The role of multiple repetitions on the size of a rumor

TL;DR

This paper introduces a generalized mathematical model for rumor spreading that accounts for multiple repetitions needed for ignorance to turn into spreading, analyzing how this affects the final size of the rumor-free population.

Contribution

It extends the classic Maki-Thompson rumor model by incorporating multiple hearing thresholds and provides a rigorous analysis of the resulting dynamical system.

Findings

The model predicts the final proportion of ignorants and stiflers based on repetition thresholds.

Limit theorems characterize the asymptotic behavior of the rumor spread.

The analysis offers insights into controlling rumor propagation in populations.

Abstract

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.