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Intrinsic Metrics on Graphs: A Survey

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Mathematical Technology of Networks

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 128))

Abstract

A few years ago various disparities for Laplacians on graphs and manifolds were discovered. The corresponding results are mostly related to volume growth in the context of unbounded geometry. Indeed, these disparities can now be resolved by using so called intrinsic metrics instead of the combinatorial graph distance. In this article we give an introduction to this topic and survey recent results in this direction. Specifically, we cover topics such as Liouville type theorems for harmonic functions, essential selfadjointness, stochastic completeness and upper escape rates. Furthermore, we determine the spectrum as a set via solutions, discuss upper and lower spectral bounds by isoperimetric constants and volume growth and study p-independence of spectra under a volume growth assumption.

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References

  1. Alon, N., Milman, V.D.: λ 1, isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory B 38(1), 73–88 (1985)

    Google Scholar 

  2. Arlotti, L., Banasiak, J.: Strictly substochastic semigroups with application to conservative and shattering solutions to fragmentation equations with mass loss, J. Math. Anal. Appl. 293(2), 693–720 (2004)

    Article  MathSciNet  Google Scholar 

  3. Azencott, R.: Behavior of diffusion semi-groups at infinity. Bull. Soc. Math. France 102, 193–240 (1974)

    MathSciNet  Google Scholar 

  4. Barlow, M.T., Bass, R.F.: Stability of parabolic Harnack inequalities. Trans. Am. Math. Soc. 356, 1501–1533 (2004)

    Article  MathSciNet  Google Scholar 

  5. Bauer, F., Hua, B., Keller, M.: On the l p spectrum of Laplacians on graphs. Adv. Math. 248(25), 717–735 (2013)

    Article  MathSciNet  Google Scholar 

  6. Bauer, F., Keller, M., Wojciechowski, R.: Cheeger inequalities for unbounded graph Laplacians. J. Eur. Math. Soc. (JEMS) 17(2), 259–271 (2015)

    Google Scholar 

  7. Boutet de Monvel, A., Lenz, D., Stollmann, P.: Schnol’s theorem for strongly local forms. Isr. J. Math. 173, 189–211 (2009)

    Google Scholar 

  8. Brooks, R.: A relation between growth and the spectrum of the Laplacian. Math. Z. 178(4), 501–508 (1981)

    Article  MathSciNet  Google Scholar 

  9. Brooks, R.: Spectral geometry and the Cheeger constant, expanding graphs. In: Proceedings of DIMACS Workshop, American Mathematical Society, pp. 5–19 (1993)

    Google Scholar 

  10. Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis (Papers dedicated to Salomon Bochner, 1969). pp. 195–199. Princeton University Press, Princeton (1970)

    Google Scholar 

  11. Chernoff, P.R.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12, 401–414 (1973)

    Google Scholar 

  12. Colin de Verdière, Y., Torki-Hamza, N., Truc, F.: Essential self-adjointness for combinatorial Schrödinger operators II–metrically non complete graphs. Math. Phys. Anal. Geom. 14(1), 21–38 (2011)

    Google Scholar 

  13. Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284(2), 787–794 (1984)

    Article  MathSciNet  Google Scholar 

  14. Dodziuk, J., Kendall, W. S.: Combinatorial Laplacians and isoperimetric inequality. In: From Local Times to Global Geometry, Control and Physics. Pitman Research Notes in Mathematics Series, vol. 150, pp. 68–74. Longman Scientific & Technical, Harlow (1986)

    Google Scholar 

  15. Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. In: Geometry of Random Motion (Ithaca, N.Y., 1987). Contemporary Mathematics, vol. 73, pp. 25–40. American Mathematical Society, Providence, RI (1988)

    Google Scholar 

  16. Feller, W.: On boundaries and lateral conditions for the Kolmogorov differential equations. Ann. Math. 65 (2), 527–570 (1957)

    Article  MathSciNet  Google Scholar 

  17. Feller, W.: Notes to my paper On boundaries and lateral conditions for the Kolmogorov differential equations. Ann. Math. 68(2), 735–736 (1958)

    Article  MathSciNet  Google Scholar 

  18. Folz, M.: Gaussian upper bounds for heat kernels of continuous time simple random walks. Electron. J. Probab. 16(62), 1693–1722 (2011)

