesses to infer higher-order relationships in these data. We use node features to construct edge and face features.
Topo'Tune finds parameter-efficceent GCCNs. By easily exploring a wide landscape of possible GCCNs for a given task, Topo helps identify models that maximize per- formance while minimizing model size. Fig. 5 illustrates this trade-off by comparing the performance and size of selected GCCNs (see Appendix I for more). On the PRO- TEINS dataset, two GCCNs using per-rank neighborhood structures (orange and black) achieve performance within 2% of the best result while requiring as little as 48% of the parameters. This reduction is due to fewer neighborhoods N, resulting fewer wn blocks per GCCN layer. Similarly,
7

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
Table 1. Cross-domain, cross-task, cross-expansion, and cross-Wx comparison of GCCN architectures with top- por-ooronnnnng CCNNs
benchmarked TopoBench (Telyatnikov et al., 2024). Best result is in bold and results within I standard deviation highlighted blue. Experiments are run with seeds. We report accuracy for classification tasks and MAE for regression.
Graph-Ieve Tasks
Node-Lever Tasks
Model
MUTAG (t)
PROTEINS (1)
NCII <t)
NC1109 (1) ZINC (U)
Cora (t)
Citeseer (1)
PubMed (t)
Graph
GNN (Best Model on TopoBench)
79.57 a 6.13
76.34 a 1.66
75.00 a 0.99
74.42 a: 0.70 0.57 a 0.04
87.21 a 1.89
75.53 1.27
89:44 1: 0.24
Cellular
CCNN (Rest Model TopoRench)
80.43 1.78
76.13 2.70
76.67 1.48
75.35 + 1,50
0.34 0.01
87.44 1.28
75.63 1,58
88.64 0.36
GCCN sny = GAT
83,40 a: 4.85
74.05 2.16
76.11 a 1.69
75.62 d: 0.76
0.38 a 0.03
88.39
0.65
74.62 4: 1.95
87.68 A 0.33
GCCN wy = GCN
85.11 = 6.73
74,41 # 1.77
76.42 # 1.67
75.62 # 0.94
0.36=00 0.01
88.51 + 0.70
75.41 + 2.00
88.18 + 0.26
GCCN sx GIN
86.38 6.49
72.54 3.00
77.65 1.11
77.19 +: 0.21
0.19 0.00
87,42 1.85
75.13 4: 1.17
88.47 0.27
GCCN wy GraphSAGE
85.53 6.80
73.62
78.23 # 1,47
77.10 0.83
0.24 0.00
88.57 + 0.58
75.89 1.84
89:40 0.57
GCCN uy
Transformer
83,83 6.49
70.97 4: 4.066
73,00 4: 1.37
73.20 à: 1.05
0.45 a 0.02
84.61 4: 1.32
75.05 4: 1.67
88.37 à: 0.22
GCCN wy =
Best GNN, Aug. Hosse graph
85,96 = 7.15
7
76.75 + 1.63
76.94 0.82
0.31 = 0.01
87.24 # 0.58
74.26 + 1.47
88.65 + 0.55
Simplicial
CCNN (Best Model on TopoBench)
76. = 6.63
75.27 = 2.14
76.60 = 1.75
77.12 1.07
0.36 = 0.02
82.27 # 1.34
71.24 1.68
88.72 + 0.50
GCCN wy = GAT
79.15 = 4.09
74.62 # 1.95
74.86 + 1.42
74.81 # 1.14
0.57 = 0.03
88.33 + 0.67
74.65 + 1.93
87.72 + 0,36
GOCN w/y= GCN
74,04 8.30
74,91 2.51
74.20
74.13 + 0.53
0.53 0.05
88,51 * 0.70
75.41 + 2,00
88.19 0.24
GCCN wy GIN
85.96 4.66
72.83 2.72
76.67 1.62
75.76 1.28
0
87.27 1.63
75.05 + 1.27
88.54 0.21
GOCN w.y
GraphSAGE
7.474
74.70 3.10
76.85 + 1.50
75.64 + 1.94 0.50 0.02
88.57 0.59
75.92 1.85
89.34 + 0.39
GCCN wy Transformer
#.00
70.97 # 4.006
70.39 + 0.96
69.99 # 1.13
0.64 0.01
84.4 1.16
74.6 = 1.88
88.55 + 0.39
GOCN wy
Best GNN,
Aug. Hasse graph
74.04 5.51
74.48 1.89
75.02 + 2.24
73.91
3,9 0.56 0.02
87,56 + 0.66
74.5 + 1.61
88.61 + 0.27
Hypergraph
CCNN (Rest Model on TopoRench)
80.43 4.09
76.63 # 1.74
75. + 1.24
74.93 + 2,50
0.51 + 0.01
88.92 # 0.44
74.93 + 1.39 W9.62 + 0.25
Table 2. We compare existing CCNNs with mm-miiified GCCN counterparts. We show the result for best choice of wxx Experiments are run with seeds.
Graph-Level Tasks
Node-Level Tusks
Model
MUTAG
PROTEINS
NCII
NCI109
Cora
Citeseer
PubMed
SCCN (Yang et al., 2022)
Renchmark results (Telyatnikov et al., 2024)
70.64 + 5.90
74.19 + 2,86
76.60 + 1.75
77.12 + 1.07
82.19 + 1.07
69.60 + 1.83
88.18 0.32
GCCN, on ensemble of strictly aug. Hasse graphs
82.13 a: 4.66
75.56 4: 2.48
75.6 à: 1.28
74.19 1: 1.44
88.06 a 0.93
74.67 4: 1.24
87.70 a: 0.19
GCCN, on aug. Hasse graph
69.79 = 4.85
74.48 2.67
74.63 = 1.76
70.71 ::: 5.50
87.62 1.62
74.86 = 1.7
87.80 0.28
CWN (Bodnar 102
Benchmark results (Telyatnikov et al., 2024)
80.43 = 1.78
76.13 + 2.70
73.93 = 1.87
73.80 +: 2.06
86.32 # 1.38
75.20 +: 1.82
88.64 0.36
GOCN, on ensemble of strictly aug. Hasse gruphs
84.26 # 8.19
75.91 + 2.75
73.87 = 1.10
73.75 + 0.49
85.64 + 1,38
74.89 + 1.45
88.40 40 0.46
GOCN, on aug. Hasse graph
81.70 5.34
75.05 * 2.39
75.14 + 0.76
75.39 # 1.01
86.44 * 1.33
74.45
1.59
88. 0.55
on ZINC, lightweight neighborhood structures (orange and dark green) are competitive with reduced parameter costs. Node-level tasks, see less benefit, likely due to the larger graph sizes and digherodimensional input features.
range. In the less clear-cut cases of PROTEINS and Cite- seer, we observe a trade-off between neighborhood structure and message function complexity. We find that larger base models (GIN, GraphSAGE) on lightweight neighborhood structures perform comparably to simpler base models (GAT, GCN) on larger neighborhood structures. This tradeoff war- rants further research on the dataset-speciffe importance of neighborhood choice, or lack thereof. We refer to Appendix H.2 for additional experiments with more advanced GNNs,
Impactfulness of GNN choice is dataset specific. Fig. 5 also provides insights into the impact of neighborhood message functions. On ZINC, GIN clearly outperforms all other models, which do not even appear in the plot's
8

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
PROTEINS
ZINC
Citeseer
Relative Perfrmance (96)
100.0
100.0
100.0
07.5
97.5
97.5
95.0
95.0
95.0-
92.5
92.5
92.5-
90.0
90.0
90.0
40
50
60
70
80
90
100
50
70
80
90
100
96
97
98
99
100
Relative Parameter Size (%)
Relative Parameter (%)
Relative Parameter Size (%)
GAT GCN
GIN GraphSAGE
NC per-rank
(Wai) A5L
[A (A
(aan
WN
Figure.5. 5. GCCN performance versus size. We compare various GCCNs across three datasets on the cellular domain, two graph-level (left, middle) and one node-level (right). Each GCCN (point) has different neighborhood strueture Nc. some of which can only be represented as per-rank structures (O in legend), and message function wy. The amount of layers is kept constant according to the best performing model. The axes are scaled relative to this model.
GATv2 (Brody et al., 2021) and PNA (Corso et al., 2020), as choices of wiy.
References
Agarwal, S., Branson, K., and Belongie, S. Higher order learning with graphs. In Proceedings of the 23rd interna- tional conference Machine learning, pp. 17-24, 2006.
7. Conclusion
This work introduces simple yet powerful graph-based methodology for constructing Generalized Combinatorial Complex Neural Networks (GCCNs), TDL architectures that generalize and subsume standard CCNNs. Additionally, we introduce TopoTTnee, the first lightweight software mod- ule for systematically and easily implementing new TDL architectures across topologiaal domains. In doing so, we have addressed, either in part or in full, 7 of the 11 open problems of the field (Papamarkou et al., 2024). Future work includes customizing GCCNs for application-specifc and potentially sparse or multimodal datasets, and leverag- ing software from state-of-the-art GNNs. TopoTune will also help bridge the gap with other fields such attentional learning and k-hop higher-order GNNs (Morris al., 2019; Maron et al., 2019).
Antelmi, A., Cordasco, G., Polato, M., Scarano, V., Spag- nuolo, C., and Yang, D. A survey on hypergraph rep- resentation learning. ACM Comput. Surv., 56(1), aug 2023. ISSN 0360-0300. doi: 10.1145/3605776. URL https://doli org. 110.0.114333910 3605776
Bandyopadhyay, S., Das, K., and Murty, M. N. Line hyper- graph convolution network: Applying graph convolution for hypergraphs. arXiv preprint arXiv:2002.03392. 2020.
Battiloro, C. Signal Processing and Learning over Topolog- ical Spaces. PhD thesis, Sapienza Unviersity of Rome, 2024.
Battiloro, C., Testa, I.., Giusti, I.., Sardellitti, S., Di Lorenzo, P., and Barbarossa, S. Generalized simplicial attention neural networks. IEEE Transactions on Signal and Infor- mation Processing over Networks, 2024.
Acknowledgements
M.P. acknowledges the support of National Science Founda- tion (NSF) CAREER 2240158 and NSF Grant 2134241, as well as from the National Science and Engineering Research Council of Canada. G.B. acknowledges support from NSF Grant 2134241. C.B. acknowledges support from the Na- tional Institutes of Health Grant EEEESS3336-00..11 N.M. acknowledges support from NSF Grant 2313150.
Battiloro, C., Karaismailoğlu, E., Tec, M., Dasoulas, G., Audirac, M., and Dominici, F. (n) equivariant topo- logical neural networks. The Thirleenth International Conference on Learning Representations (ICLR), 2025.
Bernárdez, G., Ferriol-Galmés, M., Güemes- Palau, C., Pa- pillon, M., Barlet-Ros, P., Cabellos-Aparicio, A., and Mi- olane, N. Ordered topological deep learning: network modeling case study. arXiv preprint arXiv:2503. 16746, 2025.
Impact Statement
This paper presents work whose goal to advance the field of Machine Learning. There are many potential societal consequences of our work, none which feel must be specifically highlighted here.
Bernárdez, G., Telyatnikov, L., Montagna. M., Baccini, F., Papillon, M., Galmés, M. F., Hajij, M., Papamarkou, T.

