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On Dissemination Thresholds in Regular and Irregular Graph Classes

  • I. Rapaport
  • K. Suchan
  • I. Todinca
  • J. Verstraete
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4957)

Abstract

We investigate the natural situation of the dissemination of information on various graph classes starting with a random set of informed vertices called active. Initially active vertices are chosen independently with probability p, and at any stage in the process, a vertex becomes active if the majority of its neighbours are active, and thereafter never changes its state. We show that in any cubic graph, with high probability, the information will not spread to all vertices in the graph if \(p < \frac{1}{2}\). We give families of graphs in which information spreads to all vertices with high probability for relatively small values of p.

Keywords

Random Graph Regular Graph Graph Class Active Vertex Bootstrap Percolation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Aizenman, A., Lebowitz, J.: Metastability effects in bootstrap percolation. J. Phys. A: Math. Gen. 21, 3801–3813 (1988)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Alon, N., Spencer, J.: The Probabilistic Method, 2nd edn. Wiley, Chichester (1992)zbMATHGoogle Scholar
  3. 3.
    Bollobás, B.: Random graphs, 2nd edn. Academic Press, Cambridge University Press (1985 )Google Scholar
  4. 4.
    Balogh, J., Bollobás, B.: Bootstrap percolation on the hypercube. Probab. Theory Related Fields 134(4), 624–648 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Balogh, J., Bollobás, B., Morris, J.: Majority bootstrap percolation on the hypercube. (manuscript, 2007)Google Scholar
  6. 6.
    Balogh, J., Pittel, B.: Bootstrap percolation on the random regular graph. Random Structures Algorithms 30(1-2), 257–286 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Bollobás, B., Szemerédi, E.: Girth of sparse graphs. J. Graph Theory 39(3), 194–200 (2002)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Carvajal, B., Matamala, M., Rapaport, I., Schabanel, N.: Small alliances in graphs. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 218–227. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Garmo, H.: The asymptotic distribution of long cycles in random regular graphs. Random Struct. Algorithms 15(1), 43–92 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Haynes, T.W., Hedetniemi, S.T., Henning, M.A.: Global deffensive alliances in graphs. Electronic J. Comb. 10, 139–146 (2003)MathSciNetGoogle Scholar
  11. 11.
    Holroyd, A.: Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation. Probability Theory and Related Fields 125(2), 195–224 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Kempe, D., Kleinberg, J., Tardos, E.: Maximizing the Spread of Influence through a Social Network. In: Proceedings of KDD 2003, pp. 137–146 (2003)Google Scholar
  13. 13.
    Lubotsky, A., Phillips, R., Sarnak, R.: Ramanujan graphs. Combinatorica 8, 261–278 (1988)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Habib, M., McDiarmid, C., Ramirez-Alfonsin, J., Reed, B.: Probabilistic Methods for Algorithmic Discrete Mathematics. Series: Algorithms and Combinatorics, vol. 16. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  15. 15.
    Wormald, N.: Models of random regular graphs. In: Lamb, J.D., Preece, D.A. (eds.) Surveys in Combinatorics. London Mathematical Society Lecture Note Series, vol. 276, pp. 239–298. Cambridge University Press, Cambridge (1999)Google Scholar
  16. 16.
    Janson, S., Łuczak, T., Rucinski, A.: Random graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York (2000)zbMATHGoogle Scholar
  17. 17.
    Peleg, D.: Local majorities, coalitions and monopolies in graphs: a review. Theoretical Computer Science 282, 231–257 (2002)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • I. Rapaport
    • 1
  • K. Suchan
    • 1
    • 2
  • I. Todinca
    • 3
  • J. Verstraete
    • 4
  1. 1.Departamento de Ingeniería Matemática and Centro de Modelamiento MatemáticoUniversidad de Chile 
  2. 2.Faculty of Applied MathematicsAGH - University of Science and TechnologyCracowPoland
  3. 3.LIFOUniversité d’OrléansFrance
  4. 4.University of CaliforniaSan DiegoUSA

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