    MathSciNet  Google Scholar 

  19. Folz, M.: Volume growth and stochastic completeness of graphs. Trans. Am. Math. Soc. 366(4), 2089–2119 (2014)

    Article  MathSciNet  Google Scholar 

  20. Folz, M.: Volume growth and spectrum for general graph Laplacians. Math. Z. 276(1–2), 115–131 (2014)

    Article  MathSciNet  Google Scholar 

  21. Frank, L., Lenz, D., Wingert, D.: Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory. J. Funct. Anal. 266(8), 4765–4808 (2014)

    Article  MathSciNet  Google Scholar 

  22. Fujiwara, K.: Growth and the spectrum of the Laplacian of an infinite graph. Tohoku Math. J. (2) 48(2), 293–302 (1996)

    Google Scholar 

  23. Fujiwara, K.: The Laplacian on rapidly branching trees. Duke Math. J. 83(1), 191–202 (1996)

    Article  MathSciNet  Google Scholar 

  24. Fukushima, M., Ōshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes. In: de Gruyter Studies in Mathematics, vol. 19. Walter de Gruyter, Berlin (1994)

    Google Scholar 

  25. Gaffney, M.P.: The harmonic operator for exterior differential forms. Proc. Natl. Acad. Sci. U. S. A. 37, 48–50 (1951)

    Article  MathSciNet  Google Scholar 

  26. Gaffney, M.P.: A special Stokes’s theorem for complete Riemannian manifolds. Ann. Math. (2) 60, 140–145 (1954)

    Google Scholar 

  27. Georgakopoulos, A., Haeseler, S., Keller, M., Lenz, D., Wojciechowski, R.: Graphs of finite measure. arXiv:1309.3501 to appear in J. Math. Pures Appl.

    Google Scholar 

  28. Grigor’yan, A.: Stochastically complete manifolds, Dokl. Akad. Nauk SSSR 290(3), 534–537 (1986)

    MathSciNet  Google Scholar 

  29. Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36(2), 135–249 (1999)

    Article  Google Scholar 

  30. Grigor’yan, A.: Escape rate of Brownian motion on Riemannian manifolds. Appl. Anal. 71(1–4), 63–89 (1999)

    MathSciNet  Google Scholar 

  31. Grigor’yan, A.: Analysis on Graphs. Lecture Notes. University Bielefeld, Bielefeld (2009)

    Google Scholar 

  32. Grigor’yan, A., Hsu, E.P.: Volume growth and escape rate of Brownian motion on a Cartan-Hadamard manifold. In: Sobolev Spaces in Mathematics. II. International Mathematics Series, vol. 9, pp. 209–225. Springer, New York (2009)

    Google Scholar 

  33. Grigor’yan, A., Huang, X., Masamune, J.: On stochastic completeness of jump processes. Math. Z. 271(3–4), 1211–1239 (2012)

    Article  MathSciNet  Google Scholar 

  34. Güneysu, B., Keller, M., Schmidt, M.: A Feynman-Kac-Itô formula for magnetic Schrödinger operators on graphs. arXiv:1301.1304 (preprint 2013)

    Google Scholar 

  35. Haeseler, S., Keller, M.: Generalized solutions and spectrum for Dirichlet forms on graphs. In: Lenz, D., Sobieczky, F., Woess, W. (eds.): Random Walks, Boundaries and Spectra. Progress in Probability, vol. 64, pp. 181–199. Birkhäuser Verlag, Basel (2011)

    Google Scholar 

  36. Haeseler, S., Keller, M., Lenz, D., Wojciechowski, R.K.: Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions. J. Spectr. Theory 2(4), 397–432 (2012)

    Article  MathSciNet  Google Scholar 

  37. Haeseler, S., Keller, M., Wojciechowski, R.K.: Volume growth and bounds for the essential spectrum for Dirichlet forms. J. Lond. Math. Soc. 88(3), 883–898 (2013)

    Article  MathSciNet  Google Scholar 

  38. Hempel, R., Voigt, J.: The spectrum of a Schrödinger operator in \(L_{p}(\mathbb{R}^{\nu })\) is p-independent. Commun. Math. Phys. 104, 243–250 (1986)

    Article  MathSciNet  Google Scholar 

  39. Holopainen, I., Soardi, P.M.: A strong Liouville theorem for p-harmonic functions on graphs. Ann. Acad. Sci. Fenn. Math. 22, 205–226 (1997)

    MathSciNet  Google Scholar 

  40. Hsu, E.P., Qin, G.: Volume growth and escape rate of Brownian motion on a complete Riemannian manifold. Ann. Probab. 38(4), 1570–1582 (2010)

    Article  MathSciNet  Google Scholar 

  41. Hua, B., Jost, J.: l q harmonic functions on graphs. Isr. J. Math. 202(1), 475–490 (2014)

    Google Scholar 

  42. Hua, B., Keller, M.: Harmonic functions of general graph Laplacians. Partial Differential Equations. Calc. Var 51(1–2), 343–362 (2014)