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
Bucarelli, M. S., Zaghen, O., Mathe, J., Myers, A., Ma- han, S., Lillemark, H., Vadgama, S. P., Bekkers, E. J., Doster, T., Emerson, T., Kvinge, H., Agate, K., Ahmed, N. K. Bai, P., Banf, M. Battiloro, C., Beketov, M., Bog- dan, P., Carrasco, M., Cavallo, A., Choi, Y. Y., Dasoulas, G., Elphick, M., Escalona, G., Filipiak, D., Fritze, H., Gebhart, T., Gil-Sorribes, M., Goomanee, S., Guallar, V., Imasheva, L., Irimia, A., Jin, H., Johnson, G., Kanakaris, N., Koloski, B., Kovac, V. Lecha, M., Lee, M., Leroy, P., Long, T., Magai, G., Martinez, A., Masden, M., Meznar, S., Miquel-Oliver, B., Molina, A., Nikitin, A., Nurisso, M., Piekenbrock, M.. Qin, Y., Rygiel, P., Salatiello, A., Schattauer, M., Snopov, P., Suk, J., Sánchez, V., Tec, M., Vaccarino, F., Verhellen, J., Wantiez, F., Weers, A., Zajec. P., Skrlj. B., and Miolane, N. Ieml topological deep learning challenge 2024: Beyond the graph do- main. CoRR, abs/2409.05211,, 2024. URL https: wtlli.org/1000 www.oogg/........ 48550/4555/aaa 2409.05211.
Corso, G., Cavalleri, L., Beaini, D., Liò, P., and Veličković, P. Principal neighbourhood aggregation for graph nets. In Advances in Neural Information Processing Systems, 2020.
Corso, G., Stark, H., Jegelka, S., Jaakkola, T., and Barzilay, R. Graph neural networks. Nature Reviews Methods Primers, 4(1):17, 2024.
Defferrard, M., Bresson, X., and Vandergheynst, P. Con- volutional neural networks on graphs with fast localized spectral filterin.. Advances in neural information pro- cessing systems, 29, 2016.
Ebli, S., Defferrard, M., and Spreemann, G. Simplicial neural networks. In Advances in Neural Information Pro- cessing Systems Workshop on Topological Data Analysis and Beyond, 2020.
Bodnar, C. Topological Deep Learning: Graphs, Complexes, Sheaves. PhD thesis, Cambridge University, 2023.
Eitan, Y., Gelberg, Y., Bar-Shalom, G., Frasca, F., Bronstein, M., and Maron, H. Topological blind spots: Understand- ing and extending topological deep learning through the lens of expressivity. arXiv preprint M.Xiv:2408.05449 2024.
Bodnar, C., Frasca, F., Otter, N., Wang, Y., Lio, P., Montufar, G. F., and Bronstein, M. Weisfeiler and Lehman Go Cellular: CW Networks. Advances in Neural Information Processing Systems, 14:2625-2640) 2021a.
Feng, Y., You, H., Zhang, Z., Ji, R., and Gao, Y. Hypergraph neural networks. In Proceedings of the AAAI confer- ence on artificial intelligence, volume 33, pp. 3558-3565, 2019.
Bodnar, C., Frasca, F., Wang, Y., Otter, N., Montufar, G. F., Lio, P., and Bronstein, M. Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks. In International Conference on Machine Learning, pp. 1026- 1037. PMLR, 2021b.
Fey, M. and Lenssen, J. F. Fast graph representation learning with PyTorch Geometric. International Conference on Learning Representations Workshop on Representation Learning on Graphs and Manifolds, 2019.
Bolla, M. Spectra, euclidean representations and clusterings of hypergraphs. Discrete Mathematics, 11 117(1-3):19-39 1993.
Gibson, D., Kleinberg, J., and Raghavan, Clustering cate- gorical data: An approach based on dynamical systems. The VLDB Journal, 8:222-2366 2000.
Brody, S., Alon, U., and Yahav, F. How attentive are graph attention networks? In International Conference on Learning Representations, 2021.
Gilmer, J., Schoenholz, S. S., Riley, F., Vinyals, O., and Dahl, G. E. Neural message passing for quantum chem- istry. In Proceedings of the 34th International Confer- ence on Machine Learning Volume 70, ICML'17, pp. 1263-1272. JMLR.org, 2017.
Chen, C., Cheng, Z., Li, Z., and Wang, M. Hypergraph attention networks. In 2020 IEEE 19th International Conference on Trust, Security and Privacy in Computing and Communications (TrustCom), pp. 1560-1565. IEEE, 2020a.
Giusti, L., Battiloro, C., Di Lorenzo, P., Sardellitti, S., and Barbarossa, S. Simplicial attention networks. arXiv preprint arXiv:2203.07448 2022a.
Chen, Y., Wu, L., and Zaki, M. Iterative deep graph learning for graph neural networks: Better and ro- bust node embeddings. In Larochelle, H., Ranzato, M., Hadsell, R. Balcan, M., and Lin, H. (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 19314-19326. Curran Associates, Inc., 2020b. URL https: http:///ppcceeg....... neurips. oe /
Giusti, L., Battiloro, C., Testa, L., Di Lorenzo, P., Sardellitti, S., and Barbarossa, S. Cell attention networks. arXiv preprint arXiv:2209.08179) 2022b.
Grohe, M. Descriptive complexity, canonisation, and de- finable graph structure theory, volume 47. Cambridge University Press, 2017.
pdf.
10

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
Hajij, M., Istvan, K., and Zamzmi, G. Cell complex neural networks. In Advances in Neural Information Processing Systems Workshop TDA & Beyond, 2020.
Kipf, T. N. and Welling, M. Semi-supervised classifica- tion with graph convolutional networks. In International Conference on Learning Representations (ICLR), 2017.
Hajij, M., Zamzmi, G., Papamarkou, T., Miolane, N., Guzmán-Sáenz. A., Ramamurthy, K. N., Birdal, T.,, Dey, T., Mukherjee, S., Samaga, S., Livesay, N., Walters, R., Rosen, P., and Schaub, M. Topological deep learning: Go- ing beyond graph data. arXiv preprint arXiv:1906.09000 (v33,, 2023.
Lambiotte, R., Rosvall, M., and Scholtes, I. From networks to optimal higher-order models of complex systems. Na- ture physics, 2019.
Lecha, M., Cavallo, A., Dominici, F., Isufi, E., and Battiloro, C. Higher-order topological directionality and directed simplicial neural networks. ICASSP 2025 2025 IEEE In- ternational Conference Acoustics, Speech and Signal Processing (ICASSP). 2025.
Hajij, M., Papamarkou, T., Zamzmi, G., Ramamurthy, K. N.,
Birdal, T., and Schaub, M. T. Topological Deep Learning: Going Beyond Graph Data. Online, 2024a. URL http: //tdlbook. org. Published online on August 2024.
Maggs, K., Hacker, C., and Rieck, B. Simplicial represen- tation learning with neural SkS-forms. In The Twelfth International Conference on Learning Representations, 2024. URL / http///eeeeeeeeeeeeeeeeeeeeeeeee net. forum? ja-th jH2b.
Hajij, M., Papillon, M., Frantzen, F., Agerberg, J., AlJabea, I., Ballester, R., Battiloro, C., Bernardez,, G., Birdal, T., Brent, A., et al. Topox: a suite of python packages for machine learning on topological domains. Journal of Machine Learning Research, 25(374): 2024b.
Maron, H., Ben-Hamu, H., Serviansky, H., and Lipman, Y. Provably powerful graph networks. Advances in Neural Information Processing Systems, 2019.
Hamilton, W. L.L., Ying, R., and Leskovec, J. Inductive rep- resentation learning on large graphs. In Proceedings of the s1 International Conference on Neural Informa- tion Processing Systems, NIPS' 17, pp. 1025-1035, Red Hook, NY, USA, 2017. Curran Associates Inc. ISBN 9781510860964.
Morris, C., Ritzert, M., Fey, M., Hamilton, W. I.., Lenssen, J. E., Rattan, G., and Grohe, M. Weisfeiler and leman go neural: Higher-order graph neural networks. In Pro- ceedings of the AAAI conference on artificial intelligence, 2019.
volume 33, pp.
Morris, C., Kriege, N. M., Bause, F., Kersting, K., Mutzel, P., and Neumann, M. Tudataset: A collection of bench- mark datasets for learning with graphs. arXiv preprint arXiv:2007.0800 0..60706666,, 2020.
Irwin, J. J., Sterling, T., Mysinger, M. M., Bolstad, E. S., and Coleman, R. G. ZINC: free tool to discover chem- istry for biology. Journal of Chemical Information and Modeling, 52(7): 2012.
Morris, C., Lipman. Y., Maron, H., Rieck, B., Kriege, N. M., Grohe, M., Fey, M., and Borgwardt, K. Weisfeiler and leman go machine learning: The story so far. The Jour- nal of Machine Learning Research, 24(1:11885-15923, 2023.
Jogl. F., Thiessen, M., and Gärtner, T. Reducing learning on cell complexes to graphs. In ICLR 2022 Workshop on Geometrical and Topological Representation Learning, 2022a.
Jogl, F., Thiessen, M., and Gärtner, T. Weisfeiler and leman return with graph transformations. In 18th International Workshop on Mining and Learning with Graphs, 2022b.
Papamarkou, T., Birdal, T., Bronstein, M., Carlsson, G., Curry, J., Gao, Y., Hajij, M., Kwitt, R., Liò, P., Di Lorenzo, P., et al. Position paper: Challenges and opportunities in topological deep learning. arXiv preprint arXiv:2402.08877 2024.
Jogl. F., Thiessen, M., and Gärtner, T. Expressivity- preserving GNN simulation. In Thirty-seventh Con- ference on Neural Information Processing Systems, 2023. URL https::/ httpss//ppeeeeiieeeee wtwpwrwwwwwwwweeeeeeeeeeeeeeeee ner/forum? id=ytTfonl 1Wd.
Papillon, M., Sanborn, S., Hajij, M., and Miolane, N. Ar- chitectures of topological deep learning: A survey on topological neural networks, 2023.
Jogl. F., Thiessen, M., and Gärtner, T. Expressivity- preserving gnn simulation. Advances in Neural Infor- mation Processing Systems, 36, 2024.
Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Kopf, A., Yang, F.,, De 7.,, Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., and Chintala, S. Pytorch: An imperative style, high- figh-performace deep learning library. In Advances in Neural Information Processing Systems. 2019.
Kiefer, S. Power and limits of the Weisfeiler-Leman algo- rithm. PhD thesis, Dissertation, RWTH Aachen Univer- sity, 2020, 2020.

