The Maki-Thompson Rumor Model on Infinite Cayley Trees
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The Maki-Thompson Rumor Model on Infinite Cayley Trees Valdivino V. Junior1 · Pablo M. Rodriguez2
· Adalto Speroto3
Received: 26 January 2020 / Accepted: 7 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper we study the Maki-Thompson rumor model on infinite Cayley trees. The basic version of the model is defined by assuming that a population represented by a graph is subdivided into three classes of individuals: ignorants, spreaders and stiflers. A spreader tells the rumor to any of its (nearest) ignorant neighbors at rate one. At the same rate, a spreader becomes a stifler after a contact with other (nearest neighbor) spreaders, or stiflers. In this work we study this model on infinite Cayley trees, which is formulated as a continuous-times Markov chain, and we extend our analysis to the generalization in which each spreader ceases to propagate the rumor right after being involved in a given number of stifling experiences. We study sufficient conditions under which the rumor either becomes extinct or survives with positive probability. Keywords Maki-Thompson model · Phase-transition · Homogeneous tree · Branching process · Rumor spreading Mathematics Subject Classification 60K35 · 60K37 · 82B26
Communicated by Deepak Dhar. This work has been developed with support of the Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES), Financial Code 001. This work has been supported also by FAPESP (2017/10555-0), CNPq (Grant 304676/2016-0), and CAPES (under the Program MATH-AMSUD/CAPES 88881.197412/2018-01).
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Pablo M. Rodriguez [email protected] Valdivino V. Junior [email protected] Adalto Speroto [email protected]
1
Universidade Federal de Goias, Campus Samambaia, Goiânia, GO, CEP 74001-970, Brazil
2
Universidade Federal de Pernambuco, Av. Prof. Moraes Rego, 1235. Cidade Universitária, Recife, PE, CEP 50670-901, Brazil
3
Universidade de São Paulo, Caixa Postal 668, São Carlos, SP, CEP 13560-970, Brazil
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1 Introduction Currently, there exist a wide variety of mathematical models formulated to describe in a simple way the phenomenon of information transmission on a population. A wide range of these models is formed by epidemic-like processes inspired by the Daley-Kendal and the Maki-Thompson models. The Daley-Kendal model has been formulated in the mid 60’s as an alternative, to describe information spreading, to the well-known susceptible-infectedrecovered epidemic model, see [11,12]. Later the Maki-Thompson model has appeared in [21] as a simplification of the Daley-Kendal model. Since both models behaves asymptotically equal the Maki-Thompson model, that we just refer as the MT-model, has been used as a basis for many generalizations. The MT-model assumes a homogeneously mixed population of size N + 1 subdivided into three classes of individuals: Ignorants (those not aware of the rumor), spreaders (who are spreading it), and stiflers (who know the rumor but have ceased communicating it after meeting somebody
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