Tensor Network Renormalization as an Ultra-calculus for Complex System Dynamics

  • Pouria MistaniEmail author
  • Samira Pakravan
  • Frederic Gibou
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 186)


In this chapter, following the previous one, we briefly present the modern approach to real-space renormalization group (RG) theory based on tensor network formulations which was developed during the last two decades. The aim of this sequel is to suggest a novel framework based on tensor networks in order to find the fixed points of complex systems via coarse-graining. The main result of RG is that it provides a systematic way to study the collective dynamics of a large ensemble of elements that interact according to a complex underlying network topology. RG explicitly seeks the fixed points of the complex system in the space of interactions and unravels the universality class of the complex system as well as calculates a plethora of important observables. We hope that tensor networks can particularly pave the way for better understanding of the sustainable interdependent networks (Amini et al., Sustainable interdependent networks: from theory to application, 2018) through proposing efficient computational strategies and discovering insightful features of the network behaviors.


  1. 1.
    Amini, M. H., Boroojeni, K. G., Iyengar, S. S., Pardalos, P. M., Blaabjerg, F., & Madni, A. M. (2018). Sustainable interdependent networks: From theory to application (Vol. 145). Cham: Springer.Google Scholar
  2. 2.
    Baker, S. G. (2014). A cancer theory kerfuffle can lead to new lines of research. Journal of the National Cancer Institute, 107(2), dju405.Google Scholar
  3. 3.
    Bradde, S.& Bialek, W. (2017). PCA meets RG. Journal of Statistical Physics, 167(3–4), 462–475.Google Scholar
  4. 4.
    Bridgeman, J. C., & Chubb, C. T. (2017). Hand-waving and interpretive dance: An introductory course on tensor networks. Journal of Physics A: Mathematical and Theoretical, 50(22), 223001.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Butterfield, J. (2011a). Less is different: Emergence and reduction reconciled. Foundations of Physics, 41(6), 1065–1135.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Butterfield, J. (2011b). Emergence, reduction and supervenience: a varied landscape. Foundations of Physics, 41(6), 920–959.CrossRefGoogle Scholar
  7. 7.
    Butterfield, J. (2014). Reduction, emergence, and renormalization. The Journal of Philosophy, 111(1), 5–49.CrossRefGoogle Scholar
  8. 8.
    Butterfield, J., & Bouatta, N. (2015). Renormalization for philosophers. Metaphysics in Contemporary Physics, 104, 437–485.CrossRefGoogle Scholar
  9. 9.
    Caflisch, R. E., Gyure, M., Merriman, B., Osher, S., Ratsch, C., Vvedensky, D., et al. (1999). Island dynamics and the level set method for epitaxial growth. Applied Mathematics Letters, 12(4), 13–22.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cao, T. Y., & Schweber, S. S. (1993). The conceptual foundations and the philosophical aspects of renormalization theory. Synthese, 97(1), 33–108.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chaisson, E. J., & Chaisson, E. (2002). Cosmic evolution. Cambridge: Harvard University Press.Google Scholar
  12. 12.
    Chernet, B., & Levin, M. (2013). Endogenous voltage potentials and the microenvironment: Bioelectric signals that reveal, induce and normalize cancer. Journal of Clinical & Experimental Oncology, 2013(Suppl. 1), S1-002.Google Scholar
  13. 13.
    Chernet, B. T., & Levin, M. (2014). Transmembrane voltage potential of somatic cells controls oncogene-mediated tumorigenesis at long-range. Oncotarget, 5(10), 3287.CrossRefGoogle Scholar
  14. 14.
    Clauset, A., Shalizi, C. R., & Newman, M. E. (2009). Power-law distributions in empirical data. SIAM Review, 51(4), 661–703.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Claverie, P., & Jona-Lasinio, G. (1986). Instability of tunneling and the concept of molecular structure in quantum mechanics: The case of pyramidal molecules and the enantiomer problem. Physical Review A, 33(4), 2245.CrossRefGoogle Scholar
  16. 16.
    Domínguez, A., Hochberg, D., Martín-García, J., Pérez-Mercader, J., & Schulman, L. (1999). Dynamical scaling of matter density correlations in the universe: An application of the dynamical renormalization group. Arxiv preprint astro-ph/9901208.Google Scholar
  17. 17.
    Dyson, F. J. (1949). The radiation theories of tomonaga, schwinger, and feynman. Physical Review, 75(3), 486.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dyson, F. J. (1949). The s matrix in quantum electrodynamics. Physical Review, 75(11), 1736.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Efrati, E., Wang, Z., Kolan, A., & Kadanoff, L. P. (2014). Real-space renormalization in statistical mechanics. Reviews of Modern Physics, 86(2), 647.CrossRefGoogle Scholar
  20. 20.