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
Platonov, O., Kuznedelev, D., Diskin, M., Babenko, A., and Prokhorenkova, I.. A critical look the evaluation of gnns under heterophily: Are we really making progress? In The Eleventh International Conference on Learning Representations.
Yang, M. and Isufi, E. Convolutional learning simplicial complexes. arXiv preprint arX1v:2301.11163, 2023.
Yang. R., Sala, F., and Bogdan, P. Efficient repre- sentation learning for higher-order data with simpli- cial complexes. In Rieck, B. and Pascanu, R. (eds.), Proceedings of the First Learning on Graphs Confer- ence, volume 198 of Proceedings of Machine Learn- ing Research, Pp. 13:1-13:21. PMLR, 09-12 Dec 2022. /tt:tt /ptpceedicoegesseee ml presss/ v198 v198/yangn 2a88/yang22..h... html.
Roddenberry, T. M., Glaze, N., and Segarra, S. Principled simplicial neural networks for trajectory prediction. In International Conference on Machine Learning, pp. 9020- 9029. PMLR, 2021.
Rodríguez, J. A. On the laplacian eigenvalues and metric pa- rameters of hypergraphs. Linear and Multilinear Algebra, 50(1):1-14, 2002.
Yang. Z., Cohen, W., and Salakhudinov, R. Revisiting semi-supervised learning with graph embeddings. In International conference on machine learning, pp. 40-48. PMLR, 2016.
Scarselli, F,, Gori, M., Tsoi, A. C., Hagenbuchner, M., and Monfardini, G. The graph neural network model. IEEE Transactions on Neural Networks, 2008.
Zhang, M. and Chen, Y. Link prediction based graph neu- ral networks. Advances in Neural Information Processing Systems, 2018.
Solé, P. et al. Spectra of regular graphs and hypergraphs and orthogonal polynomials. European Journal of Combina- torics, 17(5):6114477 1996.
Zhou, D., Huang, J., and Schölkopf, Learning with hypergraphs: Clustering, classification, and embedding. Advances in neural information processing systems, 19, 2006.
Telyatnikov, L., Bucarelli, M. S., Bernardez, G., Zaghen, O., Scardapane, S., and Lio, P. Hypergraph neural networks through the lens of message passing: common perspec- tive homophily and architecture design. arXiv preprint arXiv:2310.0766%. 2023.
Telyatnikov, L., Bernardez, G., Montagna, M., Vasylenko, P., Zamzmi, G., Hajij, M., Schaub, M. T., Miolane, N., Scardapane, S., and Papamarkou, T. Topobench: A frame- work for benchmarking topological deep learning. arXiv preprint arXiv:2406.06642 2024.
Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, L., and Polosukhin, I. Atten- tion all you need. In Advances in Neural Information Processing Systems, 2017.
Veličković, P. Message passing all the way up. arXiv preprint ar Xiv: 1097, 2022.
Velickovic, P., Cucurull, G., Casanova, A., Romero, A., Lio, P., Bengio, Y., et al. Graph attention networks. stat, 1050 (20): 18-550, 2017.
Xu, K., Hu, W., Leskovec, J., and Jegelka, S. How powerful are graph neural networks? In International Conference on Learning Representations, 2019a. URL https:// www.revieww.nt/forrrne rrrrr (ee.ddrryGGsis.A. 61A5Km.
Xu, K., Hu, W., Leskovec, J., and Jegelka, S. How powerful are graph neural networks? In International Conference on Learning Representations, 2019b.
Yadati, N. Neural message passing for multi-relational ordered and recursive hypergraphs. Advances in Neural Information Processing Systems, 33:3275-3288) 2020.
12

---

TopoTTne: A Framework for Generalized Combinatorial Complex Neural Networks
Traditional Discrete Domains
Domains Topological Deep Learning
Simplicial complex
Cellular complex
Combinatorial complex
Set
Graph
Hypergraph
No Relation
Pairwise Relations
Part-Whole Relations
Set-Type Relations
Nodes
Edges
part of
necessarily part of
Figure 6. Topological Deep Learning Domains. Nodes in blue, (hyper)edges in pink, and faces in dark red. Figure adopted from Papillon al. (2023).
A. Domains of Topological Deep Learning
We summarize the different discrete domains leveraged within TDL. and, in doing so, contextualize how combinatorial complexes generalize all of them. To that end, we will closely follow the description of (Papillon et al., 2023), using as well its very clarifying Figure 6. We recommend this survey for a high-level overview of TDL literature, and the more extensive work of (Hajij et al., 2023) for a detailed formulation of the field. We also refer to Appendix C of Battiloro al. (2025) for concise mathematical description of each domain. From left to right in Figure 6, the different domains in TDL are:
Traditional Discrete Domains
Set Pointcloud. A collection of points called nodes without any additional structure.
Graph. points (nodes) connected with edges that denote pairwise relationships.
Set Part-Whole Relations
Simplicial Complex. A generalization a graph that incorporates hicrarchical part-whole relations through the multi-scale construction of cells. Nodes are rank O-cells that can be combined to form edges (rank 1 cells). Edges are, turn, combined form faces (rank 2 cells), which are combined to form volumes (rank cells), and on. In particular, each cell in simplicial complex must contain all lower dimensional cells such that C Therefore, faces must be triangles, volumes must be tetrahedrons, and so forth.
Cellular Complex. A generalization of an simplicial complex in which cells not limited to simplexes, but may instead take any shape: faces can involve more than three nodes, volumes more than four faces, and so on. This flexibility endows CCs with greater expressivity than simplicial complexes (Bodnar et al., 2021a), but still edges only connect pairs of nodes.
Set Set-Type Relations
Hypergraph: A generalization of graph, in which higher-order edges called hyperedges can connect arbitrary sets of two or more nodes. Connections in HGs represent set-type relationships, in which participation in an interaction is not implied by any other relation in the system. This makes HGs an ideal choice for data with abstract and arbitrarily large interactions of equal importance, such as semantic text and citation networks.
Set PhrteWhole and Set- Syt- Relations
Combinatorial Complex: A structure that combines features of hypergraphs and cellular complexes. Like a hypergraph, edges may connect any number of nodes. Like a cellular complex, cells can be combined to form higher-ranked structures. Hence, combinatorial complexes generalize all other topological domains.
13

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
B. Proofs
B.1. Proof Generality
The proof is straightforward. It is sufficient to set (y(H' Gx) to H+ veN(o) (Ms)(()) (h', Joec in e N(o) are part of the node set Cy of the strictly augmented Hasse graph of N by definition.
B.2. Proof of Equivariance
As for GNNs, an amenable property for GCCNNs is the awareness w.r.t. relabeling of the cells. In other words, given that the order in which the cells are presented to the networks arbitrary -because CCs, like (undirected) graphs, are purely combinatorial objects-, one would expect that if the order changes, the output changes accordingly. To formalize this concept, we need the following notions.
Matrix Representation of a Neighborhood. Assume again to have a combinatorial complex C containing C := |C| cells and neighborhood function N on it. Assume again to give an arbitrary labeling to the cells in the complex, and denote the i-th cell with oe The matrix representation of the neighborhood function is a matrix Nv E RCxC such that N.₃ = if the o, E (o;)) zero otherwise. We notice that the submatrix Nv Rlexix|xxl obtained by removing all the rows and columns is the adjacency matrix of the strictly augmented Hasse graph Gex induced by .V.
Permutation Equivariance. Let C be combinatorial complex, Nc a collection of neighborhoods on it, and N INwINENs the set collecting the corresponding neighborhood matrices. Let P E RCCC be permutation matrix. Finally, J.ve function f: (H', B) + H is cell permutation equivariant / (PH', JPNyy""'eNN any permutation matrix P. Intuitively, the permutation matrix changes the arbitrary labeling of the cells, and permutation equivariant function is a function that reflects the change in its output.
denote by PH the permuted embeddings and by (PN, ('N.PP""aa J.Nee the permuted neighhorhood matrices. We say that a PS (H', (NN)NEN)) for
Proof of Proposition 4.2. We follow the approach from (Bodnar et al., 202 Given any permutation matrix P, for a cell o, let us denote its permutation as (p(() with an abuse of notation. Let h/41 be the output embedding of cell o; for the 1-th layer of a GCCN taking (H', (NA))ENNG) as input, and h be the output embedding of cell dp(s) for the same GCCN layer taking (PH', (PNyP" J.VEe) as input. To prove the permutation equivariance, it sufficient to show that h = h/1 as the update function is row-wise, i.e., it independently acts each cell. To do so, we show that the (multi-)set of embeddings being passed to the neighborhood message function, aggregation, and update functions are the same for the two cells o, and (())- The neighborhood message functions act on the strictly augmented Hasse graph of 9cx N, thus we work with the submatrix Ñv. The neighborhood message function is assumed to be node permutation equivariant, i.e., denoting again the embeddings of the cells in Gey, with H E <|CxlxFF and identifying 9c, with NA, it ( (Pe H',, Pcx NNPP CN = Py "H, NN). where Рсλ, is the submatrix of P given by the rows and the columns corresponding to the cells in Gcx-- This assumption, together with the assumption that the inger-neighborhood aggregation is assumed to be cell permutation invariant, i.e. &NENG P ex Nw) = &NEN, ( cx.- N.),
holds that w.
trivially makes the overall composition of the neighborhood message function with the neighbroig aggregation cell permutation invariant. This fact, together with the fact that the (labels of) the neighbors of the cell o, in A are given by the nonzero elements of the i-th row of Nv, or the corresponding row of NN. and that the columns and rows of NN are permuted in the same way the rows of the feature matrix H permuted, implies
Nklk [Pcy P
e (ey (oy.
(9)
thus that a, and (P)) receive the same neighhorhood message from the neighboring cells in N, for all A E Ne.
14