    Article  MathSciNet  Google Scholar 

  43. Huang, X.: On stochastic completeness of weighted graphs, Ph.D. Thesis (2011)

    Google Scholar 

  44. Huang, X.: A note on the volume growth criterion for stochastic completeness of weighted graphs. Potential Anal. 40(2), 117–142 (2014)

    Article  MathSciNet  Google Scholar 

  45. Huang, X.: Escape rate of Markov chains on infinite graphs. J. Theoret. Probab. Doi: 10.1007/s10959-012-0456-x (to appear)

    Google Scholar 

  46. Huang, X.: private communication

    Google Scholar 

  47. Huang, X., Shiozawa, Y.: Upper escape rate of Markov chains on weighted graphs. Stoch. Process. Appl. 124(1), 317–347 (2014)

    Article  MathSciNet  Google Scholar 

  48. Huang, X., Keller, M., Masamune, J., Wojciechowski, R.: A note on self-adjoint extensions of the Laplacian on weighted graphs. J. Funct. Anal. 265(8), 1556–1578 (2013)

    Article  MathSciNet  Google Scholar 

  49. Karp, L.: Subharmonic functions on real and complex manifolds. Math. Z. 179, 535–554 (1982)

    Article  MathSciNet  Google Scholar 

  50. Keller, M.: Lecture Notes SS 2012 Applications of Operator Theory - Discrete Operators. http://www.analysis-lenz.uni-jena.de/Team/Matthias+Keller.html

  51. Keller, M., Lenz, D.: Dirichlet forms and stochastic completeness of graphs and subgraphs. J. Reine Angew. Math. (Crelle’s J.) 666, 189–223 (2012)

    Google Scholar 

  52. Keller, M., Lenz, D.: Unbounded Laplacians on graphs: basic spectral properties and the heat equation. Math. Model. Natl. Phenom. 5 (4), 198–224 (2010)

    Article  MathSciNet  Google Scholar 

  53. Keller, M., Lenz, D., Schmidt, M., Wirth, M.: Diffusion determines the recurrent graph. Adv. Math. 269, 364–398 (2015)

    Article  MathSciNet  Google Scholar 

  54. Keller, M., Lenz, D., R. Wojciechowski, K.: Volume growth, spectrum and stochastic completeness of infinite graphs. Math. Z. 274 (3–4), 905–932 (2013)

    Google Scholar 

  55. Kuchment, P.: Quantum graphs. I. some basic structures. Waves Random Media 14, 107–128 (2004)

    Article  MathSciNet  Google Scholar 

  56. Lyons, R., Peres, Y.: Probability on Trees and Networks. Cambridge University Press, Cambridge (2013). http://mypage.iu.edu/~rdlyons/prbtree/book.pdf (to appear)

  57. Masamune, J.: A Liouville property and its application to the Laplacian of an infinite graph. In: Spectral Analysis in Geometry and Number Theory. Contemporary of Mathematics, vol. 484, pp. 103–115. American Mathematical Society, Providence, RI (2009)

    Google Scholar 

  58. Masamune, J., Uemura, T.: Conservation property of symmetric jump processes. Ann. Inst. Henri Poincaré Probab. Stat. 47(3), 650–662 (2011)

    Article  MathSciNet  Google Scholar 

  59. Masamune, J., Uemura, T., Wang, J.: On the conservativeness and the recurrence of symmetric jump-diffusions. J. Funct. Anal. 263(12), 3984–4008 (2012)

    Article  MathSciNet  Google Scholar 

  60. Milatovic, O.: Essential self-adjointness of discrete magnetic Schrödinger operators on locally finite graphs. Integr. Equ. Oper. Theory 71, 13–27 (2011)

    Article  MathSciNet  Google Scholar 

  61. Milatovic, O.: A Sears-type self-adjointness result for discrete magnetic Schrödinger operators. J. Math. Anal. Appl. 396, 801–809 (2012)

    Article  MathSciNet  Google Scholar 

  62. Milatovic, O., Truc, F.: Self-adjoint extensions of discrete magnetic Schrödinger operators. Ann. Henri Poincaré 15, 917–936 (2014)

    Article  MathSciNet  Google Scholar 

  63. Mokhtar-Kharroubi, M., Voigt, J.: On honesty of perturbed substochastic C 0-semigroups in L 1-spaces. J. Oper. Theory 64(1), 131–147 (2010)

