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

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 f...

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Bibliographic Details
Published inDiscrete & computational geometry Vol. 51; no. 2; pp. 300 - 322
Main Authors Jost, Jürgen, Liu, Shiping
Format Journal Article
LanguageEnglish
Published Boston Springer US 01.03.2014
Springer Nature B.V
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ISSN0179-5376
1432-0444
1432-0444
DOI10.1007/s00454-013-9558-1

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Summary: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.
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ISSN:0179-5376
1432-0444
1432-0444
DOI:10.1007/s00454-013-9558-1