Spectral Properties of Unimodular Lattice Triangulations
- 99 Downloads
- 1 Citations
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 theoryReferences
- 1.Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 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.Almendral, J.A., Daz-Guilera, A.: Dynamical and spectral properties of complex networks. New J. Phys. 9(6), 187 (2007)ADSCrossRefGoogle Scholar
- 4.Ambjørn, J., Loll, R.: Reconstructing the universe. Phys. Rev. D 72, 064014 (2005)ADSCrossRefGoogle Scholar
- 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.Aste, T., Gramatica, R., Di Matteo, T.: Exploring complex networks via topological embedding on surfaces. Phys. Rev. E 86, 036109 (2012)ADSCrossRefGoogle Scholar
- 7.Aste, T., Rivier, N.: Random cellular froths in spaces of any dimension and curvature. J. Phys. A 28(5), 1381 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 8.Aste, T., Sherrington, D.: Glass transition in self-organizing cellular patterns. J. Phys. A 32, 70497056 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Banerjee, A., Jost, J.: Graph spectra as a systematic tool in computational biology. Discret. Appl. Math. 157(10), 2425–2431 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
- 10.Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 11.Bauer, M., Golinelli, O.: Random incidence matrices: moments of the spectral density. J. Stat. Phys. 103(1–2), 301–337 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 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.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.Caputo, P., Martinelli, F., Sinclair, A., Stauffer, A.: Random lattice triangulations: structure and algorithms. Ann. Appl. Probab. 25(3), 1650–1685 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
- 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.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.Cvetković, D.M., Rowlinson, P., Simić, S.: An Introduction to the Theory of Graph Spectra. Cambridge University Press, Camebridge (2010)zbMATHGoogle Scholar
- 18.de Abreu, N.M.M.: Old and new results on algebraic connectivity of graphs. Linear Algebra Appl. 423(1), 53–73 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
- 19.De Loera, J.A., Rambau, J., Santos, F.: Triangulations. Structures for Algorithms and Applications. Springer, Berlin (2010)zbMATHGoogle Scholar
- 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 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.Du, Z., Liu, Z.: On the Estrada and Laplacian Estrada indices of graphs. Linear Algebra Appl. 435(8), 2065–2076 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
- 23.Dubertret, B., Rivier, N., Peshkin, M.A.: Long-range geometrical correlations in two-dimensional foams. J. Phys. A. 31(3), 879 (1998)ADSCrossRefzbMATHGoogle Scholar
- 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.Erdös, P., Rényi, A.: On random graphs. Publ. Math. Debrecen 6, 290–297 (1959)MathSciNetzbMATHGoogle Scholar
- 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.Erdös, P., Rényi, A.: On the strength of connectedness of a random graph. Acta Math. Acad. Sci. H. 12(1–2), 261–267 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
- 28.Estrada, E.: Characterization of 3D molecular structure. Chem. Phys. Lett. 319(56), 713–718 (2000)ADSCrossRefGoogle Scholar
- 29.Estrada, E., Rodríguez-Velázquez, J.A.: Spectral measures of bipartivity in complex networks. Phys. Rev. E 72, 046105 (2005)ADSMathSciNetCrossRefGoogle Scholar
- 30.Evangelou, S.N.: Quantum percolation and the Anderson transition in dilute systems. Phys. Rev. B 27, 1397–1400 (1983)ADSCrossRefGoogle Scholar
- 31.Evangelou, S.N.: A numerical study of sparse random matrices. J. Stat. Phys. 69(1–2), 361–383 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 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.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.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.Fiedler, M.: Algebraic connectivity of graphs. Czechoslovak Math. J. 23(2), 298–305 (1973)MathSciNetzbMATHGoogle Scholar
- 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.Grone, R.D.: Eigenvalues and the degree sequences of graphs. Linear Multilinear Algebra 39, 133–136 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
- 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.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.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.Juvan, M., Mohar, B.: Laplace eigenvalues and bandwidth-type invariants of graphs. J. Graph Theory 17(3), 393–407 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
- 42.Kaibel, V., Ziegler, G.M.: Counting lattice triangulations. Lond. Math. Soc. Lect. Note Ser. 307, 277–308 (2003)MathSciNetzbMATHGoogle Scholar
- 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.Knauf, J.F., Krüger, B., Mecke, K.: Entropy of unimodular lattice triangulations. EPL (Europhys. Lett.) 109(4), 40011 (2015)ADSCrossRefGoogle Scholar
- 45.Kownacki, J.P.: Freezing of triangulations. Eur. Phys. J. B 38(3), 485–494 (2004)ADSCrossRefGoogle Scholar
- 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.Kumar, S.: Random matrix ensembles: Wang–Landau algorithm for spectral densities. EPL (Europhys. Lett.) 101(2), 20002 (2013)ADSCrossRefGoogle Scholar
- 48.Lawson, C.L.: Transforming triangulations. Discret. Math. 3(4), 365–372 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
- 49.Lee, J.: New Monte Carlo algorithm: entropic sampling. Phys. Rev. Lett. 71, 211–214 (1993)ADSCrossRefGoogle Scholar
- 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.Mirlin, A.D., Fyodorov, Y.V.: Universality of level correlation function of sparse random matrices. J. Phys. A 24(10), 2273 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 52.Mohar, B.: Isoperimetric numbers of graphs. J. Combin. Theory Ser. B 47(3), 274–291 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
- 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.Mohar, B.: Laplace eigenvalues of graphs—a survey. Discret. Math. 109(13), 171–183 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
- 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.Monasson, R.: Diffusion, localization and dispersion relations on small-world lattices. Eur. Phys. J. B 12(4), 555–567 (1999)ADSCrossRefGoogle Scholar
- 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.Newman, M.: Networks. An Introduction. Oxford University Press, Oxford (2010)CrossRefzbMATHGoogle Scholar
- 59.Newman, M., Watts, D.: Renormalization group analysis of the small-world network model. Phys. Lett. A 263(46), 341–346 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 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.Pachner, U.: Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter semilinearer Mannigfaltigkeiten. Abh. Math. Sem. Univ. Hamburg 57, 69–85 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
- 62.Parzen, E.: On estimation of a probability density function and mode. Ann. Math. Stat. 33, 1065 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
- 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)MathSciNetCrossRefzbMATHGoogle Scholar
- 64.Rovelli, C.: Quantum Gravity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
- 65.Silverman, B.: Density Estimation for Statistics and Data Analysis. Monographs on Statistics & Applied Probability. Chapman & Hall/CRC, London (1986)CrossRefzbMATHGoogle Scholar
- 66.Song, W.M., Di Matteo, T., Aste, T.: Building complex networks with platonic solids. Phys. Rev. E 85, 046115 (2012)ADSCrossRefGoogle Scholar
- 67.Stauffer, A.: A Lyapunov function for Glauber dynamics on lattice triangulations. arXiv:1504.07980 (2015)
- 68.Sulanke, T., Lutz, F.H.: Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds. Eur. J. Comb. 30(8), 1965–1979 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
- 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.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.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.Varshney, L.: The wiring economy principle for designing inference networks. IEEE J. Select. Areas Commun. 31(6), 1095–1104 (2013)MathSciNetCrossRefGoogle Scholar
- 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.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.Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393(6684), 440–442 (1998)ADSCrossRefGoogle Scholar
- 76.Wigner, E.P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62(3), 548–564 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
- 77.Wigner, E.P.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 67(2), 325–327 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
- 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.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.Yu, A., Lu, M., Tian, F.: On the spectral radius of graphs. Linear Algebra Appl. 387, 41–49 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
- 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