Availability in Large Networks: Global Characteristics from Local Unreliability Properties

  • Hans Daduna
  • Lars Peter Saul
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7201)


We apply mean-field analysis to compute global availability in large networks of generalized SIS and voter models. The main results provide comparison and bounding techniques of the global availability depending on the local degree structure of the networks.


Reliability SIS model voter models mean field analysis stochastic ordering convex order bounding global availability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AB00]
    Anderson, H., Britton, T.: Stochastic Epedemic Models and Their Statistical Analysis. Lecture Notes in Statistics. Springer, New York (2000)CrossRefGoogle Scholar
  2. [Bai75]
    Bailey, N.J.: The Mathematical Theory of Infectiuos Diseases and Its Applications. Hafner Press, New York (1975)Google Scholar
  3. [BBV08]
    Barrat, A., Barthelemy, M., Vespignani, A.: Daynamical Processes on Complex Networks. Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar
  4. [BCFH09]
    Bakhshi, R., Cloth, L., Fokking, W., Haverkoert, B.R.: Mean-field analysis for the evaluation of gossip protocols. In: Proceedings of the Sixth International Conference on Quantitative Evaluation of Systems, pp. 247–256. IEEE Computer Society (2009)Google Scholar
  5. [BCFH11]
    Bakhshi, R., Cloth, L., Fokking, W., Haverkoert, B.R.: Mean-field framework for performance evaluation of push-pull gossip protocols. Performance Analysis 68, 157–179 (2011)Google Scholar
  6. [BGFv09]
    Bakhshi, R., Gavidia, D., Fokking, W., van Steen, M.: An analytical model of information dissemination for a gossip-based protocol. Computer Networks 53, 2288–2303 (2009)CrossRefzbMATHGoogle Scholar
  7. [BMM07]
    Boudec, J.-Y., McDonald, D., Mundinger, J.: A generic mean field convergence result for systems of interacting objects. In: Proceedings of the Fourth International Conference on Quantitative Evaluation of Systems, pp. 3–15. IEEE Computer Society (2007)Google Scholar
  8. [CM96]
    Chakka, R., Mitrani, I.: Approximate solutions for open networks with breakdowns and repairs. In: Kelly, F.P., Zachary, S., Ziedins, I. (eds.) Stochastic Networks, Theory and Applications. Royal Statistical Society Lecture Notes Series, vol. 4, ch. 16, pp. 267–280. Clarendon Press, Oxford (1996)Google Scholar
  9. [DG01]
    Daley, D.J., Gani, J.: Epidemic Modelling: An Introduction. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  10. [DM03]
    Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of Networks. Oxford University Press, Oxford (2003) (reprint 2004)CrossRefzbMATHGoogle Scholar
  11. [DM10]
    Draief, M., Massoulie, L.: Epidemics and Rumours in Complex Networks. London Mathematical Society Lecture Note Series, vol. 369. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  12. [DPR08]
    Daduna, H., Pestien, V., Ramakrishnan, S.: Throughput limits from the asymptotic profile of cyclic networks with state-dependent service rates. Queueing Systemes and Their Applications 58, 191–219 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [HMRT01]
    Haverkort, B.R., Marie, R., Rubino, G., Trivedi, K.: Performability Modeling, Technique and Tools. Wiley, New York (2001)Google Scholar
  14. [Jac08]
    Jackson, M.O.: Social and Economic Networks. Princeton University Press, Princeton (2008)zbMATHGoogle Scholar
  15. [JR07]
    Jackson, M.O., Rogers, B.W.: Relating network structure to diffusion properties through stochastic dominance. The B.E. Journal of Theoretical Economics 7(1), 1–13 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [JY07]
    Jackson, M.O., Yariv, L.: Diffusion of behavior and equilibrium properties in network games. American Economical Review (Papers and Proceedings) 97, 92–98 (2007)Google Scholar
  17. [Lam09]
    Lamberson, P.J.: Linking network structure and diffusion through stochastic dominance. In: Complex Adaptive Systems and the Threshold Effects: Views from the Natural and Social Sciences, pp. 76–82. Association for the Advancement of Artificial Intelligence (2009); Papers from the AAAI Fall Symposium: FS-09-03Google Scholar
  18. [Lig85]
    Liggett, T.M.: Interacting Particle Systems. Grundlehren der mathematischen Wissenschaften, vol. 276. Springer, Berlin (1985)zbMATHGoogle Scholar
  19. [LP06]
    Lopez-Pintado, D.: Contagion and coordination in random networks. International Journal of Game Theory 34, 371–381 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [LP08]
    Lopez-Pintado, D.: Diffusion in complex social networks. GAMES and Economic Behavior 62, 573–590 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [MS02]
    Müller, A., Stoyan, D., Liggett, T.M.: Comparison Methods for Stochastic Models and Risks. Wiley, Chichester (2002)zbMATHGoogle Scholar
  22. [PM84]
    Pinski, E., Yemini, Y.: A statistical mechanics of some interconnetion networks. In: Gelenbe, E. (ed.) Performance 1984, pp. 147–158. North-Holland, Amsterdam (1984)Google Scholar
  23. [Sau06]
    Sauer, C.: Stochastic product form networks with unreliable nodes: Analysis of performance and availability. PhD thesis, University of Hamburg, Department of Mathematics (2006)Google Scholar
  24. [SD03]
    Sauer, C., Daduna, H.: Availability formulas and performance measures for separable degradable networks. Economic Quality Control 18, 165–194 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [WDW07]
    Wang, Y., Dang, H., Wu, H.: A survey on analytic studies of Delay-Tolerant Mobile Sensor Networks. Wireless Communications and Mobile Computing 7, 1197–1208 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Hans Daduna
    • 1
  • Lars Peter Saul
    • 1
  1. 1.Department of Mathematics, Mathematical Statistics and Stochastic ProcessesUniversity of HamburgHamburgGermany

Personalised recommendations