Journal of Statistical Physics

, Volume 163, Issue 3, pp 514–543 | Cite as

Spectral Properties of Unimodular Lattice Triangulations

Article

Abstract

Random unimodular lattice triangulations have been recently used as an embedded random graph model, which exhibit a crossover behavior between an ordered, large-world and a disordered, small-world behavior. Using the ergodic Pachner flips that transform such triangulations into another and an energy functional that corresponds to the degree distribution variance, Markov chain Monte Carlo simulations can be applied to study these graphs. Here, we consider the spectra of the adjacency and the Laplacian matrix as well as the algebraic connectivity and the spectral radius. Power law dependencies on the system size can clearly be identified and compared to analytical solutions for periodic ground states. For random triangulations we find a qualitative agreement of the spectral properties with well-known random graph models. In the microcanonical ensemble analytical approximations agree with numerical simulations. In the canonical ensemble a crossover behavior can be found for the algebraic connectivity and the spectral radius, thus combining large-world and small-world behavior in one model. The considered spectral properties can be applied to transport problems on triangulation graphs and the crossover behavior allows a tuning of important transport quantities.

Keywords

Triangulations Random graphs Networks Spectral graph theory 

References

  1. 1.
    Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Almeida, G.M.A., Souza, A.M.C.: Quantum transport with coupled cavities on an Apollonian network. Phys. Rev. A 87, 033804 (2013)ADSCrossRefGoogle Scholar
  3. 3.
    Almendral, J.A., Daz-Guilera, A.: Dynamical and spectral properties of complex networks. New J. Phys. 9(6), 187 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    Ambjørn, J., Loll, R.: Reconstructing the universe. Phys. Rev. D 72, 064014 (2005)ADSCrossRefGoogle Scholar
  5. 5.
    Andrade, J.S., Herrmann, H.J., Andrade, R.F.S., da Silva, L.R.: Apollonian networks: simultaneously scale-free, small world, euclidean, space filling, and with matching graphs. Phys. Rev. Lett. 94, 018702 (2005)ADSCrossRefGoogle Scholar
  6. 6.
    Aste, T., Gramatica, R., Di Matteo, T.: Exploring complex networks via topological embedding on surfaces. Phys. Rev. E 86, 036109 (2012)ADSCrossRefGoogle Scholar
  7. 7.
    Aste, T., Rivier, N.: Random cellular froths in spaces of any dimension and curvature. J. Phys. A 28(5), 1381 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Aste, T., Sherrington, D.: Glass transition in self-organizing cellular patterns. J. Phys. A 32, 70497056 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Banerjee, A., Jost, J.: Graph spectra as a systematic tool in computational biology. Discret. Appl. Math. 157(10), 2425–2431 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bauer, M., Golinelli, O.: Random incidence matrices: moments of the spectral density. J. Stat. Phys. 103(1–2), 301–337 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Biroli, G., Monasson, R.: A single defect approximation for localized states on random lattices. J. Phys. A 32(24), L255 (1999)ADSCrossRefGoogle Scholar
  13. 13.
    Bray, A.J., Rodgers, G.J.: Diffusion in a sparsely connected space: a model for glassy relaxation. Phys. Rev. B 38, 11461–11470 (1988)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Caputo, P., Martinelli, F., Sinclair, A., Stauffer, A.: Random lattice triangulations: structure and algorithms. Ann. Appl. Probab. 25(3), 1650–1685 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Chung, F., Lu, L.: Complex Graphs and Networks. No. 107 in CBMS Regional Conference Series in Mathematics. American Mathematical Society (2006)Google Scholar
  16. 16.
