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Abstract

We study the SIR epidemic model with infections carried by k particles making independent random walks on a random regular graph. We give a edge-weighted graph reduction of the dynamics of the process that allows us to apply standard results of Erdős–Renyi random graphs on the particle set. In particular, we show how the parameters of the model produce two phase transitions: In the subcritical regime, O(ln k) particles are infected. In the supercritical regime, for a constant C determined by the parameters of the model, Ck get infected with probability C, and O(ln k) get infected with probability (1 − C). Finally, there is a regime in which all k particles are infected. Furthermore, the edge weights give information about when a particle becomes infected. We demonstrate how this can be exploited to determine the completion time of the process by applying a result of Janson on randomly edge weighted graphs.

Keywords

Random walks random graphs viral processes SIR model 

References

  1. 1.
    Abdullah, M., Cooper, C., Draief, M.: Viral Processes by Random Walks on Random Regular Graphs (2011), http://arxiv.org/abs/1104.3789
  2. 2.
    Aldous, D., Fill, J.: Reversible Markov Chains and Random Walks on Graphs, http://stat-www.berkeley.edu/pub/users/aldous/RWG/book.html (in preparation)
  3. 3.
    Baumann, H., Crescenzi, P., Fraigniaud, P.: Parsimonious Flooding in Dynamic Graphs. In: Proceedings of the 28th ACM Symposium on Principles of Distributed Computing, pp. 260–269 (2009)Google Scholar
  4. 4.
    Buscarino, A., Fortuna, L., Frasca1, M., Latora, V.: Disease Spreading in Populations of Moving Agents. EPL (Europhysics Letters) 82(3) (2008)Google Scholar
  5. 5.
    Chaintreau, A., Hui, P., Scott, J., Gass, R., Crowcroft, J., Diot, C.: Impact of Human Mobility on Opportunistic Forwarding Algorithms. IEEE Transactions Mobile Computing 6(6), 606–620 (2007)CrossRefGoogle Scholar
  6. 6.
    Cooper, C., Frieze, A.: The Cover Time of Random Regular Graphs. SIAM Journal on Discrete Mathematics 18, 728–740 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cooper, C., Frieze, A.M.: The Cover Time of the Giant Component of a Random graph. Random Structures and Algorithms 32, 401–439 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cooper, C., Frieze, A.M., Radzik, T.: Multiple Random Walks in Random Regular Graphs. SIAM Journal on Discrete Mathematics 23, 1738–1761 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Daley, D., Gani, J.: Epidemic Modelling: An Introduction. Studies in Mathematical Biology. Cambridge University Press, Cambridge (2001)Google Scholar
  10. 10.
    Dickman, R., Rolla, L., Sidoravicius, V.: Activated Random Walkers: Facts, Conjectures and Challenges. Journal of Statistical Physics 138, 126–142 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dimitriou, T., Nikoletseas, S., Spirakis, P.: The Infection Time of Graphs. Discrete Applied Mathematics 154, 18 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Draief, M., Ganesh, A.: A random Walk Model for Infection on Graphs: Spread of Epidemics and Rumours with Mobile Agents. Discrete Event Dynamic Systems 21, 1 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Draief, M., Massoulié, L.: Epidemics and Rumours in Complex Networks. London Mathematical Society Series, vol. 369. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  14. 14.
    Feller, W.: An Introduction to Probability Theory, 2nd edn., vol. I. Wiley, New York (1960)Google Scholar
  15. 15.
    Friedman, J.: A Proof of Alon’s Second Eigenvalue Conjecture and Related Problems, vol. 195. Memoirs of the American Mathematical Society, Providence (2008)Google Scholar
  16. 16.
    Janson, S.: One, Two and Three Times log n/n for Paths in a Complete Graph with Random Weights. Combinatorics, Probability and Computing 8, 347–361 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kurtz, T.G., Lebensztayn, E., Leichsenring, A.R., Machado, F.P.: Limit Theorems for an Epidemic Model on the Complete Graph. Latin American Journal of Probability and Mathematical Statistics 4, 45–55 (2008)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Lovász, L.: Random Walks on Graphs: A Survey, Combinatorics. In: Paul Erdős is Eighty, vol. 2, pp. 1–46. Bolyai Society Mathematical Studies (1993)Google Scholar
  19. 19.
    Peres, Y., Sinclair, A., Sousi, P., Stauffer, A.: Mobile Geometric Graphs: Detection, Coverage and Percolation. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 412–428 (2011)Google Scholar
  20. 20.
    Pittel, B.: On spreading a rumor. SIAM Journal of Applied Mathematics 47(1), 213–223 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rhodes, C., Nekovee, M.: The Opportunistic Transmission of Wireless Worms Between Mobile Devices. Physica A: Statistical Mechanics and Its Applications 387(27), 6837–6840 (2008)CrossRefGoogle Scholar
  22. 22.
    Pettarin, A., Pietracaprina, A., Pucci, G., Upfal, E.: Tight Bounds on Information Dissemination in Sparse Mobile Networks. In: Proceedings of the 30th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (2011)Google Scholar
  23. 23.
    Zhou, J., Liua, Z.: Epidemic Spreading in Communities with Mobile Agents. Physica A: Statistical Mechanics and its Applications 388(7), 1, 1228–1236 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Mohammed Abdullah
    • 1
  • Colin Cooper
    • 1
  • Moez Draief
    • 2
  1. 1.Department of InformaticsKing’s College LondonUK
  2. 2.Department of Electrical and Electronic EngineeringImperial College LondonUK

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