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
Increasing expressive power
GNN Hasse
k-GNN Hasse
Theorems 1-2 from Morris et al. 2019
Propositions 3,4 from Morris et al. 2019
WL Hasse
<
k-WL Hasse
Grohe 2017
Proposition B.11
Proposition B.12
Definition B.6
Definition B.8
CCWL
k-CCWL
Propositions B.13-14
Propositions B.15-16
Eq. 4
Definition B.9, Eq. 11
Eq. 8
CCNN
k-CCNN
GCCN
Proposition B.10
Proposition 3
Figure 7. Graphical summary the definitions and propositions related to the expressivity of CCNNs and GCCNs and the different
WL. tests, Neural networks expressivity is in red, and WI. test expressivity is in blue.
B.3. Proof of Expressivity
We provide the theory required to prove Proposition 4.3, i.e., to prove that GCCNs are strictly more expressive than CCNNs. The definitions and propositions from this subsection are summarized in Figure 7. This figure serves as graphical reading guide for the subsection.
B. HOMOMORPHISM AND ISOMORPHISM INDUCED BY NEIGHBORHOODS
We first recall the notion of homomorphism of a combinatorial complex (CC) from (Hajij et al., 2023) and generalize it the notions of homomorphism and isomorphism of CCs induced by neighborhood. A
Definition B.1 (CC-Homomorphism (Hajij al., 2023)). A homomorphism from CC (V1, [1,,C,,r, also called CC-homomorphism, is function f C1 C2 that satisfies the following conditions:
C1, rk.) a CC (V2, C2 rk2).
1. If o Ci satisfy C r, then (oo S S(T).
2. a E Ci, then (o) 2 (k2))
Definition B.1 proposes a CC-homomorphism that respects the incidence structures of the CCs, denoted by the symbol C in the definition above. We generalize Definition B. by allowing CC-homomorphisms to take into account labeling the cells and be defined terms of general neighborhood structures beyond incidence. We first define labeled combinatorial complex.
Definition B.2 (Labeled Combinatorial Complex). A labeled combinatorial complex (C, e) is CC C equipped with cell coloring & C + Σ with arbitrary codomain Σ. We say that E(a) is a label or color of cell C.
Next, we provide our definitions of homomorphisms.
Definition B.3 (Co-momonom induced by (N),)N2)), A homomorphism from a CC (V1, C1, rkj) with neighborhood N1 to a CC (V2, C22 rkg) with neighborhood Ag. also called a CC-homomorphism induced by (2), is a function C - C2 that satisfies: If o. r E C, are such that r E Ní then f(r) E (N)) A labeled CC-homomorphiim induced by N2) is a CC-homomorphism induced by (N1,N2) that additionally respects labeling of the cells, that is: ar C1 have the same label, then f()), f(7) C2 also have the same label.
15

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
We prove that a CC-homomorphism induced by (2) is equivalent homomorphism of the respective strictly augmented Hasse graphs GN;: ONz.
Proposition B.4. For every CC-homonorphism f from C1 to C2 induced by (N), there exists a unique graph homomorphism between their respective strictly augmented Hasse graphs GNZ and GN2-
Proof. Consider I' a CC-homomorphism from Ci to C2 induced by (2i, as in Definition B.3. Define the function f from nodes of GN, the nodes of GN2 corresponding to f, i..., f Cy2 + CN2 defined as f(ō) f(r) where the node in UN, corresponding to the cell a in C1, and f(o) is the node in GNΛ corresponding to the cell f(o) in Cz. We show that f is graph homomorphism from 9N, to ON,, i.e., a function from the nodes of 9N; to the nodes of 9N₂ that preserves edges.
By definition of the CC-homomorphism induced by (21,,N2), we have: i e (o) then f(r) N2(f(o)). Recognizing that Nı defines edges of GN₁ and N2 defines edgess of ONg, we have: if (ő, f) is an edge in 9xi then (f(Ø), f(f) is an edge in GN₂- Thus, a CC-homomorphism induced by (2) gives a homomorphism of the strictly augmented Hasse graphs.
Conversely, if f is graph homomorphism from UN; to GNa. then we similarly construct a CC-homomorphism f between C and Cg. This concludes the proof.
Lastly, we can define a notion of CC-isomorphism induced by neighborhood structures.
Definition B.5 (CC-Isomorphism induced by (N), A isomorphism from CC (Vi, C1,rki rk1) with neighborhood Mi to CC (V2, C2, rk2) with neighborhood N2. also called a CC-isomorphism induced by (22, is an invertible CC- homomorphism induced by N2) whose inverse is CC-isomorphism induced by (N2, A labeled CC-isomorphism induced by (N1,.N) is a CC-isomorphism that additionally respects labels.
B.3.2. WEISFEILER LEMAN (WL) TESTS ON COMBINATORIAL COMPLEXES
We propose two WL tests, called CCWL. and (set-based) k-CCWI. that generalize the WI. and the (set-based) k- WL. tests to labeled combinatorial complexes. We start with the generalization of the WL test to labeled combinatorial complexes.
Definition B.6 (The CC Seisfeiler-Lemant (CCWL) test on labeled combinatorial complexes). Let (C, be a labeled combinatorial complex. Let A be neighborhood on C. The scheme proceeds as follows:
Initialization: Cells are initialized with the labels given by E, i.c.: for all a C, we set: cae l(o).
Given colors of cells at iteration e, the refinement step computes the color of cell a at the next iteration !+1 using a perfect HASH function as follows:
Refinement:
(0) = (ftor Vo' eN(o))),
1+1 HASH
Termination: The algorithm stops when an iteration leaves the coloring unchanged.
Next, we generalize the set-based 6-WL test to labeled combinatorial complexes, called the k-- kcCCCL test. The set-based k-WL test is employed in (Morris al., 2019) where colors defined on kesets of nodes, opposed to k-tuples of nodes in the standard k-WL test. Specifically, we denote [C] the set of k-sets formed with cells of C. We generalize the definition of neighborhood of kesett of vertices from (Morris et al., 2019) to neighborhood of kesets of cells.
Definition B.7 (Neighborhood of k-sets of cells). Given a keset of cells or in [C] define its neighborhood as the function Ni. [c]* + P((C])) defined as:
(= - (te 11t Ius k 11
(10)
Definition B.8 (The CC k- Weisfeiler-Lemaal (k-CCWL) test on combinatorial complexes). Let (C, E) be a labeled combina- torial complex. Let N be neighborhood on C. The scheme proceeds as follows:
16

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
k-sets
H
features on other k-sets that contains o
(0)
H 8)
H' s) E RR
features on the k-set s
that contains o
cells
Figure Notations Proposition B.10. Denote |C] the number of cells. The number of k-sets that contain a given cell is equal (12-1) ) The feature one k-set that contains given cell has dimension d. Thus, H' B ||]XX for F = ((--)) d. We note that the k-sets one row not correspond the k-sets of another row. However, every row, there the same number k-sett that contain the cell characteristic of that row.
Initialization: Every k-set in [c] is initialized with color that corresponds to the CC-isomorphism type of the sub-CC defined by = [01.... } induced by M where M is the neighborhood A restricted to s. This means that two k-sets and s' get the same color if and only if there is labeled CC-isomorphism (for labeling function é) between the sub-CCs corresponding to the cells in 8 and s', respectively.
Refinement: Given colors of k-sets iteration t, the refinement step computes the color of the k-set at the next iteration ++1 using perfect HASH function, as follows: s.
(i e Is (Vi(s))) + = HASH (c.: (x(.))))))))
Termination:
The algorithm stops when an iteration leaves the coloring unchanged.
Two combinatorial complexes are deemed non-isomorphic according to the CCWL and k-CCWL respectively, their color histograms differ upon termination of the scheme. If the histograms are the same, we cannot conclude.
B.3.3. DEFINITIONS OF k-GNNs AND &-CCNNS
We generalize the definition of k-GNNs by (Morris et al., 2019) into a definition of k I-CCNNs.
Definition B.9 (k-CCNNs). Let (C, E) be labeled CC. In each k-CCNN layer I, the feature vector 11 (0) ())
Rd for each
k-set in [C]* updated into ng (t+1) (s) as follows:
b4 (++1) (s)
11 ()))))
1
Σ
wr(0)
E ERd
(11)
(.))(()
where ( wr(t) are matrices parameters for layer t, N3 the neighborhood structure on k-sets, and U an update function.
Then, we show that k-CCNNs of Definition B.9 form subclass of GCCNs. Proposition B.10. GCCNs generalize and subsume k-CCNNs.
17

