Paper Info
Title
Ollivier's Ricci Curvature, Local Clustering and Curvature-Dimension Inequalities on Graphs.
Abstract
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. ? 2013 Springer Science+Business Media New York.
Year
DOI
Venue
2014
10.1007/s00454-013-9558-1
Discrete & Computational Geometry
Keywords
DocType
Volume
Ollivier’s Ricci curvature,Curvature dimension inequality,Local clustering,Graph Laplace operator
Journal
51
Issue
ISSN
Citations 
2
14320444
19
PageRank 
References 
Authors
3.56
2
2
Name
Order
Citations
PageRank
Jürgen Jost19512.39
Shiping Liu2265.45