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Rumor Processes on \(\mathbb {N}\) and Discrete Renewal Processes

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Abstract

We study two rumor processes on \(\mathbb {N}\), the dynamics of which are related to an SI epidemic model with long range transmission. Both models start with one spreader at site \(0\) and ignorants at all the other sites of \(\mathbb {N}\), but differ by the transmission mechanism. In one model, the spreaders transmit the information within a random distance on their right, and in the other the ignorants take the information from a spreader within a random distance on their left. We obtain the probability of survival, information on the distribution of the range of the rumor and limit theorems for the proportion of spreaders. The key step of our proofs is to show that, in each model, the position of the spreaders on \(\mathbb {N}\) can be related to a suitably chosen discrete renewal process.

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Acknowledgments

This work was partially funded by CNPq grants 479313/2012-1, 302755/2010-1 and FAPESP grant 2013/03898-8. The authors thank ICMC/USP and IMECC/UNICAMP for their hospitality. We thank the anonymous referees for valuable remarks that greatly improved the presentation of this work.

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Authors and Affiliations

  1. Departamento de Métodos Estatísticos, Instituto de Matemática - Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil

    Sandro Gallo

  2. Departamento de Estatística, Instituto de Matemática, Estatística e Computação Científica - Universidade de Campinas (UNICAMP), Campinas, Brazil

    Nancy L. Garcia

  3. Instituto de Matemática e Estatística - Universidade Federal de Goiás (UFG), Goiânia, Brazil

    Valdivino Vargas Junior

  4. Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computação - Universidade de São Paulo (USP), São Carlos, Brazil

    Pablo M. Rodríguez

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  1. Sandro Gallo
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  2. Nancy L. Garcia
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  3. Valdivino Vargas Junior
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  4. Pablo M. Rodríguez
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Correspondence to Nancy L. Garcia.

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Gallo, S., Garcia, N.L., Junior, V.V. et al. Rumor Processes on \(\mathbb {N}\) and Discrete Renewal Processes. J Stat Phys 155, 591–602 (2014). https://doi.org/10.1007/s10955-014-0959-1

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  • DOI: https://doi.org/10.1007/s10955-014-0959-1

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