Topological Data Analysis in Graph Neural Networks: Surveys and Perspectives

Area of Science:

• Machine Learning
• Data Science
• Computational Topology

Background:

• Topological Data Analysis (TDA) and Deep Learning (DL) were historically separate fields.
• Integrating TDA constructs (barcodes, persistent diagrams) into DL architectures posed significant challenges.
• Recent advancements show promise in combining DL with topological learning, especially for graph data.

Purpose of the Study:

• To provide a systematic literature review of topology-driven Graph Neural Networks (GNNs).
• To explore the integration of TDA and GNNs for enhanced data analysis and representation learning.
• To establish a taxonomy and overview of state-of-the-art models in this emerging field.

Main Methods:

• Literature review and synthesis of existing research on TDA and GNN integration.
• Analysis of graph data as topological objects within the manifold paradigm.
• Categorization of topology-driven GNN models based on their methodologies.

Main Results:

• TDA-assisted GNNs demonstrate significant effectiveness in complex graph-based data representation and learning.
• The integration leverages the topological properties inherent in graph structures.
• This combination offers powerful tools for data-driven analysis and mining.

Conclusions:

• The integration of TDA and GNNs represents a promising research direction.
• This review consolidates knowledge and highlights the potential of topology-driven GNNs.
• Future research can build upon this foundation for advanced data analysis solutions.