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Doctoral Thesis
DOI
https://doi.org/10.11606/T.45.2010.tde-16112010-162236
Document
Author
Full name
Pablo Martin Rodriguez
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2010
Supervisor
Committee
Machado, Fabio Prates (President)
Garcia, Nancy Lopes
Groisman, Pablo José
Grynberg, Sebastian Pablo
Lebensztayn, Élcio
 
Title in Portuguese
Generalizações e teoremas limites para modelos estocásticos de rumores
Keywords in Portuguese
cadeias de Markov "density dependent"
modelo de Daley-Kendall
modelo de Maki-Thompson
rumores estocásticos
teoremas limites
Abstract in Portuguese
Os modelos de Daley-Kendall e Maki-Thompson são os dois modelos estocásticos para difusão de rumores mais citados até o momento. Em ambos, uma população finita fechada e totalmente misturada é subdividida em três classes de indivíduos denominados ignorantes, informantes e contidos. Depois de um rumor ser introduzido na população, difunde-se através desta seguindo determinadas regras que dependem da classe à qual a pessoa que sabe do rumor pertence. Tanto a proporção final de indivíduos que nunca chegam a conhecer o rumor quanto o tempo que este demora em ser difundido são variáveis de interesse para os modelos propostos. As técnicas encontradas na literatura para estudar modelos de rumores são o princípio de difusão de constantes arbitrárias; argumentos de martingais; o método de funções geradoras e a análise de versões determinísticas do processo. Neste trabalho apresentamos uma alternativa para essas técnicas baseando-nos na teoria de cadeias de Markov "density dependent''. O uso desta nova abordagem nos permite apresentar resultados assintóticos para um modelo geral que tem como casos particulares os famosos modelos de Daley-Kendall e Maki-Thompson, além de variações de modelos de rumores apresentados na literatura recentemente.
 
Title in English
Generalizations and limit theorems for stochastic rumour models
Keywords in English
Daley-Kendall model
density dependent Markov chains
limit theorems
Maki-Thompson model
stochastic rumours
Abstract in English
Daley-Kendall and Maki-Thompson models are the two most cited stochastic models for the spread of rumours phenomena, in scientific literature. In both, a closed homogeneously mixing population is subdivided into three classes of individuals called ignorants, spreaders and stiflers. After a rumor is introduced in the population, it spreads by following certain rules that depend on the class to which the individual who knows the rumor belongs. Both the final proportion of the population never hearing the rumor and the time it takes are variables of interest for the proposed models. The main tools used to study stochastic rumours have been the principle of the diffusion of arbitrary constants, martingale arguments, generating functions and the study of analogue deterministic versions. Relying on the theory of density dependent Markov chains, we present an alternative to these tools. This approach allows us to establish asymptotical results for a general model that has as particular cases the classical Daley-Kendall and Maki-Thompson models, and other variations for rumour models reported in the literature recently.
 
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Publishing Date
2011-05-12
 
WARNING: The material described below relates to works resulting from this thesis or dissertation. The contents of these works are the author's responsibility.
  • LEBENSZTAYN, E., MACHADO, F. P., and RODRIGUEZ, P. M.. Limit Theorems for a General Stochastic Rumour Model [doi:10.1137/100819588]. SIAM Journal on Applied Mathematics [online], 2011, vol. 71, n. 4, p. 1476.

  • LEBENSZTAYN, E., MACHADO, F.P., and RODRíGUEZ, P.M.. On the behaviour of a rumour process with random stifling [doi:10.1016/j.envsoft.2010.10.015]. Environmental Modelling & Software [online], 2011, vol. 26, n. 4, p. 517-522.

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