TopoTune : A Framework for Generalized Combinatorial Complex Neural Networks

TL;DR

This paper addresses the lack of standardized frameworks for topological deep learning (TDL) by introducing Generalized Combinatorial Complex Networks (GCCNs), a broad class of models that generalize Combinatorial Complex Neural Networks (CCNNs) and can transform any graph neural network into a TDL counterpart. It formalizes GCCNs around ensembles of strictly augmented Hasse graphs and per-rank neighborhoods, enabling principled, scalable design of topological architectures. The authors prove GCCNs subsume CCNNs, are permutation-equivariant, and can be more expressive under Weisfeiler-Leman-based analyses; they also provide time-complexity analyses and show that GCCNs achieve state-of-the-art performance with fewer parameters on diverse datasets. To accelerate adoption, the paper introduces TopoTune, a lightweight software integrated with TopoBench for defining, building, and training GCCNs, thereby democratizing access to TDL and enabling rapid benchmarking across domains. The combination of a solid theoretical framework and practical tooling suggests a tangible path toward broader application of higher-order relational learning in real-world systems.

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.

Figures (11)

• Figure 1: Generalized Combinatorial Complex Network (GCCN). The input complex $\mathcal{C}$ has neighborhoods $\mathcal{N_C} = \{\mathcal{N}_1, \mathcal{N}_2, \mathcal{N}_3$}. A. The complex is expanded into three augmented Hasse graphs $\mathcal{G}_{\mathcal{N}_i}$, $i=\{1,2,3\}$, each with features $H_{\mathcal{N}_i}$ represented as a colored disc. B. A GCCN layer dedicates one base architecture $\omega_{\mathcal{N}_i}$ (GNN, Transformer, MLP, etc.) to each neighborhood. C. The output of all the architectures $\omega_{\mathcal{N}_i}$ is aggregated rank-wise, then updated. In this example, only the complex's edge features (originally pink) are aggregated across multiple neighborhoods ($\mathcal{N}_2$ and $\mathcal{N}_3$).
• Figure 2: Augmented Hasse graphs. Expansions of a combinatorial complex $\mathcal{C}$ (middle) into two augmented Hasse graphs: (left) the Hasse graph induced by $\mathcal{N}_\mathcal{C}= \{\mathcal{N}_{I,\downarrow}\}$; (right) the augmented Hasse graph induced by $\mathcal{N}_\mathcal{C}= \{\mathcal{N}_{I,\downarrow},\mathcal{N}_{A,\uparrow}\}$. Information on cell rank is discarded (we retain rank color for illustrative purposes).
• Figure 3: Ensemble of strictly augmented Hasse Graphs. Given a complex $\mathcal{C}$ with neighborhood structure including both incidence and upper adjacency (left), this graph expansion (right) produces one augmented Hasse graph for each neighborhood.
• Figure 4: Per-rank neighborhoods. Given a complex $\mathcal{C}$ (left), we illustrate four examples of per-rank neighborhoods (right). In each case, they only include rank-specific cells.
• Figure 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 a different neighborhood structure $\mathcal{N}_\mathcal{C}$, some of which can only be represented as per-rank structures ($\square$ in legend), and message function $\omega_\mathcal{N}$. The amount of layers is kept constant according to the best performing model. The axes are scaled relative to this model.

Theorems & Definitions (21)

• Definition 2.1: CC-Homomorphism hajij2023tdl
• Definition 2.2: Labeled Combinatorial Complex
• Definition 2.3: CC-Homomorphism induced by $(\mathcal{N}_1, \mathcal{N}_2)$
• Proposition 2.4
• proof
• Definition 2.5: CC-Isomorphism induced by $(\mathcal{N}_1, \mathcal{N}_2)$
• Definition 2.6: The CC Weisfeiler-Leman (CCWL) test on labeled combinatorial complexes
• Definition 2.7: Neighborhood of $k$-sets of cells
• Definition 2.8: The CC $k$-Weisfeiler-Leman ($k$-CCWL) test on combinatorial complexes
• Definition 2.9: $k$-CCNNs