nLab Ollivier-Ricci curvature

Idea

Ollivier-Ricci curvature (ORC) is an analogue of Ricci curvature applicable also to graphs, based on optimal transport theory. ORC compares the counting distance between vertices to the Wasserstein distance between their neighbourhood probability measures.

References

The original articles:

• Yann Ollivier: Ricci curvature of metric spaces, Comptes Rendus Mathematique, 345 11 (2007) 643–646 [doi:10.1016/j.crma.2007.10.041, pdf]

• Yann Ollivier: Ricci curvature of markov chains on metric spaces, Journal of Functional Analysis 256 3 (2009) 810–864 [doi:10.1016/j.jfa.2008.11.001, arXiv:math/0701886]

See also:

• Corinna Coupette, Sebastian Dalleiger, Bastian Rieck: Ollivier-Ricci Curvature for Hypergraphs: A Unified Framework, ICLR (2023) [arXiv:2210.12048]

On discretized gravity via Ollivier-Ricci curvature of piecewise flat spacetimes:

• Carlo A. Trugenberger: Networks as the fundamental constituents of the universe, J. Phys. Complex. 6 (2025) 042001 [arXiv:2512.17676]