Ollivier’s Ricci Curvature, Local Clustering and Curvature-Dimension Inequalities on Graphs

• Published in: Discrete & computational geometry Vol. 51; no. 2; pp. 300 - 322
• Main Authors: Jost, Jürgen, Liu, Shiping
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.03.2014; Springer Nature B.V
• Subjects: Clustering; Coefficients; Combinatorial analysis; Combinatorics; Computational mathematics; Computational Mathematics and Numerical Analysis; Curvature; Geometry; Graphs; Inequalities; Mathematical analysis; Mathematics; Mathematics and Statistics; Triangles; Curvature dimension inequality; Local clustering; Ollivier’s Ricci curvature; Graph Laplace operator
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-013-9558-1

In this paper, we explore the relationship between one of the most elementary and important properties of graphs, the presence and relative frequency of triangles, and a combinatorial notion of Ricci curvature. We employ a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin–Yau. In analogy with curvature notions in Riemannian geometry, we interpret this Ricci curvature as a control on the amount of overlap between neighborhoods of two neighboring vertices. It is therefore naturally related to the presence of triangles containing those vertices, or more precisely, the local clustering coefficient, that is, the relative proportion of connected neighbors among all the neighbors of a vertex. This suggests to derive lower Ricci curvature bounds on graphs in terms of such local clustering coefficients. We also study curvature-dimension inequalities on graphs, building upon previous work of several authors.