TopoTune : A Framework for Generalized Combinatorial Complex Neural Networks (2410.06530v4)

Abstract: Graph Neural Networks (GNNs) excel in learning from relational datasets as they preserve the symmetries of the graph domain. However, many complex systems -- such as biological or social networks -- involve multiway complex interactions that are more naturally represented by higher-order topological domains. The emerging field of Topological Deep Learning (TDL) aims to accommodate and leverage these higher-order structures. Combinatorial Complex Neural Networks (CCNNs), fairly general TDL models, have been shown to be more expressive and better performing than GNNs. However, differently from the GNN ecosystem, TDL lacks a principled and standardized framework for easily defining new architectures, 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.

• The paper introduces TopoTune which systematically transforms graph-based neural networks into GCCNs via an ensemble of augmented Hasse graphs.
• It develops novel topological architectures with cell permutation equivariance, demonstrating competitive performance and improved computational efficiency.
• The framework integrates with TopoBenchmarkX for standardized evaluations, offering a practical tool to advance complex system modeling in topological deep learning.

Overview of TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks

The paper introduces TopoTune, a novel framework designed to generalize combinatorial complex neural networks (CCNNs) into a broader category termed Generalized Combinatorial Complex Networks (GCCNs). This framework seeks to address several limitations within topological deep learning (TDL), particularly the lack of standardized methodologies that allow for easy creation and exploration of new neural network architectures within higher-order topological domains.

Problem Statement

Graph Neural Networks (GNNs) are limited in their ability to capture more complex, higher-order interactions in systems such as biological networks, due to their reliance on pairwise edge representations. While Combinatorial Complex Neural Networks (CCNNs) can model such interactions, they suffer from restricted accessibility and applicability due to the absence of a standardized framework.

Proposed Solution

The authors propose GCCNs to systematically transform any neural network, originally designed for graph-based data, into a TDL counterpart. Key contributions include:

1. Systematic Generalization: The transformation of combinatorial complexes into a collection of graphs via a new expansion mechanism allows existing neural networks to be seamlessly adapted to topological domains.
2. General Architectures: A novel class of TDL architectures is developed, characterized by their cell permutation equivariance and comparable expressiveness to CCNNs.
3. Implementation and Benchmarking: The introduction of a lightweight module, TopoTune, enables researchers to define and implement GCCNs easily within their existing workflows. It integrates with the TopoBenchmarkX platform for standardized evaluation.

Methodological Innovations

The GCCNs leverage two primary methodological innovations:

• Ensemble of Strictly Augmented Hasse Graphs: Combinatorial complexes are expanded into an ensemble of graphs based on neighborhood function, which are then processed by a base model, such as GNN or Transformer.
• Per-rank Neighborhoods: Allows for ranking of cells, enabling efficient message-passing through selected neighborhoods, thus reducing computational complexity.

Results

Experimental results demonstrate that GCCNs:

• Consistently meet or surpass the performance of existing CCNNs across different datasets, domains, and tasks.
• Offer computational efficiency, often requiring fewer parameters than comparable CCNN architectures.

The experiments span various tasks, including node classification and graph-level regression, showcasing GCCNs' capability to handle diverse datasets while retaining model expressiveness.

Implications and Future Directions

The development of the TopoTune framework signifies a crucial step toward the standardization of TDL methodologies. By democratizing the development of TDL architectures, it opens pathways for broader applications beyond the specialized domain.

Future work could focus on further enhancing the adaptability of GCCNs to various neural network architectures and exploring applications in sparse or multimodal datasets. Additionally, extending GCCNs to incorporate recent advancements in GNN design could ensure the continued relevance and utility of this framework.

In conclusion, TopoTune provides a systematic and accessible approach to elevating the standard of TDL, potentially leading to significant advancements in modeling complex systems with multiway interactions.