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Rumor Processes on N

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Valdivino V. Junior ,
Fábio P. Machado  and
Mauricio Zuluaga
Valdivino V. Junior*
Affiliation:
Universidade Federal de Goias
Fábio P. Machado*
Affiliation:
Universidade de São Paulo
Mauricio Zuluaga*
Affiliation:
Universidade de São Paulo
*
∗ Postal address: Institute of Mathematics and Statistics, Universidade Federal de Goias, Campus Samambaia, CEP 74001-970, Goiânia, GO, Brazil. Email address: [email protected]
∗∗ Postal address: Institute of Mathematics and Statistics, Universidade de São Paulo, Rua do Matão 1010, CEP 05508-090, São Paulo, SP, Brazil.
∗∗ Postal address: Institute of Mathematics and Statistics, Universidade de São Paulo, Rua do Matão 1010, CEP 05508-090, São Paulo, SP, Brazil.
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Abstract

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We study four discrete-time stochastic systems on N, modeling processes of rumor spreading. The involved individuals can either have an active or a passive role, speaking up or asking for the rumor. The appetite for spreading or hearing the rumor is represented by a set of random variables whose distributions may depend on the individuals. Our goal is to understand - based on the distribution of the random variables - whether the probability of having an infinite set of individuals knowing the rumor is positive or not.

Keywords

Coverage of spaceepidemic modeldisk percolationrumor model

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60G50: Sums of independent random variables; random walks
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 624 - 636
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Athreya, S., Roy, R. and Sarkar, A. (2004). On the coverage of space by random sets. Adv. Appl. Prob. 36, 1–18.Google Scholar
[2] Bertacchi, D. and Zucca, F. (2008). Critical behaviors and critical values of branching random walks on multigraphs. J. Appl. Prob. 45, 481–497.Google Scholar
[3] Bonar, D. D. and Khoury, M. J. Jr. (2006). Real Infinite Series. Mathematical Association of America, Washington, DC.Google Scholar
[4] Bremaud, P. (1999). {Markov Chains} (Texts Appl. Math. 31). Springer, New York.Google Scholar
[5] Daley, D. and Kendall, D. G. (1965). Stochastic rumors. J. Inst. Math. Appl. 1, 42–55.Google Scholar
[6] Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. John Wiley, New York.Google Scholar
[7] Fort, T. (1930). Infinite Series. Clarendon Press, Oxford.Google Scholar
[8] Grimmett, G. (1999). Percolation, 2nd edn. Springer, New York.Google Scholar
[9] Isham, V., Harden, S. and Nekovee, M. (2010). Stochastic epidemics and rumors on finite random networks. Physica A 389, 561–576.Google Scholar
[10] Kesten, H. and Sidoravicius, V. (2005). The spread of a rumor or infection in a moving population. Ann. Prob. 33, 2402–2462.Google Scholar
[11] Kostka, J., Oswald, Y. A. and Wattenhofer, R. (2008). Word of mouth: rumor dissemination in social networks. In Structural Information and Communication Complexity (Lecture Notes Comput. Sci. 5058), Springer, Berlin, pp. 185–196.Google Scholar
[12] Kurtz, T. G., Lebensztayn, E., Leichsenring, A. and Machado, F. P. (2008). Limit theorems for an epidemic model on the complete graph. Alea 4, 45–55.Google Scholar
[13] Lebensztayn, E. and Rodriguez, P. M. (2008). The disk-percolation model on graphs. Statist. Prob. Lett. 78, 2130–2136.Google Scholar
[14] Lebensztayn, E., Machado, F. P. and Martinez, M. Z. (2006). Self-avoiding random walks on homogeneous trees. Markov Process. Relat. Fields 12, 735–745.Google Scholar
[15] Maki, D. P. and Thompson, M. (1973). Mathematical Models and Applications. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[16] Pearce, C. E. M. (2000). The exact solution of the general stochastic rumours. Math. Comput. Modelling 31, 289–298.Google Scholar
[17] Pittel, B. (1990). On a Daley–Kendall model of random rumors. J. Appl. Prob. 27, 14–27.Google Scholar
[18] Sudbury, A. (1985). The proportion of the population never hearing a rumor. J. Appl. Prob. 22, 443–446.Google Scholar