    MathSciNet  Google Scholar 

  64. Ohno, Y., Urakawa, H.: On the first eigenvalue of the combinatorial Laplacian for a graph. Interdiscip. Inform. Sci. 1(1), 33–46 (1994)

    MathSciNet  Google Scholar 

  65. Ouyang, S.: Volume growth, comparison theorem and escape rate of diffusion process. arXiv:1310.3996 (Preprint 2013)

    Google Scholar 

  66. Pinsky, M.A.: Mean exit time of a diffusion process from a small sphere. Proc. Am. Math. Soc. 93(1), 157–158 (1985)

    Article  MathSciNet  Google Scholar 

  67. Reuter, G.E.H.: Denumerable Markov processes and the associated contraction semigroups on l. Acta Math. 97, 1–46 (1957)

    Google Scholar 

  68. Rigoli, M., Salvatori, M., Vignati, M.: Subharmonic functions on graphs. Isr. J. Math. 99, 1–27 (1997)

    Article  MathSciNet  Google Scholar 

  69. Roelcke, W.: Über den Laplace-Operator auf Riemannschen Mannigfaltigkeiten mit diskontinuierlichen Gruppen. Math. Nachr. 21, 131–149 (1960)

    Article  MathSciNet  Google Scholar 

  70. Schmidt, M.: Global properties of Dirichlet forms on discrete spaces. Diplomarbeit 2011. arXiv:1201.3474v2

    Google Scholar 

  71. Shnol’, I. E.: On the behaviour of the eigenfunctions of Schrödinger’s eqution. Mat. Sb. 42, 273–286 (1957). Erratum 46(88), 259 (1957)

    Google Scholar 

  72. Simon, B.: Spectrum and continuum eigenfunctions of Schrödinger operators. J. Funct. Anal. 42, 347–355 (1981)

    Article  MathSciNet  Google Scholar 

  73. Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc. 7(3), 447–526 (1982)

    Article  Google Scholar 

  74. Strichartz, R.S.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52(1), 48–79 (1983)

    Article  MathSciNet  Google Scholar 

  75. Sturm, K.-T.: On the lp-spectrum of uniformly elliptic operators on riemannian manifolds. J. Funct. Anal. 118, 442–453 (1993)

    Article  MathSciNet  Google Scholar 

  76. Sturm, K.-T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and L p Liouville properties. J. Reine Angew. Math. 456, 173–196 (1994)

    Google Scholar 

  77. Weidmann, J.: Linear operators in Hilbert spaces. In: Graduate Texts in Mathematics, vol. 68. Springer, New York (1980)

    Google Scholar 

  78. Woess, W.: Random walks on infinite graphs and groups. In: Cambridge Tracts in Mathematics, vol. 138. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  79. Wojciechowski, R.K.: Stochastic completeness of graphs. ProQuest LLC, Ann Arbor, MI, Ph.D. Thesis, University of New York (2008)

    Google Scholar 

  80. Wojciechowski, R.K.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. 58, 1419–1441 (2009)

    Article  MathSciNet  Google Scholar 

  81. Wojciechowski, R.K.: Stochastically incomplete manifolds and graphs. In: Lenz, D., Sobieczky, F., Woess, W. (eds.) Random Walks, Boundaries and Spectra. Progress in Probability, vol. 64, pp. 163–179. Birkhäuser Verlag, Basel (2011)

    Chapter  Google Scholar 

  82. Yau, S.T.: Some function-theoretic properties of complete Riemannian manifold and their application to geometry. Indiana Univ. Math. J. 25(7), 659–670 (1976)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author expresses his gratitude to Daniel Lenz for introducing him to the guiding theme of this survey that intrinsic metrics are the answer to many questions. Furthermore, several inspiring discussions with Xueping Huang resulted in various insights that improved this article. Moreover, the author appreciates the support and the kind invitation to the conference on ‘Mathematical Technology of Networks—QGraphs 2013’ at the ZiF in Bielefeld by Delio Mugnolo. Finally, funding by the DFG is acknowledged.

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  1. Mathematisches Institut, Friedrich Schiller Universität Jena, 07743, Jena, Germany

    Matthias Keller

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  1. Lehrgebiet Analysis, FernUniversität in Hagen, Hagen, Germany

    Delio Mugnolo

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Keller, M. (2015). Intrinsic Metrics on Graphs: A Survey. In: Mugnolo, D. (eds) Mathematical Technology of Networks. Springer Proceedings in Mathematics & Statistics, vol 128. Springer, Cham. https://doi.org/10.1007/978-3-319-16619-3_7

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