    Evenbly, G. (2017). Algorithms for tensor network renormalization. Physical Review B, 95(4), 045117.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Evenbly, G., & Vidal, G. (2011). Tensor network states and geometry. Journal of Statistical Physics, 145(4), 891–918.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Evenbly, G., & Vidal, G. (2015). Tensor network renormalization. Physical Review letters, 115(18), 180405.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Fisher, M. E. (1974). The renormalization group in the theory of critical behavior. Reviews of Modern Physics, 46(4), 597.CrossRefGoogle Scholar
  24. 24.
    Franklin, A., & Knox, E. (2018). Emergence without limits: The case of phonons. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics.Google Scholar
  25. 25.
    Frisch, U., Hasslacher, & Pomeau, Lattice-gas automata for the navier-stokes equation. Physical Review Letters, 56(14), 1505.Google Scholar
  26. 26.
    Gibou, F., Ratsch, C., Gyure, M., Chen, S., & Caflisch, R. (2001). Rate equations and capture numbers with implicit islands correlations. Physical Review B, 63(11), 115401.CrossRefGoogle Scholar
  27. 27.
    Goldenfeld, N. (2018). Lectures on phase transitions and the renormalization group. Boca Raton: CRC Press.CrossRefGoogle Scholar
  28. 28.
    Gu, Z.-C., Levin, M., & Wen, X.-G.. Tensor-entanglement renormalization group approach as a unified method for symmetry breaking and topological phase transitions. Physical Review B, 78(20), 205116.Google Scholar
  29. 29.
    Jaffe, L. F., & Nuccitelli, R. (1977). Electrical controls of development. Annual Review of Biophysics and Bioengineering, 6(1), 445–476.CrossRefGoogle Scholar
  30. 30.
    Kadanoff, L. P. (1966). Scaling laws for ising models near t (c). Physics, 2, 263–272.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Kadanoff, L. P., & Wegner, F. J. (1971). Some critical properties of the eight-vertex model. Physical Review B, 4(11), 3989.CrossRefGoogle Scholar
  32. 32.
    Laughlin, R. B., & Pines, D. (2000). The theory of everything. Proceedings of the National Academy of Sciences of the United States of America, 97(1), 28–31.MathSciNetCrossRefGoogle Scholar
  33. 33.
    Levin, M. (2007). Large-scale biophysics: Ion flows and regeneration. Trends in Cell Biology, 17(6), 261–270.CrossRefGoogle Scholar
  34. 34.
    Levin, M., & Nave, C. P. (2007). Tensor renormalization group approach to two-dimensional classical lattice models. Physical Review Letters, 99(12), 120601.CrossRefGoogle Scholar
  35. 35.
    Longo, G. (2017). The biological consequences of the computational world: Mathematical reflections on cancer biology (2017). arXiv preprint arXiv:1701.08085.Google Scholar
  36. 36.
    Lund, E. (1925). Experimental control of organic polarity by the electric current. V. The nature of the control of organic polarity by the electric current. Journal of Experimental Zoology Part A: Ecological Genetics and Physiology, 41(2), 155–190.CrossRefGoogle Scholar
  37. 37.
    Lund, E. J. (1947). Bioelectric fields and growth (Vol. 64). Philadelphia: LWW.Google Scholar
  38. 38.
    Mathews, A. P. (1903). Electrical polarity in the hydroids. American Journal of Physiology–Legacy Content, 8(4), 294–299.CrossRefGoogle Scholar
  39. 39.
    Mistani, P., Guittet, A., Bochkov, D., Schneider, J., Margetis, D., Ratsch, C., et al. (2018). The island dynamics model on parallel quadtree grids. Journal of Computational Physics, 361, 150–166.MathSciNetCrossRefGoogle Scholar
  40. 40.
    Mistani, P., Guittet, A., Poignard, C., & Gibou, F. (February 2018). A parallel voronoi-based approach for mesoscale simulations of cell aggregate electropermeabilization. ArXiv e-prints.Google Scholar
  41. 41.
    Mora, T., & Bialek, W. (2011). Are biological systems poised at criticality? Journal of Statistical Physics, 144(2), 268–302.MathSciNetCrossRefGoogle Scholar
  42. 42.
    Nagel, E., & Hawkins, D. (1961). The structure of science. American Journal of Physics, 29, 716.CrossRefGoogle Scholar
  43. 43.
    Nakamoto, N., & Takeda, S. (2016). Computation of correlation functions by tensor renormalization group method. Sciece Reports of Kanazawa University, 60, 11–25MathSciNetGoogle Scholar
  44. 44.
