Interaction of Networks

  • Ilya Gertsbakh
  • Yoseph Shpungin
  • Radislav Vaisman
Chapter
Part of the SpringerBriefs in Electrical and Computer Engineering book series (BRIEFSELECTRIC)

Abstract

The simplest form of two interacting networks is sharing the same set of nodes by two independent networks. For example, the power supply and water supply networks in the same geographic area share the same set of nodes (houses or residencies). Section 3.1 presents several simple results concerning the size of the set of nodes which receive “full” supply, i.e. are adjacent to edges of both types. Here we use some basic facts from the theory of large random Erdos-Renyi or Poisson networks. Section 3.2 considers a system of two or more finite interacting networks. Here the interaction means that a node \(v_a\) of network A delivers “infection” to a randomly chosen node \(v_B\) in B which in turn, bounces back and infects another randomly chosen node \(w_a\) in network A, and so on. As a result, random number \(Y\) of nodes in B gets “infected” and fails. We compute, using D-spectra technique, the DOWN probability for network B. This model is generalized to the case of several peripheral networks attacking one “central” network B. In this “attack”, some of nodes in B will receive more than one hit. The use of DeMoivre combinatorial formula together with D-spectra technique allows obtaining an expression for network B DOWN probability in a close form. Finally, Sect. 3.3 extends the results of Sect. 3.2 to the case when the “central” network is ternary. In that case, we must take into account different node behavior that are hit once or more. It is assumed that a node hit only once changes its state from up to mid. When this node receives another hit, it turns into down and remains in it forever. Network DOWN probability for this case can be estimated by an efficient Monte Carlo algorithm.

Keywords

Networks with colored links Giant component Poisson network  Network Interaction Star-type configuration D-spectra Network DOWN probability Ternary network DeMoivre formula 

References

  1. 1.
    Buldyrev SV, Parshani R, Stanley HF, Havlin S (2010) Catastrophic cascade of failures in interdependent networks. Nature 464:1025–1028CrossRefGoogle Scholar
  2. 2.
    David FN, Barton DE (1962) Combinatorial chance. Charles Griffin and Co Ltd, LondonGoogle Scholar
  3. 3.
    Dickinson M, Havlin S (2012) Epidemics on interconnected networks. Phys Rev E85:066109Google Scholar
  4. 4.
    Elperin T, Gertsbakh IB (1991) Estimation of network reliability using graph evolution models. IEEE Trans Reliab 40(5):572–581CrossRefMATHGoogle Scholar
  5. 5.
    Gao J et al (2012) Robustness of network formed by \(n\) interdependent networks with one-to-one correspondence between nodes. Phys Rev E85:066134Google Scholar
  6. 6.
    Gertsbakh I, Shpungin Y (2009) Models of network reliability: analysis combinatorics and Monte Carlo. CRC Press, Boca RatonGoogle Scholar
  7. 7.
    Gertsbakh I, Shpungin Y (2011) Network reliability and resilience. Springer briefs in electrical and computer engineering. Springer, BerlinGoogle Scholar
  8. 8.
    Gertsbakh I, Shpungin Y (2012) Combinatorial approach to computing importance indices of coherent systems. Probab Eng Inf Sci 26:117–128Google Scholar
  9. 9.
    Gertsbakh I, Shpungin Y (2012) Failure development in a system of two connected networks. Transp Commun 13(4):255–260Google Scholar
  10. 10.
    Gertsbakh I, Shpungin Y (2012) Stochastic models of network survivability. Qual Technol Quant Manag 9(1):45–58MathSciNetGoogle Scholar
  11. 11.
    Gertsbakh I, Shpungin Y (2014) Single-source epidemic process in a system of two interconnected networks. In: Frenkel I, Lisniansky A, Karagrigoriu A, Kleyner A (eds) Chapter 13, in Applied reliability engineering and risk analysis. Wiley, New YorkGoogle Scholar
  12. 12.
    Lewis TG (2009) Network science. Theory and applications. John Wiley & Sons Inc, New YorkGoogle Scholar
  13. 13.
    Li W, Bashan A et al (2012) Cascading failures in interconnected lattice networks. Phys Rev Lett 108(22):228702CrossRefGoogle Scholar
  14. 14.
    Newman MEJ (2010) Networks: an introduction. Oxford University Press, New YorkGoogle Scholar
  15. 15.
    Samaniego FJ (1985) On closure under ifr formation of coherent systems. IEEE Trans Reliab 34:69–72CrossRefMATHGoogle Scholar
  16. 16.
    Samaniego FJ (2007) System signatures and their application in engineering reliability. Springer, New YorkGoogle Scholar
  17. 17.
    Wolfram S (1991) Mathematica: a system for doing mathematics by computer, 2nd edn. Addison-Wesley Publishing Company, New YorkGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Ilya Gertsbakh
    • 1
  • Yoseph Shpungin
    • 2
  • Radislav Vaisman
    • 3
  1. 1.Department of MathematicsBen Gurion UniversityBeer-ShevaIsrael
  2. 2.Department of Software EngineeringShamoon College of EngineeringBeer-ShevaIsrael
  3. 3.School of Mathematics and PhysicsUniversity of QueenslandBrisbaneAustralia

Personalised recommendations