Coarse Ricci curvature of weighted Riemannian manifolds

Key Points

• The aim is to understand how the generalized Ricci tensor can be estimated in weighted Riemannian manifolds using Wasserstein distances.
• Analyzed the generalized Ricci tensor in weighted complete Riemannian manifolds.
• Utilized scaled metric derivatives of Wasserstein 1-distances.
• Applied findings to random geometric graphs from Poisson point processes.
• Demonstrated asymptotic retrieval of generalized Ricci tensor from metric derivatives.
• Showed convergence of coarse curvature in graphs to the generalized Ricci tensor.