The manifold-based analysis presented in the paper provides a robust framework for understanding the generalization capabilities of Graph Neural Networks (GNNs) when applied to graph data derived from continuous topological spaces. This framework can be extended to real-world graph data, such as social networks and biological networks, by considering the underlying structures of these networks as manifolds.
In social networks, for instance, the relationships between individuals can be modeled as points on a manifold where the connections (edges) represent social interactions. By applying the manifold theory, we can analyze how GNNs generalize to unseen nodes (individuals) based on the manifold's properties, such as curvature and dimensionality. The insights regarding the generalization gap decreasing with the number of nodes can inform strategies for sampling and training on larger social networks, ensuring that GNNs can effectively learn from the available data while maintaining predictive accuracy on unseen nodes.
Similarly, in biological networks, such as protein-protein interaction networks or metabolic networks, the manifold perspective can help in understanding the complex relationships and interactions among biological entities. By treating these networks as sampled points from a manifold, we can leverage the theoretical results to design GNNs that can generalize well across different biological contexts. This approach can also facilitate the integration of multi-modal data, where different types of biological data can be represented as different manifolds, allowing for a more comprehensive analysis of biological systems.