Heat Flow and Concentration of Measure on Directed Graphs with a Lower Ricci Curvature Bound

• Published in: Potential analysis Vol. 59; no. 3; pp. 955 - 969
• Main Authors: Ozawa, Ryunosuke, Sakurai, Yohei, Yamada, Taiki
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-10-2023; Springer Nature B.V
• Subjects: Curvature; Functional Analysis; Geometry; Graph theory; Graphs; Heat transfer; Heat transmission; Mathematics; Mathematics and Statistics; Potential Theory; Probability Theory and Stochastic Processes; Primary 05C20; Concentration of measure; 05C12; 53C23; 53C21; Ricci curvature; Heat flow; 05C81; Gradient estimate; Directed graph; Functional inequality
• ISSN: 0926-2601; 1572-929X
• DOI: 10.1007/s11118-022-09994-9

In a previous work (Ozawa et al. Calc. Var. Partial Diff. Equ.
59
(4), 39
2020
), the authors introduced a Lin-Lu-Yau type Ricci curvature for directed graphs referring to the formulation of the Chung Laplacian. The aim of this note is to provide a von Renesse-Sturm type characterization of our lower Ricci curvature bound via a gradient estimate for the heat semigroup, and a transportation inequality along the heat flow. As an application, we will conclude a concentration of measure inequality for directed graphs of positive Ricci curvature.