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Distance bounds for graphs with some negative Bakry-Émery curvature. (English) Zbl 1416.05093

Summary: 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.

MSC:

05C12 Distance in graphs
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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[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. · Zbl 1304.35310
[2] Dominique Bakry and Michel Émery. Diffusions hypercontractives. In Séminaire de Probabilités XIX 1983/84, pages 177-206. Springer, 1985. · Zbl 0561.60080
[3] Frank Bauer, Bobo Hua, and Matthias Keller. On the l^p spectrum of Laplacians on graphs. Advances in Mathematics, 248:717-735, 2013. · Zbl 1283.05162
[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. · Zbl 1323.35189
[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. · Zbl 1297.05143
[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. · Zbl 1309.05059
[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. · Zbl 1309.05117
[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. · Zbl 0755.58048
[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. · Zbl 1362.60084
[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. · Zbl 1329.53059
[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. · Zbl 1256.53028
[12] Max Fathi and JanMaas. Entropic Ricci curvature bounds for discrete interacting systems. The Annals of Applied Probability, 26(3): 1774-1806, 2016. · Zbl 1345.60076
[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. · Zbl 1244.60099
[14] Matthew Folz. Volume growth and stochastic completeness of graphs. Transactions of American Mathematical Society, 366(4): 2089-2119, 2014. · Zbl 1325.60069
[15] Max Fathi and Yan Shu. Curvature and transport inequalities for Markov chains in discrete spaces. Bernoulli, 24(1): 672- 698, 2018. · Zbl 1396.60084
[16] Alexander Grigor’yan, Xueping Huang, and JunMasamune. On stochastic completeness of jump processes. Mathematische Zeitschrift, 271(3-4): 1211-1239, 2012. · Zbl 1408.60076
[17] Chao Gong and Yong Lin. Equivalent properties for CD inequalities with unbounded Laplacians. Chinese Annals of Mathematics B, 38(5): 1059-1070, 2017. · Zbl 1432.58020
[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. · Zbl 1298.31009
[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. · Zbl 1435.35400
[20] Bobo Hua and Yong Lin. Stochastic completeness for graphs with curvature dimension conditions. Advances in Mathematics, 306: 279-302, 2017. · Zbl 1364.35402
[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). · Zbl 1432.35213
[22] Xueping Huang. On stochastic completeness of weighted graphs. PhD Thesis, Bielefeld University, 2011. · Zbl 1229.60088
[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. · Zbl 1294.05061
[24] Jürgen Jost. Riemannian geometry and geometric analysis. Universitext. Springer-Verlag, Berlin, fifth edition, 2008. · Zbl 1143.53001
[25] Christian Ketterer. On the geometry of metric measure spaces with variable curvature bounds. The Journal of Geometric Analysis, 27(3): 1951-1994, 2017. · Zbl 1375.53057
[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. · Zbl 1424.58014
[27] Yong Lin, Linyuan Lu, and Shing-Tung Yau. Ricci curvature of graphs. Tohoku Mathematical Journal, Second Series, 63(4): 605-627, 2011. · Zbl 1237.05204
[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. · Zbl 1394.53047
[29] John Lott and Cédric Villani. Ricci curvature for metric-measure spaces via optimal transport. Annals of Mathematics, 169: 903-991, 2009. · Zbl 1178.53038
[30] Yong Lin and Shing-Tung Yau. Ricci curvature and eigenvalue estimate on locally finite graphs. Mathematical Research Letters, 17(2): 343-356, 2010. · Zbl 1232.31003
[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. · Zbl 1282.60072
[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. · Zbl 1400.05219
[33] Florentin Münch. Remarks on curvature dimension conditions on graphs. Calculus of Variations and Partial Differential Equations, 56(1): no.11, 2017. · Zbl 1362.53048
[34] Yann Ollivier. Ricci curvature of Markov chains on metric spaces. Journal of Functional Analysis, 256(3): 810-864, 2009. · Zbl 1181.53015
[35] Peter Petersen. Riemannian geometry, volume 171 of Graduate Texts in Mathematics. Springer, Cham, third edition, 2016. · Zbl 1417.53001
[36] Peter Petersen and Chadwick Sprouse. Integral curvature bounds, distance estimates and applications. Journal of Differential Geometry, 50(2): 269-298, 1998. · Zbl 0969.53017
[37] Michael Schmuckenschläger. Curvature of nonlocal markov generators. Convex geometric analysis (Berkeley, CA, 1996), 34: 189-197, 1998. · Zbl 0943.60088
[38] Karl-Theodor Sturm. On the geometry of metric measure spaces, I. Acta Mathematica, 196(1): 65-131, 2006. · Zbl 1105.53035
[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. · Zbl 1214.35075
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