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
Proof. Let be given a k-CCNN defined by (II). We show that we can recover (11) by an appropriate choice of feature dimensionality F, update function and sub-module wi in (8) defining GCCNs, and thus that any k-CCNN can be expressed as GCCN.
For simplicity of notations, we assume that the layers of the k-CCNN have same feature dimensionality, denoted d. Given that there are ((=-)) k-sets containing a given cell a, we define F = ( d to be the feature dimensionality of the layers of GCCN. Denote (o) the row of H' containing F-dimensional feature corresponding to cell as well as H (a, s) the subrow containing the d-dimensional feature corresponding to one k-set to which a belongs. Figure 8 illustrates these notations. We then define:
H = & (H' (H'))
by defining & and w on (a, s--okokks of the matrix H'. Specifically, we have:
H1 (o, s) = D(a,)) (H' (a. 8), ((o,)) (H'))
=U (', w +
Σ (ENi(()
Ho, u) w!O)
where:
W(,,))H))
Σ
H' u) w
(A, = U(A w( + B).
(12)
(*)
In other words, first use defined as a sequence of w(r.s) to update each (o, sl-bock of H' into auxiliary feature B = H.' Then, we use e as a sequence of ((a.)) to perform a block-wise operations. Thus, we have built a GCCN that reproduces the computations of the k-CCNN. Therefore, GCCNs generalize and subsume k-CCNNs.
B.3.4. RELATIONSHIPS BETWEEN CCWL/GCWL TESTS AND CCNNS/GCCNS
We prove relationships between the expressivity of the WL tests and the expressivity of the corresponding neural networks. We first recall results on WL tests on graphs and GNNs (Morris al., 2019). In what follows, (G, E) is labeled graph, and W(()) denote the parameters of GNN up layer We encode the initial labels {(v), for vertex v, by vectors (1))) (v) E Rixd
WL//GNSs and k-WL/K-GNNs Theorem I in (Morris al., 2019) states that, for every encoding of the graph labels {()) as devectors (0)) (v), and for every choice of parameters W(O), the coloring c(t), of the WL test always refines the coloring htt) induced by the GNN parameterized by W Theorem 2 in (Morris et al., 2019) states that there exists parameter matrices ( such that GNNs have exactly the same power as the WL test. Consequently, we say that GNNs have the same expressivity as the WL test. Similarly, Propositions and 4 from (Morris et al., 2019) show that k- G-GNNs have the same expressivity as the k-WL test.
CCWL/CCNNs and k-CCWL/GCCNS We generalize the equivalence between WL. tests and GNNs to the framework of CCs. First, we prove two propositions establishing equivalence of WL tests between CCs and Hasse graphs
Proposition B.11 (CCWL and WL on the Hasse graph). Let (C, E) be a labeled CC. Let N be one neighborhood on this CC and GNv the associated strictly augmented Hasse graph. The CCWL test defined in Def. B.6 is equivalent to the WL test defined on GN-
Proof. We prove the equivalence between the CCWL and the WL on GN-
Equivalence of initializations. The CCWL test initializes cell colors using the labels given by E. The labeling function l labels cells of C and therefore its restriction to Cy labels nodes of the associated Hasse graph Gy. This turns GN into labeled graph
We initialize the WI. test on Gy with colors from fcx-
Equivalence of refinements. By construction of the strictly augmented Hasse graph 9Ns nodes in GNr are cells in Cy and edges in Gx are neighbors in Cy for the neighborhood N. Thus, the refinement equation of the CCWL test is equal to the refinement equation of the WL. test on Gx. This proves that CCWL. and the WI. on ON are equivalent.
18

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
Proposition B.12 (k-CCWL and k-WL on the Hasse graph). Let (C, E) be a labeled CC. Let N be one neighborhood on this CC and GN the associated strictly augmented Hasse graph. The k- D-CCWL defined in Def. B.8 is equivalent the k-WL. test GN.
Proof. We prove the equivalence between the k-CCWL and the k- WL on GN-
Equivalence of initializations. The k-CCWL test initializes colors of k-sets based on the CC-isooorphism class of every sub-CC defined by every k-set. Using Proposition B.4, the CC-isomorphism class of a sub-CC 8 corresponds to the graph isomorphism class on the associated subgraph in the strictly augmented Hasse graph. We initialize the k-WL test on GN with colors on k-sets associated with this isomorphism class.
Equivalence of refinements. By construction of the strictly augmented Hasse graph 9y, k-sets of nodes in 9N are k-sets of cells in Cx, and the neighborhoods of k-sets of nodes defined in (Morris et al., 2019) are the neighborhoods of k-sets of cells defined in Definition B.7. Thus, the refinement equation of the A-CCWL test is equal to the refinement equation of the k-WL test on Cx.
This proves that k-CCWL and the &-WL on 9N are equivalent.
Given the equivalence between the computations in C and in 9N provided by Proposition B.11, we can pull the results from Theorems I and 2 from (Morris et al., 2019) and provide the following propositions.
Proposition B.13. Let (C, be a labeled CC. Then for all 2 0 and for all choices of initial colorings h consistent with
f, and weights ((t)), (1) [ te99, i.e., the coloring (1) induced by the CCWL test refines the coloring induced by the CCNN (t())
Cf
Proposition B.14. Let E) be a labeled CC. Then for all L 2 0 there exists a sequence of weights W(O, and a CCNN architecture such that ((0) = h(0),, i.e., the coloring of the CCWL and the CCNN are equivalent.
Consequently, CCNNs have the same power as the CCWL. Next, we measure the power of k-CCNNS using the k-CCWL. Proposition B.15. Let E) be a labeled CC and let l 2 2. Then, for all t 2 for all choices of initial colorings hy consistent with l and for all weights W(1), S.k. C lig i.e,, the coloring (t) S.k. induced by the k-CCWL test refines the
(0)
coloring induced by the k-CCNN hi
Proposition B.16. Let (C, E) be a labeled CC and let l 2 Then, for all t t2 0 there exists sequence of weights W(1), and a k-CCNN architecture such that (t)
c,,k = 1/4
Propositions B.15 and 1.16 are given by Proposition 1. and Propositions and from Morris et al. (2019). They show that k-CCNNs have the same power as the k-CCWL
B.3.5. PROOF
We now provide the proof for Proposition 4.3 that states that GCCNs are strictly more expressive than CCNNs.
Proof. We prove that GCCNs are strictly more powerful than CCNNs in distinguishing non-isomorphic combinatorial complexes. We leverage the propositions of this subsection summarized on Figure 7.
By Proposition 10 GCCN have at least the same expressive power as k-CCNNs. By Propositions B. B.15-B.16. 16, k-CCNNs have the same expressive power as the k-CCWL. By Proposition 12, the k-CCWL test equivalent to the k-Wl. test on the associated strictly augmented Hasse graph. is known (e.g., (Grohe, 2017)) that the k-WL test on graph strictly more powerful than the WL test. Thus, the kL test on the strictly augmented Hasse graph is strictly more powerful than the WI. test on that same graph. By Proposition 1.1 the WI. test on the strictly augmented Hasse graph is equivalent to the CCWL test on the corresponding CC. By Propositions B.13-B.144 14,, the CCWL test on the CC has the same expressive power as CCNNs.
Consequently, we have shown that GCCN are strictly more powerful than CCNNs in distinguishing nonisomorphic CCs.
Additionally, we construct two combinatorial complexes C₁ and C2 that are indistinguishable by CCNNs but distinguishable by GCCNs.
19

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
C1
C2
Combinatorial complex
91 = -N (C1)
9 = 9N (C2)
b. Strictly augmented Hasse graph
N N2 A,
Figure 9. Pair of combinatorial complexes: Ci is an icosahedron polygon, and C2 is five tetrahedrons. b. Strietly augmented Hasse graphs corresponding each combinatorial complex, given a choice of neighborhood N2.1-
Let Cj and C2 be two combinatorial complexes with neighborhood structure NC Na, (down-adjacency of faces). These complexes are illustrated in Figure 9a.
The corresponding strictly augmented Hasse graphs Gi and G2 (Fig. 9b) represent the 20 faces of each complex as nodes, where each node has degree 3. Thus:
Both G1 and G2 are 3-regular graphs.
is known that regular graphs of the same order are indistinguishable by the WL test (see, e.g., (Kiefer, 2020; Morris et al., 2023)).
Every pair of graphs with nodes are distinguishable by the n- n-WL. test (Morris et al., 2023).
Since CCWL is equivalent to WL on Gy (Proposition B.11), the two complexes C₁ and C2 are indistinguishable by CCWL. Since CCWL has the same expressive power CCNNs (Propositions B. B.13-B.14), the two complexes C₁ and C2 are indistinguishable by CCNNs.
Since &-CCWL is equivalent to k-WL on GN (Proposition B.12), the two complexes C1 and C2 are distinguishable by k-CCWL. Since &-CCWL has the same expressive power as k-CCNNs (Propositions B. B..5-B.6), the two complexes C₁ and C2 distinguishable by k-CCNNs. Since GCCNs generalize and subsume k- k-CCNNs (Proposition B.) C₁ and C2 distinguishable by GCCNs.
Thus, we have constructed two combinatorial complexes C₁ and C2 that are indistinguishable by CCNNs, but are distinguish- able by GCCNs.
20

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
Time Complexity
To analyze the time complexity (in terms of FLOPs) the Generalized Combinatorial Complex Neural Network (GCCN), we derive the complexity of its submodule wx and then compute the complexity of GCCN layer. We then compare it with GNN and CCNN complexity.
C.1. Key Definitions
Message Complexity (M): The complexity of a single message computation along route (e.g., node node). For example, in Graph Convolutional Network (GCN), single message is defined as:
Myy+y = Cary Ony O,
where hy a IxF vector, è is an FxF weight matrix, and awy is scalar. This involves a matrix-vector multiplication, contributing complexity of (F22) per message.
Update Complexity (U): The complexity of the update function in the reference GNN. For simplicity, we assume the update is an element-wise function, giving O((NV) where |N] is the number of nodes.
C.2. Complexity of ww
Assuming each WN submodule is a single-layer GNN, the complexity of wy can be decomposed into three components: message computation, aggregation, and update.
CwN Ciessagge
C
aggregaatin
Cupdate
This breaks down as:
Cuy - + Σ A INUU,
nEN
where:
E Number of edges in the graph,
M: Complexity per message (O()²)).
deg(n): Degree of node n,
A: Complexity of aggregation (e.g., assuming sum/average, O(F)),
U: Complexity of the update function (O(1) per node).
Substituting assumptions for convolutional message passing, summation aggregation, and constant node degree d:
Cux F2 Σ deg(n)FF
Cux 2| + |N ONNI),
Cuy O(|E|F2 + [N]FF
C.3. Complexity Using Combinatorial Complex Notations
Up until now, have expressed Cux in terms of the nodes and edges making up the strictly expanded Hassse graph it receives as input. To be able write the complexity of a whole GCCN layer, we must express Cax in terms of the original cells represented nodes in the graph. Specifically, we will denote the source cells (cells sending messages) as cells of rank r and the destination cells (cells receiving messages) as cells of rank r'. The relationships governing adjacency between the nodes representing these cells will come from the neighborhood A to which the submodule wN is assigned.
Rewriting in terms of combinatorial complex notations, where:
21

