Optical Memory and Neural Networks

, Volume 26, Issue 2, pp 87–95 | Cite as

Polynomial algorithm for exact calculation of partition function for binary spin model on planar graphs

  • Ya. M. Karandashev
  • M. Yu. Malsagov


In this paper we propose and realize an algorithm for exact calculation of partition function for planar graph models with binary variables. The complexity of the algorithm is O(N 2) Experiments show good agreement with Onsager’s analytical solution for the two-dimensional Ising model of infinite size.


Planar graph Ising model partition function binary model polynomial algorithm 


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  1. 1.
    Dixon, J.M., Tuszynski, J.A., and Carpenter, E.J., Analytical expressions for energies, degeneracies and critical temperatures of the 2D square and 3D cubic Ising models, Phys. A, 2005, vol. 349, pp. 487–510. doi 10.1016/j.physa.2004.10.029MathSciNetCrossRefGoogle Scholar
  2. 2.
    Martin, O.C., Monasson, R., and Zecchina, R., Statistical mechanics methods and phase transitions in optimization problems, Theor. Comput. Sci., 2001, vol. 265, p. 3. Scholar
  3. 3.
    Hinton, G.E., Osindero, S., and Teh, Y.W., A fast learning algorithm for deep belief nets, Neural Comput., 2006, vol. 18, p. 1527.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Wainwright, M.J., Jaakkola, T., and Willsky, A.S., A new class of upper bounds on the log partition function, IEEE Trans. Inf. Theory, 2005, vol. 51, pp. 2313–2335.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Yedidia, J.S., Freeman, W.T., and Weiss, Y., Constructing free-energy approximations and generalized belief propagation approximations, IEEE Trans. Inf. Theory, 2005, vol. 51, pp. 2282–2312.CrossRefzbMATHGoogle Scholar
  6. 6.
    Wainwright, M.J. and Jordan, M.I., Graphical models, exponential families, and variational inference, Technical Report, UC Berkeley Dept. of Statistics, 2003.Google Scholar
  7. 7.
    Wainwright, M.J., Jaakkola, T., and Willsky, A.S., Tree-based reparameterization framework for analysis of sum-product and related algorithms, IEEE Trans. Inform. Theory, 2003, vol. 49, no. 5, pp. 1120–1146.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kryzhanovsky, B. and Litinskii, L., Approximate method of free energy calculation for spin system with arbitrary connection matrix, J. Phys. Conf. Ser., 2015, p. 574, 012017. Scholar
  9. 9.
    Kryzhanovsky, B. and Litinskii, L., Generalized approach to energy distribution of spin system, Opt. Mem. Neural Networks, 2015, vol. 24, p. 165. Scholar
  10. 10.
    Kryzhanovsky, B. and Litinskii, L., Applicability of n-vicinity method for calculation of free energy of Ising model, Phys. A: Statistical Mechanics and Its Appl., ISSN 0378–4371. Cited November 3, 2016. doi 10.1016/j.physa.2016.10.074Google Scholar
  11. 11.
    Kac, M. and Ward, J., A combinatorial solution of the two-dimensional Ising model, Phys. Rev., 1952, vol. 88, no. 6.Google Scholar
  12. 12.
    Sherman, S., Combinatorial aspects of the Ising model for ferromagnetism. i: A conjecture of Feynman on paths and graphs, J. Math. Phys., 1960, vol. 1, no. 3.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kasteleyn, P., Dimer statistics and phase transitions, J. Math. Phys., 1963, vol. 4, no. 2.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fisher, M., On the dimer solution of planar Ising models, J. Math. Phys., 1966, vol. 7, no. 10.CrossRefGoogle Scholar
  15. 15.
    Schraudolph, N. and Kamenetsky, D., Efficient exact inference in planar Ising models, In NIPS, 2008.Google Scholar
  16. 16.
    Johnson, J.K., Oyen, D., Chertkov, M., and Praneeth Netrapalli, Learning Planar Ising Models, arXiv:1502.00916, 2015.zbMATHGoogle Scholar
  17. 17.
    Onsager, L., Crystal statistics, i: A two-dimensional model with an order-disorder transition, Phys. Rev., 1944, vol. 65, no. 3–4.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lovas, L. and Plammer, M., Applied problems of the graph theory, The Pair Matching Theory in Mathematics, Physics, Chemistry, Mir, 1998.Google Scholar
  19. 19.
    Middleton, A., Middleton, T., and Creighton, K., Matching Kasteleyn Cities for Spin Glass Ground States, Physics, 2007, p. 180. Scholar
  20. 20.
    Liers, F. and Pardella, G., A simple MAX-CUT algorithm for planar graphs, Technical Report, 2008, p. 16.Google Scholar
  21. 21.
    Davis, T.A., Direct Methods for Sparse Linear Systems, Philadelphia: SIAM, 2006.Google Scholar

Copyright information

© Allerton Press, Inc. 2017

Authors and Affiliations

  1. 1.Center of Optical Neural Technologies Scientific Research Institute for System Analysis RASMoscowRussia

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