Elcio Lebensztayn

LIMIT THEOREMS FOR A GENERAL STOCHASTIC RUMOUR MODEL

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.

On the behaviour of a rumour process with random stifling

We propose a realistic generalization of the Maki-Thompson rumour model by assuming that each spreader ceases to propagate the rumour right after being involved in a random number of stifling experiences. We consider the process with a general initial configuration and establish the asymptotic behaviour (and its fluctuation) of the ultimate proportion of ignorants as the population size grows to ∞. Our approach leads to explicit formulas so that the limiting proportion of ignorants and its variance can be computed.

Random walks systems with killing on ℤ

We study random walks systems on ℤ whose general description follows. At time zero, there is a number of particles at each vertex of ℕ, all being inactive, except for those placed at the vertex one. Each active particle performs a simple random walk on ℤ and, up to the time it dies, it activates all inactive particles that it meets along its way. An active particle dies at the instant it reaches a certain fixed total of jumps (L ≥ 1) without activating any particle, so that its lifetime depends strongly on the past of the process. We investigate how the probability of survival of the process depends on L and on the jumping probabilities of the active particles.

A process of rumour scotching on finite populations

Rumour spreading is a ubiquitous phenomenon in social and technological networks. Traditional models consider that the rumour is propagated by pairwise interactions between spreaders and ignorants. Only spreaders are active and may become stiflers after contacting spreaders or stiflers. Here we propose a competition-like model in which spreaders try to transmit an information, while stiflers are also active and try to scotch it. We study the influence of transmission/scotching rates and initial conditions on the qualitative behaviour of the process. An analytical treatment based on the theory of convergence of density-dependent Markov chains is developed to analyse how the final proportion of ignorants behaves asymptotically in a finite homogeneously mixing population. We perform Monte Carlo simulations in random graphs and scale-free networks and verify that the results obtained for homogeneously mixing populations can be approximated for random graphs, but are not suitable for scale-free networks. Furthermore, regarding the process on a heterogeneous mixing population, we obtain a set of differential equations that describes the time evolution of the probability that an individual is in each state. Our model can also be applied for studying systems in which informed agents try to stop the rumour propagation, or for describing related susceptible–infected–recovered systems. In addition, our results can be considered to develop optimal information dissemination strategies and approaches to control rumour propagation.

A large deviations principle for the Maki-Thompson rumour model

Abstract We consider the stochastic model for the propagation of a rumour within a population which was formulated by Maki and Thompson [20] . Sudbury [22] established that, as the population size tends to infinity, the proportion of the population never hearing the rumour converges in probability to 0.2032. Watson [23] later derived the asymptotic normality of a suitably scaled version of this proportion. We prove a corresponding large deviations principle, with an explicit formula for the rate function.

Random Walks Systems with Finite Lifetime on \mathbb {Z}

We consider a non-homogeneous random walks system on

Limit theorems for an epidemic model on the com- plete graph

\mathbb {Z}

Nonhomogeneous random walks systems on ℤ

Z in which each active particle performs a nearest neighbor random walk and activates all inactive particles it encounters up to a total amount of L jumps. We present necessary and sufficient conditions for the process to survive, which means that an infinite number of random walks become activated.