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
W Total number of relationships in W (i.e. number of nonzero entries in matrix corresponding to NV),
Tyy Number of r'-cells.
d Assumed constant degree of r'-cells,
The complexity becomes:
Cux O(I + nro + +nr), = C nrows((V)d,FFF nrows(N\).
C.4. Complexity of GCCN Layer
GCCN layer is composed of set of wN''s, s, one for each N E Nc. The complexity GCCN layer the sum of all the complexities of its submodules, plus the complexity of the module responsible for aggregating the outputs of each neighborhood, i.e. the inter-neighborhood aggregation. We assume this inter-aggregation to be a sum. The layer complexity
is:
CoCCN =
CuN + Cinter-agg NENe
Σ
where:
Cinter-agg
ny F,
['E(ORR]
and nw,, is the number of neighborhoods sending messages to -cels..
C.5. Takeaways
GNN Comparison: GCCNs increase complexity compared to traditional GNNs due to
the introduction of multiple neighborhoods. A GCCN considers many N E Ne, going beyond the simple node-level adjacency Ne = Ao of a GNN. This is what allows TDL models (GCCNs and CCNNs) to operate on richer topological space than GNNs.
inter-neighborhood aggregation.
CCNN Comparison: Unlike traditional CCNNs, GCCNs allow per-rank neighborhoods, enabling many smaller
possible sets of neighborhoods. Ne. This more selective inclusion of neighborhoods reduces redundancy. Concretely, this means the sum Evere Cuxy be smaller.
Tradcoff: GCCNs' time complexity are compromise between GNNs and CCNNs. While they do introduce Cinteragg (like CCNNs) and additional clements to the sum ΣNENG Cwyy they can introduce less elements to this sum than CCNNs.
22

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
D. Software
Algorithm shows how the Topo module instantiates a GCCN by taking a choice of model w/ and neighborhoods No as input. Given an imput complex 2, TopoTune first expands into an ensemble of strictly augmented Hasse graphs that are then passed their respective wN models within each GCCN layer.
Remark. We decided to design the software module of Tone i.e., how to implement GCCNs, as we did for mainly two reasons: (i) the full compatibility with TopoBench (implying consistency of the combinatorial complex instantiations and the benchmarking pipeline), and (ii) the possibility of using GNNs as neighborhood message functions that are not necessarily implemented with a specific library. However, the practitioner is interested in entirely wrapping the GCCN implementation into Pytorch Geometric or DGL,, they can do by noticing that GCCN is equivalent GNN where the heterogeneous graph the whole augmented Hasse graph, with node types given by the rank of the cell (e.g. O-cells, I-cells, and 2-cells) while the edge type is given by the per-rank neighborhood function (e.g. "O-cells to I-cells" or "2-cells to 1-cells" for NP, and N7 respectively).
heterogeneous
Algorithm 1 Tune
Class Topo'T www.ttorrc.....Monee
0: procedure INIr(neighborhoods, Wn. wn params, layers)
0: self.omega submodells +-
0: for 1 +- 1 to layers do
0: layer models +-
0:
for each nà in neighborhoods do
0:
model + Wn (wn params)
0:
0: end for
0: sel wme.gg...tddmode....aenlleeeeeee
0: end for
0: end procedure
0: procedure FORWARD(I))
0: for each layer in www.ff.maa......ummmoo. do
0: outputs
0: for each wn model in layer do
0:
hasse hasse-graph + self.eeeaand to strictlyy aug hasse caasse.graph(11
0:
0: end for
0:
self * waw.grrgooee.ruku.kkaaaaaubbbbb.
0: end for
0: return æ
0: end procedure
Example Instantiation:
0: neighborhoods [[[0, 0), up adjacency), [[2, 1), incidence]
0: Wn + tor.tt.gooooeeeee www.hgggcc.c...nnnen.. mod.ls.....AA
0: params + (numeayers: 2, heads 4}
0: loyers t-
0: model +- Topo TupoTuneeneighhoorroodds wh, un params, layers)
=0
23

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
E. Additional details on experiments
In this section, we delve into the details the datasets, hyperparameter search methodology, and computational resources utilized for conducting the experiments.
E.1. Neighborhood Structures
In order to build broad class of GCCNs, we consider X different neighborhood structures on which we perform graph expansion. Importantly, three of these structures are lightweight, per-rank neighborhood structures, as proposed in Section 4. The neighborhood structures are:
N Nai A.t (NRmoNie (Wan:Wa)) Wan. (Wat) (Nat:N11 (Nat (Wat:Wra3 (Wan.Na Wan
E.2. Datasets
Dataset statisties
Table 3 provides the statistics for each dataset lifted to three topological domains: simplicial complex, cellular complex, and hypergraph. The table shows the number of O-cells (nodes), 1-cells (edges), and 2-cells (faces) of each dataset after the topology lifting procedure. We recall that:
the simplicial clique complex lifting is applied to lift the graph to simplicial domain, with maximum complex dimension equal to 2;
the cellular cycle-based lifting is employed to lift the graph into the cellular domain, with maximum complex dimension set well.
Table Descriptive summaries the datasets used the experiments.
Dataset
Domain
# O-cell
# 1-cell
# 2-cell
Cellular
2,708
5,278
2,648
Cora
Simplicial
2,708
5,278
1,630
Cellular
3,327
4,552
1,663
Citeseer
Simplicial
3,327
4,552
1,167
Cellular
19,717
44,324
23,605
PubMed
Simplicial
19,717
44,324
12,520
Cellular
3,371
3,721
538
MUTAG
Simplicial
3,371
3,721
Cellular
122,747
132,753
14,885
NCII
Simplicial
122,747
132,753
186
Cellular
122,494
132,604
15,042
NCI109
Simplicial
122,494
132,604
183
Cellular
43,471
81,044
38,773
PROTHINS
Simplicial
43,471
81,044
30,501
Cellular
277,864
298,985
33,121
ZINC (subset)
Simplicial
277,864
298,985
769
24

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
Dataset selection and limitations
The datasets employed in this work and other TDL studies are predominantly adapted from the GNN literature. Among these, molecular datasets stand out due to the inherent importance of cycles and hyperedges, which effectively capture chemical rings and functional groups. These structures that are naturally represented in topological domains.
While TDL methods are not intrinsically constrained to these datasets, the lifting procedures used to construct higher- order cells introduce computational bottlenecks, particularly in memory usage. For instance, operations such as cycle detection and clique cnumeration, required for constructing cellular complexes or simplicial complexes, respectively, become computationally prohibitive for large or densely connected graphs.
To address these limitations, ongoing research is focused developing scalable lifting procedures that extend TDL methods broader datasets, including those with more complex structures or larger scales, For example, Bernárdez et al. (2024) propose innovative topological liftings, paving the way for more scalable and applicable datasets in TDL.
E.3. Hyperparameter search
Five splits are generated for each dataset to ensure fair evaluation of the models across domains. Each split comprises 50% training data, 25% validation data, and 25% test data. An exception is made for the ZINC dataset, where predefined splits are used (Irwin et al., 2012).
To avoid the combinatorial explosion of possible hyperparameter sets, we fix the values of all hyperparameters beyond GCCNs: hence, to name a few relevant parameters, we set the learning rate to 0.01, the batch size to the default value of TopoBench for each dataset, and the cell hidden state dimension to 32. Regarding the internal GCCN hyperparameters, grid-search strategy is employed to find the optimal set for each model and dataset. Specifically, we consider 10 different neighborhood structures (see Section E.1), and the number of GCCN layers varied over {2, 4, 8). For GNN-based neighborhood message functions, we vary over (GCN,GAT,GIN, GraphSSaee models from PyTorch Geometric, and for each of them consider either 1 or 2 number of layers. For the Transformer-baan -aaed neighborhood message function (Transformer Encoder model from PyTorch), we vary the number of heads over {2, 4), and the deed-fowward neural network dimension over (64, 128).
For node-level task datasets, validation is conducted after each training epoch, continuing until either the maximum number of epochs is reached or the optimization metric fails to improve for 50 consecutive validation epochs. The minimum number of epochs is set to 50. Conversely, for graph-level tasks, validation is performed every 5 training epochs, with training halting if the performance metric does not improve on the validation set for the last 10 validation epochs. To optimize the models, torch. opt im. Adam is combined with torch. opt im. 1r ir.scheduler. StepLR wherein the step size was set to 50 and the gamma value to 0.5. The optimal hyperparameter set is generally selected based on the best average performance over five validation splits. For the ZINC dataset, five different initialization seeds are used obtain the average performance.
E.4. Hardware
The hyperparameter search is executed on a Linux machine with 256 cores, 1TB of system memory, and NVIDIA A100 GPUs, each with 80GB of GPU memory.
25

