Bakry–Émery Curvature Functions on Graphs

Cushing, David ; Liu, Shiping ; Peyerimhoff, Norbert

Canadian journal of mathematics, Tome 72 (2020) no. 1, pp. 89-143

Résumé

We study local properties of the Bakry–Émery curvature function ${\mathcal{K}}_{G,x}:(0,\infty ]\rightarrow \mathbb{R}$ at a vertex $x$ of a graph $G$ systematically. Here ${\mathcal{K}}_{G,x}({\mathcal{N}})$ is defined as the optimal curvature lower bound ${\mathcal{K}}$ in the Bakry–Émery curvature-dimension inequality $CD({\mathcal{K}},{\mathcal{N}})$ that $x$ satisfies. We provide upper and lower bounds for the curvature functions, introduce fundamental concepts like curvature sharpness and $S^{1}$-out regularity, and relate the curvature functions of $G$ with various spectral properties of (weighted) graphs constructed from local structures of $G$. We prove that the curvature functions of the Cartesian product of two graphs $G_{1},G_{2}$ are equal to an abstract product of curvature functions of $G_{1},G_{2}$. We explore the curvature functions of Cayley graphs and many particular (families of) examples. We present various conjectures and construct an infinite increasing family of 6-regular graphs which satisfy $CD(0,\infty )$ but are not Cayley graphs.

Détail

DOI : 10.4153/CJM-2018-015-4

Mots-clés : Bakry–Emery curvature, curvature-dimension inequality, Cayley graph, Cartesian product

Bibliographie

[1] Alon, N. and Roichman, Y., Random Cayley graphs and expanders . Random Structures Algorithms 5(1994), no. 2, 271–284. . Google Scholar | DOI

[2] Bakry, D., Functional inequalities for Markov semigroups. In: Probability measures on groups: recent directions and trends. Tata Inst. Fund. Res., Mumbai, 2006, pp. 91–147. Google Scholar

[3] Bakry, D. and Émery, M., Diffusions hypercontractives . Lecture Notes in Math. 1123, Springer, Berlin, 1985, pp. 177–206.. . Google Scholar | DOI

[4] Bauer, F., Horn, P., Lin, Y., Lippner, G., Mangoubi, D., and Yau, S.-T., Li-Yau inequality on graphs . J. Differential Geom. 99(2015), no. 3, 359–405. Google Scholar

[5] Brouwer, A. E. and Haemers, W. H., Spectra of graphs . Universitext, Springer, New York, 2012. . Google Scholar | DOI

[6] Chung, F. R. K., Lin, Y., and Yau, S.-T., Harnack inequalities for graphs with non-negative Ricci curvature . J. Math. Anal. Appl. 415(2014), 25–32. . Google Scholar | DOI

[7] Chung, F. R. K. and Yau, S.-T., Logarithmic Harnack inequalities . Math. Res. Lett. 3(1996), no. 6, 793–812. . Google Scholar | DOI

[8] Cushing, D., Kangaslampi, R., Lipläinen, V., Liu, S., and Stagg, G. W., The graph curvature calculator and the curvatures of cubic graphs. arxiv:1712.03033. Google Scholar

[9] Davis, M. W., The geometry and topology of Coxeter groups . London Mathematical Society Monographs Series, 32. Princeton University Press, Princeton, NJ, 2008. Google Scholar

[10] Friedman, J., Murty, R., and Tillich, J.-P., Spectral estimates for abelian Cayley graphs . J. Combin. Theory Ser. B 96(2006), no. 1, 111–121. . Google Scholar | DOI

[11] Godsil, C. and Royle, G., Algebraic graph theory . Graduate Texts in Mathematics, 207. Springer-Verlag, New York, 2001. . Google Scholar | DOI

[12] Horn, P., Lin, Y., Liu, Shuang, and Yau, S.-T., Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for nonnegative curvature graphs. J. Reine Angew Math. . Google Scholar | DOI

[13] Hua, B. and Lin, Y., Stochastic completeness for graphs with curvature dimension conditions . Adv. Math. 306(2017), 279–302. . Google Scholar | DOI

[14] Hua, B. and Lin, Y., Graphs with large girth and nonnegative curvature dimension condition . Comm. Anal. Geom., to appear. arxiv:1608.07000. Google Scholar

[15] Jost, J. and Liu, S., Ollivier’s Ricci curvature, local clustering and curvature-dimension inequalities on graphs . Discrete Comput. Geom. 51(2014), no. 2, 300–322. . Google Scholar | DOI

[16] Klartag, B., Kozma, G., Ralli, P., and Tetali, P., Discrete curvature and abelian groups . Canad. J. Math. 68(2016), 655–674. . Google Scholar | DOI

[17] Kolesnikov, A. V. and Milman, E., Brascamp–Lieb type inequalities on weighted Riemannian manifolds with boundary . J. Geom. Anal. 27(2017), no. 2, 1680–1702. . Google Scholar | DOI

[18] Lakzian, A. and Mcguirk, Z., A global Poincaré inequality on graphs via a conical curvature-dimension condition . Anal. Geom. Metr. Spaces 6(2018), 32–47. . Google Scholar | DOI

[19] Lin, Y. and Yau, S.-T., Ricci curvature and eigenvalue estimate on locally nite graphs . Math. Res. Lett. 17(2010), no. 2, 343–356. . Google Scholar | DOI

[20] Liu, S., Münch, F., and Peyerimhoff, N., Curvature and higher order Buser inequalities for the graph connection Laplacian . SIAM J. Discrete Math., to appear. arxiv:1512.08134. Google Scholar

[21] Liu, S., Münch, F., and Peyerimhoff, N., Bakry-Émery curvature and diameter bounds on graphs . Calc. Var. Partial Differential Equations 57(2018), 57–67. arxiv:1608.07778 . Google Scholar | DOI

[22] Liu, S. and Peyerimhoff, N., Eigenvalue ratios of nonnegatively curved graphs . Combinatorics, Probability and Computing. . Google Scholar | DOI

[23] Münch, F., Li-Yau inequality on finite graphs via non-linear curvature dimension conditions. arxiv:1412.3340. Google Scholar

[24] Münch, F., Remarks on curvature dimension conditions on graphs . Calc. Var. Partial Differential Equations 56(2017), no. 1. Art. 11, 8 pp. . Google Scholar | DOI

[25] Ohta, S., (K, N)-convexity and the curvature-dimension condition for negative . N. J. Geom. Anal. 26(2016), no. 3, 2067–2096. . Google Scholar | DOI

[26] Qian, Z., Estimates for weighted volumes and applications . Quart. J. Math. Oxford Ser. (2) 48(1997), no. 190, 235–242. . Google Scholar | DOI

[27] Schmuckenschläger, M., Curvature of nonlocal Markov generators. In: Convex geometric analysis. Math. Sci. Res. Inst. Publ., 34. Cambridge Univ. Press, Cambridge, 1999, pp. 189–197. Google Scholar