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Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel

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Distance Bounds for Graphs with Some Negative Bakry-Émery Curvature

Shiping Liu
  • Corresponding author
  • School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui Province, China
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/ Florentin Münch / Norbert Peyerimhoff / Christian Rose
Published Online: 2019-03-22 | DOI: https://doi.org/10.1515/agms-2019-0001


We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the tubular neighborhood with an explicit radius around the non-positively curved vertices. Those results seem to be the first assuming non-constant Bakry-Émery curvature assumptions on graphs.

Keywords: Bakry-Émery curvature; discrete Bonnet-Myers theorem; intrinsic metric; heat semigroup

MSC 2010: Primary: 51K10; Secondary: 51F99, 05C12


  • [1]

    Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré. Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Mathematical Journal, 163(7): 1405–1490, 2014.CrossrefWeb of ScienceGoogle Scholar

  • [2] Dominique Bakry and Michel Émery. Diffusions hypercontractives. In Séminaire de Probabilités XIX 1983/84, pages 177–206. Springer, 1985.Google Scholar

  • [3] Frank Bauer, Bobo Hua, and Matthias Keller. On the lp spectrum of Laplacians on graphs. Advances in Mathematics, 248:717–735, 2013.Web of ScienceGoogle Scholar

  • [4] Frank Bauer, Paul Horn, Yong Lin, Gabor Lippner, Dan Mangoubi, Shing-Tung Yau. Li-Yau inequality on graphs. Journal of Differential Geometry, 99(3): 359–405, 2015.Google Scholar

  • [5] Frank Bauer, Jürgen Jost, and Shiping Liu. Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator. Mathematical Research Letters, 19(6): 1185–1205, 2012.CrossrefGoogle Scholar

  • [6] Frank Bauer, Matthias Keller, and Radosław K. Wojciechowski. Cheeger inequalities for unbounded graph Laplacians. Journal of the European Mathematical Society, 17(2): 259–271, 2015.CrossrefGoogle Scholar

  • [7] Fan Chung, Yong Lin, and Shing-Tung Yau. Harnack inequalities for graphs with non-negative Ricci curvature. Journal of Mathematical Analysis and Applications, 415(1): 25–32, 2014.Google Scholar

  • [8] K. D. Elworthy, Manifolds and graphs with mostly positive curvatures. Stochastic analysis and applications (Lisbon, 1989), 96–110, Progress in Probability, 26, Birkhäuser Boston, Boston, MA, 1991.Google Scholar

  • [9] Matthias Erbar, Christopher Henderson, Georg Menz, Prasad Tetali. Ricci curvature bounds for weakly interacting Markov chains. Electronic Journal of Probability, 22: no. 40, 2017.Web of ScienceGoogle Scholar

  • [10] Matthias Erbar, Kazumasa Kuwada, and Karl-Theodor Sturm. On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces. Inventiones mathematicae, 201(3): 993–1071, 2015.Google Scholar

  • [11] Matthias Erbar and Jan Maas. Ricci curvature of finite Markov chains via convexity of the entropy. Archive for Rational Mechanics and Analysis, 206(3): 997–1038, 2012.Google Scholar

  • [12] Max Fathi and JanMaas. Entropic Ricci curvature bounds for discrete interacting systems. The Annals of Applied Probability, 26(3): 1774–1806, 2016.CrossrefGoogle Scholar

  • [13] Matthew Folz. Gaussian upper bounds for heat kernels of continuous time simple random walks. Electronic Journal of Probability, 16: no. 62, 1693–1722, 2011.Web of ScienceCrossrefGoogle Scholar

  • [14] Matthew Folz. Volume growth and stochastic completeness of graphs. Transactions of American Mathematical Society, 366(4): 2089–2119, 2014.Google Scholar

  • [15] Max Fathi and Yan Shu. Curvature and transport inequalities for Markov chains in discrete spaces. Bernoulli, 24(1): 672– 698, 2018.CrossrefGoogle Scholar

  • [16] Alexander Grigor’yan, Xueping Huang, and JunMasamune. On stochastic completeness of jump processes. Mathematische Zeitschrift, 271(3-4): 1211–1239, 2012.Web of ScienceGoogle Scholar

  • [17] Chao Gong and Yong Lin. Equivalent properties for CD inequalities with unbounded Laplacians. Chinese Annals of Mathematics B, 38(5): 1059–1070, 2017.Google Scholar

  • [18] Bobo Hua and Matthias Keller. Harmonic functions of general graph Laplacians. Calculus of Variations and Partial Differential Equations, 51(1-2): 343–362, 2014.Google Scholar