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
F. Model Size
provide details on model size for reported results Section 6.
Table Model size corresponding results reported in Table
Graph-Level
Node-Level Tasks
Model
MUTAG
PROTEINS
NCII
NCI109
ZINC
Cora
Citeseer
PubMed
Cellular
(Best Model on TopoBench)
101.12K
17.67K
88.06K
1032.84K
163.72K
GCCN by GAT
15.11K
46.27K
68.99K
49.63K
39.78K
341.54K
1677.32K
344.83K
GCCN 40 GCN
45.44K
45.25K
65.92K
30.69K
29.54K
801.66
1507.590
443.91K
wx
23.49K
49,03K
66.79K
64.35K
669.58K
1674.25K
211.9K
GCCN un GraphSAGE
44.42K
76.99K
47.49K
115.17K
79.71K
1195.14K 14K
741.5K
640,51K
GCCN 4x Transformer
112.26K
78.79K
82.05K
115.43K
317.02K
249.51K
468.29
331.59K
GCCN w GNN,
graph
15.91K
Simplicial
CCNN (Best Model TopoBench)
398.85K
10.24K
13184K
135.75K
617.86K
144.22K
737.29K
134.40K
GCCN wx
15.11K
46.27K
68.99K
49.63K
67.42K
341.45K
1677.32K
344.83K
GCCN ux = GCN
45.44K
45.25K
65.92K
30.69K
64.35K
801.16K
1507.59K
443.91K
GCCN 4/y GIN
63.62K
23.49K
49.03K
66.79K
118.11K
669.58K
1674.25K
211.97K
GCCN w GraphSAGE
44.42K
76.99K
47.49K
115.17K
147.30K
1195.14K
741.51K
640.51K
GCCN wn = Transformer
113.15K
213.70K
82.05K
166.24K
148.83K
284.58K
468.29K
331,59K
GCCN wy Best GNN, 1 Hasse graph
19.07K
14.66K
31,11K
15.91K
29.54K
150.12K
367.88K
66.50K
Hypergraph
CCNN (Best Model on TopoBench)
84. 10K
14.34K
88. 19K
88.32K
22.53K
60.26K
258.50K
280.83K
Table 5. Model sizes corresponding to results in Table 2.
Graph-Level Tasks
Node-Level Tasks
Model
MUTAG
PROTEINS
NCI1
NCI109
Cora
Citeseer
PubMed
SCCN
TopoBench
398.85K
397.31K
13184K
135.75K
155.88K
782.34K
457.99K
Hasse graph WN Best(GNN)
852.74K
851.97K
248.58K
291.39K
159.46K
791.56K
510.47K
Hasse graph for (A), w/ Besi(GNN)
104.32K
153.09K
71.17K
54.85K
143.66K
741.51K
376.58K
CWN
TopoBench
334.72K
101.12K
124.10K
412.29K
343.11K
1754.50K
163.72K
Hasse graph/ / wy Best(GNNN)
350.46K
353.54K
95.75K
465.28K
900.23K
177.10K
159.56K
Hasse graph for (A), w/ Best(GNN)
219.65K
283.91K
78.85K
264.45K
138.95K
163.94K
138.95K
26

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
G. Model Training Time
provide training times for all experiments reported on in Section 6. We measure these training times by running each experiment on single A30 NVIDIA GPU. We note that these times include the on-the- fly graph expansion method, which slows down the model forward proportionally dataset size. We plan on moving this process into data preprocessing the future.
Table 6. Model training time (scconds) corresponding results reported in Table
Graph-Level Tasks
Node-Level Tasks
Model
MUTAG (1) PROTEINS (t) NCII (1) NC1109 (1)
ZINC (L)
Cora (1) Citeseer (1)
PubMed (t)
Cellular
CCNN (Best Model TopoBench)
100
132 19
238 89
254 39
228 44
75
57 + 4.4
128 50
GCCN uuy GAT
80 11
64 10
778 118
486 = 75
3173 954
46 +
63 +
202 = 22
GCCN w/y GCN
67
544
495 108
4013 620
46
65
149 =
GCCN w.y GIN
61 + 18
39 18
$23 119
386 + 76
3301 440
64 =
77
207 = 33
GCCN w. GraphSAGE
2 12
4
691
102
2863 262
49
60
GCCN wyy Transformer
50 19
786 = 147
1005 + 27
1484 = 181
15320 = 5386
121 =
94 = 20
5459 + 1374
GCCN wy Best GNN, Aug. Hasse graph
451 123
130
3162 340
4
194 35
Simplicial
CCNN (Best Model on TopoBench)
123 57
104 + 28
172 50
183 =
= 86
=
75
114 =
GCCN GAT
25 5
70 17
755 158
794 151
2242 275
49 +
68 +
192 38
GCCN w.y GCN
+ 7
138 26
548 185
603 181
2428 833
49
67 #
167 + 22
GCCN GIN
61 7
66 21
904 180
538 39
3603 475
71 6
77 8
210 + 42
GCCN w. = GraphSAGE
61 + 27
572 124
511 74
1721 201
3
74
221 #
GCCN Transformer
35 5
947 333
1386 404
1360 410
7979 1373
146 58
77
5281 827
GCCN Best GNN, Aug. Hasse graph
+
78
598 +
312
2681 910
52 =:4
72
156 =
Hypergraph
CCNN (Best Model on TopoBench)
127 + 48
96 = 20
220 = 74
128 =
387 + 105
121 = 38
48 4841
177 =1
Table 7. Model training times (scconds) corresponding results in Table 2.
Graph-Level Tasks
Node-Level Tasks
Model
MUTAG
PROTHINS
NCTI
NCT109
Cora
Citeseer
PubMed
SCCN (Yang et al., 2022)
Benchmark results (Telyatnikov al., 2024)
11 2
60 # 18
247 65
311 83
102
101 41
143 35
GCCN, on ensemble of strictly aug. Hasse graphs *2, dig
14
75 8
413 # 120
298 15
121 #
172 + 6
285 a 20
GCCN, on aug. Hasse graph *2, dig
283
110
166
CWN (Bodnar et al., 2021a)
Benchmark results (Telyatnikov et al., 2024)
+
=
240 1:00:0
252 92
+ 25
52
119 +
GCCN, ensemble of strictly aug. Hasse graphs *2, dig
12
73 10
536 38
426 90
91 +: 17
49
125 19
GCCN, on aug. Hasse graph +2, dig
11 +
11
573 # 107
410 64
96 4:
46
130 4: 20
27

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
H. Additional experiments
H.1. Larger node-level datasets
Table 8 additionally presents the experimental results on heterophilic datasets introduced in (Platonov et al.) (Amazon Ratings, Roman Empire, Minesweeper, and Questions). These represent larger node-level classification tasks than those shown in the main Table 1, with up to 48,921 nodes and 153,540 edges in the case of the Questions graph. Except this precise dataset, which models from Telyatnikov et al. (2024). We observe that overall GCCNs achieve similar performance than regular CCNNs, and they outperform them by significant margin on Minesweeper.
considered in previous TDL. literature, we compare the results against CCNNs and hypergraph
Amazon Ratings
Roman Empire
Minesweeper
Questions
Best GCCN Cell Best CCNN Cell
50.17 = 0.71
84.48 + 0.29
94.02 = 0.28
78.04 + 1.34
51.90 = 0.15
82.14 + 0.00
89.42 = 0.00
Best GCCN Simplicial
50.53 0.64
88.24 + 0.51
94.06 + 0.32
77.43 + 1.33
Best CCNN Simplicial
0OM
89.15 + 0.32
90.32 # 0.11
Best Hypergraph Model
50.50 0.27
81.01 0.24
84.52 + 0.05
Table 8. Results on larger node level datascts, cach experiment with 5 seeds. We report accuracy for Amazon Ratings and Roman Empire, and AUC-ROC for Minesweeper and Questions. The values for the best CCNNs and hypergraph models are extracted from TopoBench (Telyatnikov et al., 2024).
H.2. More advanced GNNs
We include subset of experiments with the same protocol as in Table using GATv2 (Brody al., 2021) and PNA (Corso et al., 2020) in the cellular domain. Results show how the GCCNs built with these models perform consistently well across node-level and graph-level tasks on the cell domain, often < la of the best (standard-GN) GCCN as reported in Table 1, but only outperform them on MUTAG.
Table Cross-domain, cross-task, cross-expansion, and cross-wjy comparison GCCN architectures built with GATv2 (Brody al., 2021) and PNA (Corso et al., 2020) with lop-performing GCCNs from Table 1 and benchmarked on TopoBench (Telyatnikov et al., 2024). Best result is in bold and results within 1 standard deviation are highlighted blue Experiments run with 5 seeds. We report accuracy for classification tasks and MAE regression.
Graph-L Tasks
Node-Level Tasks
Model
MUTAG (f) PROTHINS (t)
NCH (f)
NCT109 (*)
Cora (1)
Citeseer (f) PuhMed (t)
Cellular
CCNN (Best Model on TopoBench)
80.43 1.78
76.13 2.70
76.67 = 1.48
75.35 + 1.50
87.44 1.28
75.63 + 1.58
88.64 + 0.36
GCCN w/ Best Standard GNN
86.38 6.49
74.41 + 1.77
78.23 + 1.47
77.10 + 0.83
88.57
0.58
75.89 + 1.84
89.40 + 0.57
GCCN ww PNA
83.83 6.31
73.91
77.24 1.72
76.5
87.27 = 0.64
74.63 = 1.32
86.34 + 0.38
GCCN wM GATv2
86.38 4.15
72.54 +
77.78 0.94
77.04 0.63
85.11 0.46
72.03
2.54
88.32 + 0.38
I. Performance versus Size Complexity
In, Fig. 10, we extend Fig. 5 to all benchmark datasets. As before, we keep GCCN layers and GNN sublayers in cach subplot constant, matching those of the best model of that dataset.
28

---

TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks
NCI1
NC1109
Relativee Performance (%)
100
100
98
96-
96
94-
92
92
90
40
50
70
80
90
100
50
60
70
93
100
PROTEINS
MUTAG
Relative Performance (%)
100
98
96
96
90
88
92
06
100
100
amazon, ratings
PubMed
100
Relative Performance (9)
95
90
85
94
80-
75
90
40
50
60
100
90
92
96
96
100
102
104
citeseer
Cora
Relative Performance (96)
300
100
90
99
96
94
92
92
90
90
90
92
94
98
100
102
104
50
60
70
80
90
100
Relative Parameter (%)
Relative Parameter Size (%)
Neighborhoods
GNN used WN
N2. .2. NL.
No. .I.
Na. 7. AZ.
No. W.. .VL v. .V1 n. N. N,
NP. A. .M. M7.
GCN
N2. 1. Af, y .V. .V MP.
Nº. A. NR. AP
GIN
GraphSAGE
Nº. NT.
N.... 2. N7 N
MI. Aa WT.
Figure 10. Performance size, scaled best model. We consider models within 10% of the best performance each dataset.
29