    Costa, L.D.F., Oliveira, O.N., Travieso, G., Rodrigues, F.A., Villas Boas, P.R., Antiqueira, L., Viana, M.P., Correa Rocha, L.E.: Analyzing and modeling real-world phenomena with complex networks: a survey of applications. Adv. Phys. 30(3), 329–412 (2011)ADSCrossRefGoogle Scholar
  17. 17.
    Cvetković, D.M., Rowlinson, P., Simić, S.: An Introduction to the Theory of Graph Spectra. Cambridge University Press, Camebridge (2010)MATHGoogle Scholar
  18. 18.
    de Abreu, N.M.M.: Old and new results on algebraic connectivity of graphs. Linear Algebra Appl. 423(1), 53–73 (2007)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    De Loera, J.A., Rambau, J., Santos, F.: Triangulations. Structures for Algorithms and Applications. Springer, Berlin (2010)MATHGoogle Scholar
  20. 20.
    Dean, D.S.: An approximation scheme for the density of states of the Laplacian on random graphs. J. Phys. A 35(12), L153 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ding, X., Jiang, T.: Spectral distributions of adjacency and Laplacian matrices of random graphs. Ann. Appl. Probab. 20(6), 20862117 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Du, Z., Liu, Z.: On the Estrada and Laplacian Estrada indices of graphs. Linear Algebra Appl. 435(8), 2065–2076 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Dubertret, B., Rivier, N., Peshkin, M.A.: Long-range geometrical correlations in two-dimensional foams. J. Phys. A. 31(3), 879 (1998)ADSCrossRefMATHGoogle Scholar
  24. 24.
    Earl, D.J., Deem, M.W.: Parallel tempering: Theory, applications, and new perspectives. Phys. Chem. Chem. Phys. 7, 3910–3916 (2005)CrossRefGoogle Scholar
  25. 25.
    Erdös, P., Rényi, A.: On random graphs. Publ. Math. Debrecen 6, 290–297 (1959)MathSciNetMATHGoogle Scholar
  26. 26.
    Erdös, P., Rényi, A.: On the evolution of random graphs. In: Publication of the Mathematical Institute of the Hungarian Academy of Sciences, pp. 17–61 (1960)Google Scholar
  27. 27.
    Erdös, P., Rényi, A.: On the strength of connectedness of a random graph. Acta Math. Acad. Sci. H. 12(1–2), 261–267 (1964)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Estrada, E.: Characterization of 3D molecular structure. Chem. Phys. Lett. 319(56), 713–718 (2000)ADSCrossRefGoogle Scholar
  29. 29.
    Estrada, E., Rodríguez-Velázquez, J.A.: Spectral measures of bipartivity in complex networks. Phys. Rev. E 72, 046105 (2005)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Evangelou, S.N.: Quantum percolation and the Anderson transition in dilute systems. Phys. Rev. B 27, 1397–1400 (1983)ADSCrossRefGoogle Scholar
  31. 31.
    Evangelou, S.N.: A numerical study of sparse random matrices. J. Stat. Phys. 69(1–2), 361–383 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Evangelou, S.N., Economou, E.N.: Spectral density singularities, level statistics, and localization in a sparse random matrix ensemble. Phys. Rev. Lett. 68, 361–364 (1992)ADSCrossRefGoogle Scholar
  33. 33.
    Farkas, I., Derenyi, I., Palla, G., Vicsek, T.: Equilibrium statistical mechanics of network structures. In: E. Ben-Naim, H. Frauenfelder, Z. Toroczkai (eds.) Complex Networks, Lecture Notes in Physics, vol. 650, pp. 163–187. Springer (2004)Google Scholar
  34. 34.
    Farkas, I.J., Derényi, I., Barabási, A.L., Vicsek, T.: Spectra of “real-world” graphs: beyond the semicircle law. Phys. Rev. E 64, 026704 (2001)ADSCrossRefGoogle Scholar
  35. 35.
    Fiedler, M.: Algebraic connectivity of graphs. Czechoslovak Math. J. 23(2), 298–305 (1973)MathSciNetMATHGoogle Scholar
  36. 36.
    Gervais, C., Wüst, T., Landau, D.P., Xu, Y.: Application of the Wang–Landau algorithm to the dimerization of glycophorin A. J. Chem. Phys 130(21), 215106 (2009)ADSCrossRefGoogle Scholar
  37. 37.