  • [19] Xueping Huang, Matthias Keller, Jun Masamune, and Radoslaw K. Wojciechowski. A note on self-adjoint extensions of the Laplacian on weighted graphs. Journal of Functional Analysis, 265(8): 1556–1578, 2013.Google Scholar

  • [20] Bobo Hua and Yong Lin. Stochastic completeness for graphs with curvature dimension conditions. Advances in Mathematics, 306: 279–302, 2017.Web of ScienceGoogle Scholar

  • [21] Paul Horn, Yong Lin, Shuang Liu, and Shing-Tung Yau. Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for nonnegative curvature graphs. Journal für die reine und angewandteMathematik, ahead of print, (2017-10-17).Google Scholar

  • [22] Xueping Huang. On stochastic completeness of weighted graphs. PhD Thesis, Bielefeld University, 2011.Google Scholar

  • [23] Jürgen Jost and Shiping Liu. Ollivier’s Ricci curvature, local clustering and curvature-dimension inequalities on graphs. Discrete & Computational Geometry, 51(2): 300–322, 2014.CrossrefGoogle Scholar

  • [24] Jürgen Jost. Riemannian geometry and geometric analysis. Universitext. Springer-Verlag, Berlin, fifth edition, 2008.Google Scholar

  • [25] Christian Ketterer. On the geometry of metric measure spaces with variable curvature bounds. The Journal of Geometric Analysis, 27(3): 1951-1994, 2017.Web of ScienceGoogle Scholar

  • [26] Yong Lin and Shuang Liu. Equivalent properties of CD inequality on graph. Acta Mathematica Sinica, Chinese Series, 61(3): 431-440, 2018. https://arxiv.org/abs/1512.02677.

  • [27] Yong Lin, Linyuan Lu, and Shing-Tung Yau. Ricci curvature of graphs. Tohoku Mathematical Journal, Second Series, 63(4): 605–627, 2011.Google Scholar

  • [28] Shiping Liu, Florentin Münch, and Norbert Peyerimhoff. Bakry-Émery curvature and diameter bounds on graphs. Calculus of Variations and Partial Differential Equations, 57(2): no.67, 2018.Google Scholar

  • [29] John Lott and Cédric Villani. Ricci curvature for metric-measure spaces via optimal transport. Annals of Mathematics, 169: 903–991, 2009.Web of ScienceGoogle Scholar

  • [30] Yong Lin and Shing-Tung Yau. Ricci curvature and eigenvalue estimate on locally finite graphs. Mathematical Research Letters, 17(2): 343–356, 2010.Google Scholar

  • [31] Alexander Mielke. Geodesic convexity of the relative entropy in reversible Markov chains. Calculus of Variations and Partial Differential Equations, 48(1-2): 1-31, 2013.Google Scholar

  • [32] Florentin Münch. Li-Yau inequality on finite graphs via non-linear curvature dimension conditions. Journal de Mathématiques Pures et Appliquées, 120: 130-164, 2018.Google Scholar

  • [33] Florentin Münch. Remarks on curvature dimension conditions on graphs. Calculus of Variations and Partial Differential Equations, 56(1): no.11, 2017.Google Scholar

  • [34] Yann Ollivier. Ricci curvature of Markov chains on metric spaces. Journal of Functional Analysis, 256(3): 810–864, 2009.Google Scholar

  • [35] Peter Petersen. Riemannian geometry, volume 171 of Graduate Texts in Mathematics. Springer, Cham, third edition, 2016.Google Scholar

  • [36] Peter Petersen and Chadwick Sprouse. Integral curvature bounds, distance estimates and applications. Journal of Differential Geometry, 50(2): 269–298, 1998.CrossrefGoogle Scholar

  • [37] Michael Schmuckenschläger. Curvature of nonlocal markov generators. Convex geometric analysis (Berkeley, CA, 1996), 34: 189–197, 1998.Google Scholar

  • [38] Karl-Theodor Sturm. On the geometry of metric measure spaces, I. Acta Mathematica, 196(1): 65–131, 2006.Google Scholar

  • [39] Andreas Weber. Analysis of the physical Laplacian and the heat flow on a locally finite graph. Journal of Mathematical Analysis and Applications, 370(1): 146–158, 2010.Google Scholar

About the article

Received: 2018-04-02

Accepted: 2019-01-29

Published Online: 2019-03-22

Published in Print: 2019-03-01

Citation Information: Analysis and Geometry in Metric Spaces, Volume 7, Issue 1, Pages 1–14, ISSN (Online) 2299-3274, DOI: https://doi.org/10.1515/agms-2019-0001.

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© 2019 Shiping Liu et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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