Ollivier's Ricci curvature, local clustering and curvature dimension inequalities on graphs

Details Ollivier's Ricci curvature, local clustering and curvature dimension inequalities on graphs Ollivier's Ricci curvature, local clustering and curvature dimension inequalities on graphs

2011-03-21

arxiv.org/abs/1103.4037v2

Mathematics

Combinatorics

18 pages, 1 figures. We add a remark in Cor. 1 and the Remark 17, and two references along with them. We also correct some typ

Scientific paper

In Riemannian geometry, Ricci curvature controls how fast geodesics emanating from a common source are diverging on average, or equivalently, how fast the volume of distance balls grows as a function of the radius. Recently, such ideas have been extended to Markov processes and metric spaces. Employing a definition of generalized Ricci curvature proposed by Ollivier and applied in graph theory by Lin-Yau, we derive lower Ricci curvature bounds on graphs in terms of local clustering coefficients, that is, the relative proportion of connected neighbors among all the neighbors of a vertex. This translates the above Riemannian ideas into a combinatorial setting. We also study curvature dimension inequalities on graphs, building upon previous work of several authors.

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