Topological Data Analysis in Graph Neural Networks: Surveys and Perspectives

For many years, topological data analysis (TDA) and deep learning (DL) have been considered separate data analysis and representation learning approaches, which have nothing in common. The root cause of this challenge comes from the difficulties in building, extracting, and integrating TDA constructs, such as barcodes or persistent diagrams, within deep neural network architectures. Therefore, the powers of these two approaches are still on their islands and have not yet combined to form more powerful tools for dealing with multiple complex data analysis tasks. Fortunately, we have witnessed several remarkable attempts to integrate DL-based architectures with topological learning paradigms in recent years. These topology-driven DL techniques have notably improved data-driven analysis and mining problems, especially within graph datasets. Recently, graph neural networks (GNNs) have emerged as a popular deep neural architecture, demonstrating significant performance in various graph-based analysis and learning problems. Explicitly, within the manifold paradigm, the graph is naturally considered as a topological object (e.g., the topological properties of the given graph can be represented by the edge weights). Therefore, integrating TDA and GNN is considered an excellent combination. Many well-known studies have recently presented the effectiveness of TDA-assisted GNN-based architectures in dealing with complex graph-based data representation analysis and learning problems. Motivated by the successes of recent research, we present systematic literature about this nascent and promising research direction in this article, which includes general taxonomy, preliminaries, and recently proposed state-of-the-art topology-driven GNN models and perspectives.