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Dynamical attraction in parallel network models

Author

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  • Aledo, Juan A.
  • Diaz, Luis G.
  • Martinez, Silvia
  • Valverde, Jose C.

Abstract

In this work, we give a characterization of attractors in parallel deterministic network models, which evolve by means of maxterm and minterm Boolean functions and provide a method to obtain their basins of attraction. In order to do that, we distinguish the two possible cases: attractive fixed points and attractive 2-periodic orbits. Furthermore, we state necessary and sufficient conditions to know when a fixed point or a 2-periodic orbit is globally attractive. This makes possible to obtain a detailed description of their phase diagrams. Besides, we provide optimal upper bounds for the transient in such models, i.e., for the maximum number of iterations required to reach one of the periodic orbits. Moreover, we establish patterns that allow us to obtain a PDS on a maxterm or minterm Boolean function for which any given optimal upper bound for the transient is reached.

Suggested Citation

  • Aledo, Juan A. & Diaz, Luis G. & Martinez, Silvia & Valverde, Jose C., 2019. "Dynamical attraction in parallel network models," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 874-888.
    • Handle: RePEc:eee:apmaco:v:361:y:2019:i:c:p:874-888
    DOI: 10.1016/j.amc.2019.05.048
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    References listed on IDEAS

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    1. Barrett, Chris L & Chen, William Y.C & Zheng, Michelle J, 2004. "Discrete dynamical systems on graphs and Boolean functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 66(6), pages 487-497.
    2. Chiaselotti, G. & Gentile, T. & Oliverio, P.A., 2014. "Parallel and sequential dynamics of two discrete models of signed integer partitions," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 1249-1261.
    3. Andrew Wuensche, 1998. "Discrete Dynamical Networks and Their Attractor Basins," Working Papers 98-11-101, Santa Fe Institute.
    4. Juan A. Aledo & Silvia Martinez & Jose C. Valverde, 2015. "Parallel Dynamical Systems over Graphs and Related Topics: A Survey," Journal of Applied Mathematics, Hindawi, vol. 2015, pages 1-14, March.
    5. Juan A. Aledo & Luis G. Diaz & Silvia Martinez & Jose C. Valverde, 2017. "On the Periods of Parallel Dynamical Systems," Complexity, Hindawi, vol. 2017, pages 1-6, January.
    6. Toroczkai, Zoltán & Guclu, Hasan, 2007. "Proximity networks and epidemics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 378(1), pages 68-75.
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    Cited by:

    1. Giacopelli, G. & Migliore, M. & Tegolo, D., 2020. "Graph-theoretical derivation of brain structural connectivity," Applied Mathematics and Computation, Elsevier, vol. 377(C).

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