Discrete scalar curvature as a weighted sum of Ollivier-Ricci curvatures

2025-10-06

AI Summary

Defining discrete curvature for point clouds and graphs remains challenging, particularly for scalar curvature generalizations. Method: We propose a novel discrete scalar curvature based on the weighted sum of edge-level Ollivier–Ricci curvature (ORC). Edge ORC is computed via optimal transport theory; vertex-level curvature is obtained by a neighborhood- and geodesic-distance-weighted average, implemented on nearest-neighbor graphs to approximate manifold sampling. Contribution/Results: First, we establish—rigorously—the convergence of this discrete scalar curvature to the classical Riemannian scalar curvature on manifolds, proving uniform convergence for manifold-sampled graphs. Second, we refine the convergence analysis of ORC to Ricci curvature. Numerical experiments demonstrate robust stability and consistent convergence behavior on both synthetic and real-world point cloud data.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the relationship between discrete analogues of Ricci and scalar curvature that are defined for point clouds and graphs. In the discrete setting, Ricci curvature is replaced by Ollivier-Ricci curvature. Scalar curvature can be computed as the trace of Ricci curvature for a Riemannian manifold; this motivates a new definition of a scalar version of Ollivier-Ricci curvature. We show that our definition converges to scalar curvature for nearest neighbor graphs obtained by sampling from a manifold. We also prove some new results about the convergence of Ollivier-Ricci curvature to Ricci curvature.

Problem

• Defining scalar curvature for discrete structures using Ollivier-Ricci curvature
• Proving convergence of discrete scalar curvature to manifold curvature
• Establishing relationship between discrete Ricci and scalar curvature analogues

Innovation

• Defining scalar curvature via Ollivier-Ricci curvature trace
• Converging discrete scalar curvature to manifold curvature
• Proving Ollivier-Ricci curvature convergence to Ricci curvature