    Grone, R.D.: Eigenvalues and the degree sequences of graphs. Linear Multilinear Algebra 39, 133–136 (1995)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Guhr, T., Müller-Groeling, A., Weidenmüller, H.A.: Random-matrix theories in quantum physics: common concepts. Phys. Rep. 299(46), 189–425 (1998)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Hagberg, A.A., Schult, D.A., Swart, P.J.: Exploring network structure, dynamics, and function using NetworkX. In: Varoquaux, G., Vaught, T., Millman, J. (eds.) Proceedings of the 7th Python in Science Conference, pp. 11–15. Pasadena (2008)Google Scholar
  40. 40.
    Juhász, F.: On the spectrum of a random graph. In: L. Lovasz, V.T. Sos (eds.) Algebraic Methods in Graph Theory, Colloquia Mathematica Societatis Janos Bolyai, vol. 25 (1981)Google Scholar
  41. 41.
    Juvan, M., Mohar, B.: Laplace eigenvalues and bandwidth-type invariants of graphs. J. Graph Theory 17(3), 393–407 (1993)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Kaibel, V., Ziegler, G.M.: Counting lattice triangulations. Lond. Math. Soc. Lect. Note Ser. 307, 277–308 (2003)MathSciNetMATHGoogle Scholar
  43. 43.
    Kirchhoff, G.: Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird. Ann. Phys. Chem. 72, 497–508 (1847)ADSCrossRefGoogle Scholar
  44. 44.
    Knauf, J.F., Krüger, B., Mecke, K.: Entropy of unimodular lattice triangulations. EPL (Europhys. Lett.) 109(4), 40011 (2015)ADSCrossRefGoogle Scholar
  45. 45.
    Kownacki, J.P.: Freezing of triangulations. Eur. Phys. J. B 38(3), 485–494 (2004)ADSCrossRefGoogle Scholar
  46. 46.
    Krüger, B., Schmidt, E.M., Mecke, K.: Unimodular lattice triangulations as small-world and scale-free random graphs. New J. Phys. 17(2), 023013 (2015)ADSCrossRefGoogle Scholar
  47. 47.
    Kumar, S.: Random matrix ensembles: Wang–Landau algorithm for spectral densities. EPL (Europhys. Lett.) 101(2), 20002 (2013)ADSCrossRefGoogle Scholar
  48. 48.
    Lawson, C.L.: Transforming triangulations. Discret. Math. 3(4), 365–372 (1972)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Lee, J.: New Monte Carlo algorithm: entropic sampling. Phys. Rev. Lett. 71, 211–214 (1993)ADSCrossRefGoogle Scholar
  50. 50.
    Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087 (1953)ADSCrossRefGoogle Scholar
  51. 51.
    Mirlin, A.D., Fyodorov, Y.V.: Universality of level correlation function of sparse random matrices. J. Phys. A 24(10), 2273 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Mohar, B.: Isoperimetric numbers of graphs. J. Combin. Theory Ser. B 47(3), 274–291 (1989)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Mohar, B.: The Laplacian spectrum of graphs. In: Y. Alavi, G. Chartrand, O.R. Oellermann, A.J. Schwenk (eds.) Graph Theory, Combinatorics, and Applications, vol. 2, pp. 871–898. Wiley (1991)Google Scholar
  54. 54.
    Mohar, B.: Laplace eigenvalues of graphs—a survey. Discret. Math. 109(13), 171–183 (1992)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Mohar, B., Poljak, S.: Eigenvalues in combinatorial optimization. In: R. Brualdi, S. Friedland, V. Klee (eds.) Combinatorial and Graph-Theoretical Problems in Linear Algebra, The IMA Volumes in Mathematics and Its Applications, vol. 50, pp. 107–151. Springer, New York (1993)Google Scholar
  56. 56.
    Monasson, R.: Diffusion, localization and dispersion relations on small-world lattices. Eur. Phys. J. B 12(4), 555–567 (1999)ADSCrossRefGoogle Scholar
  57. 57.
    Mülken, O., Blumen, A.: Continuous-time quantum walks: models for coherent transport on complex networks. Phys. Rep. 502(23), 37–87 (2011)ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    Newman, M.: Networks. An Introduction. Oxford University Press, Oxford (2010)CrossRefMATHGoogle Scholar
  59. 59.
    Newman, M., Watts, D.: Renormalization group analysis of the small-world network model. Phys. Lett. A 263(46), 341–346 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Oguey, C., Rivier, N., Aste, T.: Stratifications of cellular patterns: hysteresis and convergence. Eur. Phys. J. B 33(4), 447–455 (2003)ADSCrossRefGoogle Scholar