Reviews:
[{"review_id": "soWiDSZNjK", "reviewer": "Reviewer_hN8B", "summary": "This paper introduces generalized combinatorial complex neural networks (GCCNs), which provide a general technique for turning any existing (graph) neural network architecture into a topological network, which operates on combinatorial complexes. Their method operates by turning a combinatorial complex into a series of graphs, where each one is defined by a neighborhood function of the combinatorial complex. The proposed GCCN is proven to be more expressive than the existing CCNNs, preserves the appropriate permutation symmetries, and also tends to outperform CCNNs on TopoBenchmark graph tasks. In computational complexity, it interpolates between GNNs and CCNNs. The paper also presents a codebase, TopoTune, for implementing GCCNs.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n**(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.”\n\n**(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.\n\n**(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.\n\n**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).\n\n**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.\n\nIn 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 ($\\omega_\\mathcal{N}$).\n\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.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "The benchmark dataset makes sense, as well as the comparison to CCNNs. However, it would have also been helpful to compare to non-CCNN baselines, as in the original TopoBenchmark paper.", "theoretical_claims": "I did not check correctness of proofs", "experimental_designs_or_analyses": "No issues found", "relation_to_broader_scientific_literature": "This paper builds naturally on Hajij et all, which introduced combinatorial complexes and architectures for them (CCNNs). The new class of architectures, GCCNs, extends CCNNs. It also answers open challenges in TDL as posed by Papamarkou et al 2024. It uses, as a base architecture for the message passing on each created augmented Hasse graph, methods like GIN, GCN, GraphSAGE, GAT, etc, thereby building on those papers.", "essential_references_not_discussed": "n/a", "other_strengths_and_weaknesses": "Strengths: The paper is clearly written and outlines their contributions clearly, which is a general architectural framework that seems to address open challenges in TDL. Moreover, the paper produces a library, TopoTune, which promises to be a useful mechanism for fast prototyping of TDL architectures on TopoBenchmark and for reproducing their results with GCCNs. Armed with TopoTune, the experimental results cover a wide range of GCCN architectures, which often outperform existing CCNNs. Although I am not very familiar with TDL, these seems like a valuable contribution to the field.\n\nWeaknesses: The paper only compares GCCNs to CCNNs, rather than other non-topological, perhaps more standard architectures (vanilla GNNs, transformers, etc). Also, their GCCN can be instantiated with many different choices of base graph architecture, but it is not clear how to choose one in practice.", "other_comments_or_suggestions": "As someone who is not very embedded in the TDL community, it would expand the reach of your paper to better motivate (briefly) the field of TDL in the intro. The first paragraph does a nice job of this, conceptually; are there empirical works that you can cite, which demonstrate the benefits of capturing these multi-way relationships? For example, what are some datasets on which TDL is SOTA?", "questions_for_authors": "1. Is there a reason for only comparing with CCNNs on the tasks in TopoBenchmark? \n2. Relatedly, the best MAE reported for ZINC in Table 1 is 0.19. paperswithcode reports the SOTA number as 0.056, as achieved by chromatic self-attention (Menegaux et al 2023), a non-topological or combinatorial architecture. Does GCCN achieve SOTA on any of the datasets in TopoBenchmark, when considering architectures other than CCNNs? If not, how do the authors view the utility of the benchmark, and their GCCN architecture, in a broader context?\n3. Based on Table 1, the specific architecture choice of $\\omega_{\\mathcal{N}}$ that performs best varies by task. In practice, how then should one choose $\\omega_{\\mathcal{N}}$? Is there any way to avoid exhaustive search and retraining, e.g. by using task-specific insights?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "gPIr3dM49h", "reviewer": "Reviewer_7uco", "summary": "The paper introduces Generalized Combinatorial Complex Neural Networks (GCCNs) extending Topological Deep Learning (TDL) models to the combinatorial domain. It generalizes Combinatorial Complex Neural Networks (CCNNs), offering improved expressivity and performance, often with reduced model complexity. To facilitate the design and training of these TDL models, they present TopoTune, a lightweight software framework that simplifies the creation of TDL architectures.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your helpful feedback. We address comments and questions below.\n\n**(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.\n\n**(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 ($\\omega_\\mathcal{N}$).\n\n**(Weakness 3) Open problems:** \n- (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).”\n- (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.”\n\n**Questions**\n1. 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),...”\n\n2. 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.\n\nThank 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.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "No", "experimental_designs_or_analyses": "Yes", "relation_to_broader_scientific_literature": "- This work builds on prior Topological Deep Learning (TDL) work by generalizing Combinatorial Complex Neural Networks (CCNNs), which are more expressive than GNNs.\n- The authors prove their method, i.e., Generalized CCNNs (GCCNs) subsume CCNNs, achieving comparable or better performance with lower model complexity.\n- The paper introduces TopoTune, a software framework that simplifies the design and training of TDL models, similar to how PyG and DGL standardized GNNs.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "- The paper achieves multi-level information aggregation through an ensemble of augmented hasse graphs. Could a similar effect be obtained using combinatorial or Hodge Laplacians, as explored in https://arxiv.org/pdf/2403.06687 and https://arxiv.org/abs/2309.12971, which aim to integrate information across different ranked simplices via spectral filtering?\n\n- While the proposed Generalized Combinatorial Complex Network (GCCN) is theoretically broad, the experiments are limited to simplicial and cellular complexes—potentially due to memory constraints or challenges in the lifting procedure. Given this, would it be useful to compare GCCNs with models for cell and simplicial complexes, such as MPSN or CWN, to evaluate the method’s efficacy across different topological representations?\n\n- The paper claims to address Open Problem 1 from the TDL position paper. However, the experiments primarily focus on simple graph and node classification tasks, lacking diversity in application domains. This is also valid for open problem 3 as the paper proposes a method not a benchmarking software like TopoBenchmark. The authors should be cautious in making such claims, as the current scope of evaluation may not fully substantiate solving the broader open problem, potentially leading to misleading interpretations.", "other_comments_or_suggestions": "See weaknesses and questions.", "questions_for_authors": "- Are the hyper-parameters of baselines also optimized as they are done for the GCCN?\n- How does the transformer version of GCCN compare to this https://arxiv.org/pdf/2405.14094? It seems it has already solved the open problem 11, as mentioned in the TDL position paper.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "VR34RFMtIc", "reviewer": "Reviewer_DsXg", "summary": "This paper aims to further topological deep learning by allowing for the easy adaption of any GNN into network for cell complexes. The basis for their method is representing the cell complexes with augmented hesse graphs, running GNNs on these graphs separately and then combining features from each of the graphs\n\nThey show that their class of networks has the same expressive power as CCNN, unlike other graph based methods, but allow for easy plug-and-play GNN backbones (on the Hesse graphs). This allows for users to more easily access TDL methods without the loss of expressivity. Although this idea is fairly simple, it has the potential to make a significant impact.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review. We are happy to read that you believe there is potential for significant impact and acceleration of TDL.\n\nWe 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.\nAs 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). \n\nWe 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.”\n\nPlease let us know if you have any other questions.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "I skimmed but did not thoroughly check the proofs. I view the theory as fairly \"standard\"/\"unsurpising\", but I am reasonably confident it is likely correct (or at least not substantially wrong)", "experimental_designs_or_analyses": "I did not check this", "relation_to_broader_scientific_literature": "This should help accelerate research into TDL", "essential_references_not_discussed": "None that I am aware of", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "Can GNNs that don't fit the strict definition of message passing, e.g. ChebNet be incorporated into your software?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4UhoE5fXjF", "reviewer": "Reviewer_478c", "summary": "The paper suggests a general way to lift GNN architectures to work with simplicial and cell complexes. They implemented their project within a well-known benchmarking suite. The projected is completed with a benchmarking effort covering training on multiple classical graph datasets.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n**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.\n\n**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.\n\n**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.”\n\n**Weaknesses**\n- 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).\n- 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.\n\n**Questions**\n1. Addressing in two parts:\n- 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.\n- 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).\n2. We improve the strengths of our empirical results. \n- 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). \n- 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. \n\n3. 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). \n \nTo 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.", "claims_and_evidence": "The empirical power of the method is only somewhat supported - the experiments are run on extremely small and outdated datasets and it might be a good idea to mention that pure GNNs do a lot better on those. (For example a simple GIN from 2018 achieves more or less identical performance on MUTAG and newer methods are not even tested on them any more). This may be a problem of the field of topological deep learning that modern benchmarks are lacking.\n\nFurther claims (general method etc) are well-supported.", "methods_and_evaluation_criteria": "I would like to argue that the datasets are heavily outdated and should no longer be used for any scientific claims.\n\nAlso the base models are a bit simplistic, GATv2 (or better transformerconv) as message passing, as well as architectures such as GatedGCN or PNA typically work better than the basic ones used in the experiments here.", "theoretical_claims": "The theoretical claims look ok, but I have not checked the proofs in the appendix. It would have been nice to mention the main proof idea in one sentence in the main paper, indicating whether the proof is straightforward and what the property hinges on.", "experimental_designs_or_analyses": "I did not verify the experiments myself as the code is not yet available.\n\nThe authors state that they did only use default configurations for all methods and did not perform any hyperparameter tuning. I would like to argue that this is often a very bad idea leading to incomparable numbers between architectures that are too often not representative of the methods' true performance.", "relation_to_broader_scientific_literature": "looks good to me, but I am also from the GNN side and not the TDL side.", "essential_references_not_discussed": "-", "other_strengths_and_weaknesses": "Strengths:\n- relatively easy to read, also for non-experts in TDL\n\nWeaknesses:\n- the general construction that is suggested is not highlighted in terms of any structure, its just part of section 4 and the key observation/construction could have been highlighted a lot more. Also an illustration of how any GCN architecture is turned into a TDL architecture could have been nice.\n- the experiments (as mentioned earlier)", "other_comments_or_suggestions": "none", "questions_for_authors": "How did you choose datasets and base models?\n\nWould it be possible to re-run the experiments on e.g. molPCBA and malnet-tiny including automatic hyperparameter tuning?\n\nIs it really common that the whole lifting procedure does not help with the performance? As the absolute values reported in table 1 are more or less what one would expect of a pure GNN on the same datasets (at least if there is still data on those).", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The authors present TopoTune, a framework that generalizes Graph Neural Networks to Topological Neural Networks by introducing Generalized Combinatorial Complex Neural Networks (GCCNs). All reviewers gave a positive rating. It is the AC’s opinion that the work makes a nice contribution to Topological Deep Learning by providing a standardized, accessible framework. The software implementation further enhances the paper's impact by democratizing TDL research.  I recommend accepting this paper.
Decision: Accept (poster)