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Rumor Processes on \mathbb {N} and Discrete Renewal Processes |
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| Authors: | Sandro Gallo Nancy L Garcia Valdivino Vargas Junior Pablo M Rodríguez |
| Institution: | 1. Departamento de Métodos Estatísticos, Instituto de Matemática - Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil 2. Departamento de Estatística, Instituto de Matemática, Estatística e Computa??o Científica - Universidade de Campinas (UNICAMP), Campinas, Brazil 3. Instituto de Matemática e Estatística - Universidade Federal de Goiás (UFG), Goiania, Brazil 4. Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computa??o - Universidade de S?o Paulo (USP), S?o Carlos, Brazil |
| 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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