Phase Transition for the Maki–Thompson Rumour Model on a Small-World Network

Agliari, Elena; Pachón, Angélica; Rodríguez, Pablo M.; Tavani, Flavia

doi:10.1007/s10955-017-1892-x

Cited by 14 publications

(12 citation statements)

References 36 publications

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“…= µ. We point out that the interchange of limit and summation is guarantee by the Dominated Convergence Theorem, see [31,Theorem 9,1,p. 26].…”

Section: 2mentioning

confidence: 94%

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Stochastic rumors on random trees

Junior,

Rodriguez,

Speroto

2021

Preprint

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The Maki-Thompson rumor model is defined by assuming that a population represented by a graph is subdivided into three classes of individuals; namely, 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 the model on random trees. As usual we define a critical parameter of the model as the critical value around which the rumor either becomes extinct almost-surely or survives with positive probability. We analyze the existence of phase-transition regarding the survival of the rumor, and we obtain estimates for the mean range of the rumor. The applicability of our results is illustrated with examples on random trees generated from some well-known discrete distributions.

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“…= µ. We point out that the interchange of limit and summation is guarantee by the Dominated Convergence Theorem, see [31,Theorem 9,1,p. 26].…”

Section: 2mentioning

confidence: 94%

“…Note that Z d is an infinite graph. On the other hand, [1] studies the Maki-Thompson model on a smallworld graph. The graph is constructed by starting from a k-regular ring and by inserting, in the average, c additional links in such a way that k and c are tuneable parameters for the population structure; see Figure 1.1(c).…”

Section: Introductionmentioning

confidence: 99%

Stochastic rumors on random trees

Junior,

Rodriguez,

Speroto

2021

Preprint

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“…We point out that all these works deal with the case of homogeneous mixed populations, which is the same to say that the population is represented by a complete graph. For the case of a population represented by another type of graph we refer the reader to [1,9] for rigorous results based on probabilistic methods and to [3,22,23,28,29] for approximation results based on mean-field arguments and computational simulations.…”

Section: Introductionmentioning

confidence: 99%

The Maki-Thompson rumor model on infinite Cayley trees

Junior,

Rodriguez,

Speroto

2020

Preprint

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In this paper we study the Maki-Thompson rumor model on homogeneous trees. The basic version of the model may be 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 homogeneous 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.

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“…(1. 1) This means that if the process is in state (i, j) at time t, then the probabilities that it jumps to states (i − 1, j + 1) or (i, j − 1) at time t + h are, respectively, i j h + o(h) and j(N − 1 − i) h + o(h), where o(h) represents a function such that lim h→0 o(h)/h = 0. This describes exactly the situation in which individuals interact by contacts initiated by the spreaders, and the two possible transitions in (1.1) correspond to spreader-ignorant, and spreader-(spreader or stifler) interactions.…”

Section: Introductionmentioning

confidence: 99%

The role of multiple repetitions on the size of a rumor

Rada¹,

Coletti²,

Lebensztayn³

et al. 2020

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We propose a mathematical model to measure how multiple repetitions may influence in the ultimate proportion of the population never hearing a rumor during a given outbreak. The model is a multi-dimensional continuous-time Markov chain that can be seen as a generalization of the Maki-Thompson model for the propagation of a rumor within a homogeneously mixing population. In the well-known basic model, the population is made up of "spreaders", "ignorants" and "stiflers", and any spreader attempts to transmit the rumor to the other individuals via directed contacts. In case the contacted individual is an ignorant, it becomes a spreader, while in the other two cases the initiating spreader turns into a stifler. The process in a finite population will eventually reach an equilibrium situation, where individuals are either stiflers or ignorants. We generalize the model by assuming that each ignorant becomes a spreader only after hearing the rumor a predetermined number of times. We identify and analyze a suitable limiting dynamical system of the model, and we prove limit theorems that characterize the ultimate proportion of individuals in the different classes of the population.

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Phase Transition for the Maki–Thompson Rumour Model on a Small-World Network

Cited by 14 publications

References 36 publications

Stochastic rumors on random trees

Stochastic rumors on random trees

The Maki-Thompson rumor model on infinite Cayley trees

The role of multiple repetitions on the size of a rumor

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