Limit Theorems for a General Stochastic Rumour Model

Abstract

Abstract. We study a general stochastic rumour model in which an ignorant individual has a certain probability of becoming a stifler immediately upon hearing the rumour. We refer to this special kind of stifler as an uninterested individual. Our model also includes distinct rates for meetings between two spreaders in which both become stiflers or only one does, so that particular cases are the classical Daley-Kendall and Maki-Thompson models. We prove a Law of Large Numbers and a Central Limit Theorem for the proportions of those who ultimately remain ignorant and those who have heard the rumour but become uninterested in it.

Laws of large numbers for the frog model on the complete graph

“…The previous numerical analysis indicates that, for the Maki-Thompson model in which all spreaders act simultaneously, the limiting fraction of ignorants in the population would be equal to 0.174545. For more on the definitions and limit theorems for stochastic rumour models, we refer to Daley and Gani [11, Chapter 5], Lebensztayn [28], and Lebensztayn et al [32].…”

Asymptotic behavior for a modified Maki-Thompson model with directed inter-group interactions

“…Indeed, the first rigorous results for these models are limit theorems for the remaining proportion of ignorants when the process ends, and it has been proved that such a proportion approximates to 20% of the population, see [19,21]. We refer the reader to [7,12,13,14] for a review of results and generalizations of these models. Also, many modifications of these models have been considered assuming that the population is not necessarily homogeneous nor totally mixing, see [1,3,5,16,17] and the references therein.…”

“…The motivation of this work is to propose and analyze a general Maki-Thompson model which incorporates directed inter-group interactions. As far as we know no rigorous results exist for this type of model and therefore it may be seen as a contribution to increase the familiy of general models like those considered recently by [13,14]. In our model we assume the existence of two groups of individuals, say A-individuals and B-individuals.…”

A large deviations principle for the Maki–Thompson rumour model

“…The limit theorems proved by Sudbury [22] and Watson [23] were generalized by Lebensztayn et al [17] for a Maki-Thompson rumour model with general initial configuration and in which a spreader becomes a stifler only after being involved in a random number of unsuccessful telling meetings. In Lebensztayn et al [16], these limit theorems are also established for a general stochastic rumour model defined in terms of parameters that determine the rates at which the different interactions between individuals occur. This definition allows a quantitative formulation of various behavioural mechanisms of the people involved in the dissemination of the rumour.…”