Nonlinear threshold Boolean automata networks and phase transitions

Preprint English OPEN
Demongeot, Jacques ; Sené, Sylvain (2010)
  • Subject: Computer Science - Discrete Mathematics | Nonlinear Sciences - Cellular Automata and Lattice Gases | Mathematics - Dynamical Systems

In this report, we present a formal approach that addresses the problem of emergence of phase transitions in stochastic and attractive nonlinear threshold Boolean automata networks. Nonlinear networks considered are informally defined on the basis of classical stochastic threshold Boolean automata networks in which specific interaction potentials of neighbourhood coalition are taken into account. More precisely, specific nonlinear terms compose local transition functions that define locally the dynamics of such networks. Basing our study on nonlinear networks, we exhibit new results, from which we derive conditions of phase transitions.
  • References (14)
    14 references, page 1 of 2

    [AHS85] D. H. Ackley, G. E. Hinton, and T. J. Sejnowski. A learning algorithm for Boltzmann machines. Cognitive Science, 9:147-169, 1985.

    [Dem81] J. Demongeot. Asymptotic inference for Markov random field on Zd. Springer Series in Synergetics, 9:254-267, 1981.

    [DJS08] J. Demongeot, C. J´ez´equel, and S. Sen´e. Boundary conditions and phase transitions in neural networks. Theoretical results. Neural Networks, 21(7):971-979, 2008.

    [Dob68a] R. L. Dobrushin. Gibbsian random fields for lattice systems with pairwise interactions. Functional Analysis and Its Applications, 2(4):292-301, 1968.

    [Dob68b] R. L. Dobrushin. The description of a random field by means of conditional probabilities and conditions of its regularity. Theory of Probability and its Applications, 13(2):197-224, 1968.

    [Dob68c] R. L. Dobrushin. The problem of uniqueness of a Gibbsian random field and the problem of phase transitions. Functional Analysis and Its Applications, 2(4):302-312, 1968.

    [Dob69] R. L. Dobrushin. Gibbsian random fields. The general case. Functional Analysis and Its Applications, 3(1):22-28, 1969.

    J. Demongeot and S. Sen´e. Boundary conditions and phase transitions in neural networks. Simulation results. Neural Networks, 21(7):962-970, 2008.

    F. Harary. Graph Theory. Addison-Wesley, 1969.

    E. Ising. Beitrag zur theorie des ferromagnetismus. Zeitschrift fu¨r Physics, 31(1):253-258, 1925.

  • Metrics
    No metrics available
Share - Bookmark