    Onuki, A. (2002). Phase transition dynamics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  45. 45.
    Orús, R. (2014). A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of Physics, 349, 117–158.MathSciNetCrossRefGoogle Scholar
  46. 46.
    Perez-Mercader, J. (2004). Coarsegraining, scaling and hierarchies. In Nonextensive Entropy-Interdisciplinary Applications (pp. 357–376). Oxford: Oxford University Press.Google Scholar
  47. 47.
    Pietak, A., & Levin, M. (2017). Bioelectric gene and reaction networks: computational modelling of genetic, biochemical and bioelectrical dynamics in pattern regulation. Journal of the Royal Society Interface, 14(134), 20170425 (2017).Google Scholar
  48. 48.
    Robertson, D., Miller, M. W., & Carstensen, E. L. (1981). Relationship of 60-hz electric-field parameters to the inhibition of growth ofpisum sativum roots. Radiation and Environmental Biophysics, 19(3), 227–233.CrossRefGoogle Scholar
  49. 49.
    Rodriguez-Laguna, J. (2002). Real space renormalization group techniques and applications. arXiv preprint cond-mat/0207340.Google Scholar
  50. 50.
    Rozenfeld, H. D., Song, C., & Makse, H. A. (2010). Small-world to fractal transition in complex networks: A renormalization group approach. Physical Review Letters, 104(2), 025701.CrossRefGoogle Scholar
  51. 51.
    Ruderman, D. L., & Bialek, W. (1994). Statistics of natural images: Scaling in the woods. In Advances in Neural Information Processing Systems, pp. 551–558.Google Scholar
  52. 52.
    Schuch, N., Wolf, M. M., Verstraete, F., & Cirac, J. I. (2007). Computational complexity of projected entangled pair states. Physical Review Letters, 98(14), 140506.MathSciNetCrossRefGoogle Scholar
  53. 53.
    Simon, H. A. (1996). The sciences of the artificial.Google Scholar
  54. 54.
    Song, C., Havlin, S., & Makse, H. A. (2005). Self-similarity of complex networks. Nature, 433(7024), 392.CrossRefGoogle Scholar
  55. 55.
    Soto, A. M., Longo, G., Miquel, P.-A., Montévil, M., Mossio, M., et al. (2016). Toward a theory of organisms: Three founding principles in search of a useful integration. Progress in Biophysics and Molecular Biology, 122(1), 77–82.CrossRefGoogle Scholar
  56. 56.
    Soto, A. M., & Sonnenschein, C. (2011). The tissue organization field theory of cancer: A testable replacement for the somatic mutation theory. Bioessays, 33(5), 332–340.CrossRefGoogle Scholar
  57. 57.
    Wegscheid, B., Condon, C., & Hartmann, R. K. (2006). Type a and b rnase p rnas are interchangeable in vivo despite substantial biophysical differences. EMBO Reports, 7(4), 411–417 (2006).Google Scholar
  58. 58.
    Weinberg, S. (1997). What is quantum field theory, and what did we think it is? arXiv preprint hep-th/9702027.Google Scholar
  59. 59.
    White, S. R. (1992). Density matrix formulation for quantum renormalization groups. Physical Review Letters, 69(19), 2863.CrossRefGoogle Scholar
  60. 60.
    White, S. R. (1993). Density-matrix algorithms for quantum renormalization groups. Physical Review B, 48(14), 10345.CrossRefGoogle Scholar
  61. 61.
    Wilson, K. G. (1971). Renormalization group and critical phenomena. I. Renormalization group and the kadanoff scaling picture. Physical Review B, 4(9), 3174.Google Scholar
  62. 62.
    Wilson, K. G. (1975). The renormalization group: Critical phenomena and the kondo problem. Reviews of Modern Physics, 47(4), 773.MathSciNetCrossRefGoogle Scholar
  63. 63.
    Wilson, K. G., & Kogut, J. (1974). The renormalization group and the e expansion. Physics Reports, 12(2), 75–199.CrossRefGoogle Scholar
  64. 64.
    Yang, S., Gu, Z.-C., & Wen, X.-G. (2017). Loop optimization for tensor network renormalization. Physical Review Letters, 118(11), 110504.CrossRefGoogle Scholar
  65. 65.
    Yeong, C. L. Y., & Torquato, S. (1998). Reconstructing random media. Physical Review E, 57, 495–506.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Pouria Mistani
    • 1
    Email author
  • Samira Pakravan
    • 1
  • Frederic Gibou
    • 1
    • 2
  1. 1.Department of Mechanical EngineeringUniversity of California Santa BarbaraSanta BarbaraUSA
  2. 2.Department of Computer ScienceUniversity of California Santa BarbaraSanta BarbaraUSA

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