A Picture for Complex Stochastic Boolean Systems: The Intrinsic Order Graph

  • Luis González
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3993)


Complex stochastic Boolean systems, depending on a large number n of statistically independent random Boolean variables, appear in many different scientific, technical or social areas. Each one of the 2 n binary states associated to such systems is denoted by its corresponding binary n-tuple of 0s and 1s, \(\left( u_{1},\ldots,u_{n}\right) \), and it has a certain occurrence probability \(\Pr\left\{ \left( u_{1},\ldots ,u_{n}\right) \right\} \). The ordering between the 2 n binary n-tuple probabilities, \(\Pr\left\{ \left( u_{1},\ldots,u_{n}\right) \right\} \), can be illustrated by a directed graph which “scales” them by decreasing order, the so-called intrinsic order graph. In this context, this paper provides a simple algorithm for iteratively drawing the intrinsic order graph, for any complex stochastic Boolean system and for any number n of independent random Boolean variables. The presentation is self-contained.


Directed Graph Occurrence Probability Binary String Lexicographic Order Boolean Variable 


  1. 1.
    González, L.: A new method for ordering binary states probabilities in reliability and risk analysis. In: Sloot, P.M.A., Tan, C.J.K., Dongarra, J., Hoekstra, A.G. (eds.) ICCS-ComputSci 2002. LNCS, vol. 2329, pp. 137–146. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    González, L.: N-tuples of 0s and 1s: Necessary and sufficient conditions for intrinsic order. In: Kumar, V., Gavrilova, M.L., Tan, C.J.K., L’Ecuyer, P. (eds.) ICCSA 2003. LNCS, vol. 2667, pp. 937–946. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    González, L., Galván, B., García, D.: Sobre el análisis computacional de funciones Booleanas estocásticas de muchas variables. In: González, L., Sendra, J.R. (eds.) Proc. Primer Encuentro de Álgebra Computacional y Aplicaciones (EACA 1995), Santander, pp. 45–55 (1995)Google Scholar
  4. 4.
    González, L., García, D., Galván, B.J.: An intrinsic order criterion to evaluate large, complex fault trees. IEEE Trans. Reliability 53(3), 297–305 (2004)CrossRefGoogle Scholar
  5. 5.
    Stanley, R.P.: Enumerative Combinatorics, vol. 1. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
  6. 6.
    Stuart, A., Ord, J.K.: Kendall’s Advanced Theory of Statistics, Distribution Theory, vol. 1. Oxford University Press, New York (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Luis González
    • 1
  1. 1.Department of Mathematics, Research Institute IUSIANIUniversity of Las Palmas de Gran CanariaLas Palmas de Gran CanariaSpain

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