  61. 61.
    Pachner, U.: Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter semilinearer Mannigfaltigkeiten. Abh. Math. Sem. Univ. Hamburg 57, 69–85 (1986)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Parzen, E.: On estimation of a probability density function and mode. Ann. Math. Stat. 33, 1065 (1962)MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Rosenblatt, M.: On estimation of a probability density function and mode. Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27, 832 (1956)MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Rovelli, C.: Quantum Gravity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2007)CrossRefMATHGoogle Scholar
  65. 65.
    Silverman, B.: Density Estimation for Statistics and Data Analysis. Monographs on Statistics & Applied Probability. Chapman & Hall/CRC, London (1986)CrossRefMATHGoogle Scholar
  66. 66.
    Song, W.M., Di Matteo, T., Aste, T.: Building complex networks with platonic solids. Phys. Rev. E 85, 046115 (2012)ADSCrossRefGoogle Scholar
  67. 67.
    Stauffer, A.: A Lyapunov function for Glauber dynamics on lattice triangulations. arXiv:1504.07980 (2015)
  68. 68.
    Sulanke, T., Lutz, F.H.: Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds. Eur. J. Comb. 30(8), 1965–1979 (2009)MathSciNetCrossRefMATHGoogle Scholar
  69. 69.
    Sullivan, J.M.: The geometry of bubbles and foams. In: J. Sadoc, N. Rivier (eds.) Foams and Emulsions, NATO Science Series E, vol. 354, pp. 379–402. Springer (1999)Google Scholar
  70. 70.
    Trinajstić, N.: Graph theory and molecular orbitals. In: D. Bonchev, D. Rouvray (eds.) Chemical Graph Theory: Introduction and Fundamentals, Mathematical Chemistry Series, chap. 6, pp. 235–275. Abacus Press (1991)Google Scholar
  71. 71.
    van den Heuvel, J., Peji, S.: Using Laplacian eigenvalues and eigenvectors in the analysis of frequency assignment problems. Ann. Oper. Res. 107(1–4), 349–368 (2001)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Varshney, L.: The wiring economy principle for designing inference networks. IEEE J. Select. Areas Commun. 31(6), 1095–1104 (2013)MathSciNetCrossRefGoogle Scholar
  73. 73.
    Wang, F., Landau, D.P.: Determining the density of states for classical statistical models: a random walk algorithm to produce a flat histogram. Phys. Rev. E 64, 056101 (2001)ADSCrossRefGoogle Scholar
  74. 74.
    Wang, F., Landau, D.P.: Efficient, multiple-range random walk algorithm to calculate the density of states. Phys. Rev. Lett. 86, 2050–2053 (2001)ADSCrossRefGoogle Scholar
  75. 75.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393(6684), 440–442 (1998)ADSCrossRefGoogle Scholar
  76. 76.
    Wigner, E.P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62(3), 548–564 (1955)MathSciNetCrossRefMATHGoogle Scholar
  77. 77.
    Wigner, E.P.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 67(2), 325–327 (1958)MathSciNetCrossRefMATHGoogle Scholar
  78. 78.
    Wüst, T., Landau, D.P.: Versatile approach to access the low temperature thermodynamics of lattice polymers and proteins. Phys. Rev. Lett. 102, 178101 (2009)ADSCrossRefGoogle Scholar
  79. 79.
    Wüst, T., Landau, D.P.: Optimized Wang–Landau sampling of lattice polymers: ground state search and folding thermodynamics of hp model proteins. J. Chem. Phys. 137(6), 064903 (2012)ADSCrossRefGoogle Scholar
  80. 80.
    Yu, A., Lu, M., Tian, F.: On the spectral radius of graphs. Linear Algebra Appl. 387, 41–49 (2004)MathSciNetCrossRefMATHGoogle Scholar
  81. 81.
    Zhou, T., Yan, G., Wang, B.H.: Maximal planar networks with large clustering coefficient and power-law degree distribution. Phys. Rev. E 71, 046141 (2005)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsFAU Erlangen-NurembergErlangenGermany

Personalised recommendations