The Maki-Thompson model with random awareness

TL;DR

This paper extends the Maki-Thompson rumor model to an infinite-dimensional setting, analyzing the asymptotic behavior of rumor spread with individuals becoming spreaders after random exposures.

Contribution

It introduces an infinite-dimensional extension of the classical law of large numbers for density-dependent models and applies it to a generalized rumor spreading process.

Findings

Derived asymptotic proportions of individuals in each state.

Characterized maximum spreading proportions.

Explored conditions for wave-like rumor propagation.

Abstract

We propose a rumor propagation model in which individuals within a homogeneously mixed population can assume one of infinitely many possible states. To analyze this model, we extend the classical law of large numbers for density-dependent population models in $\mathbb{R}^d$ (Ethier \& Kurtz in 2005) to an infinite--dimensional setting. Specifically, we prove a convergence result for stochastic processes in the space of $p$-summable real sequences, generalizing the finite-dimensional theory. We apply this framework to an infinite-dimensional continuous-time Markov chain that can be seen as a generalization of the Maki--Thompson model, where each ignorant individual becomes a spreader only after hearing the rumor a random number of times. We derive the asymptotic proportions of individuals in each state at the end of a rumor outbreak. Furthermore, we characterize the maximum proportion of individuals actively spreading the rumor and explore conditions under which the model exhibits waves of rumor propagation.