The maximum proportion of spreaders in stochastic rumor models

TL;DR

This paper analyzes stochastic rumor models, including variants of Daley--Kendall and Maki--Thompson, proving the almost sure convergence of the maximum proportion of spreaders and determining the asymptotic rumor peak.

Contribution

It establishes the almost sure convergence of the maximum spreader proportion and derives the asymptotic rumor peak for these stochastic models.

Findings

Maximum proportion of spreaders converges almost surely as population grows.

Asymptotic rumor peak is approximately 0.3069 for classical models.

Results apply to , p variants of well-known rumor models.

Abstract

We examine a general stochastic rumor model characterized by specific parameters that govern the interaction rates among individuals. Our model includes the \((\alpha, p)\)-probability variants of the well-known Daley--Kendall and Maki--Thompson models. In these variants, a spreader involved in an interaction attempts to transmit the rumor with probability \(p\); if successful, any spreader encountering an individual already informed of the rumor has probability \(\alpha\) of becoming a stifler. We prove that the maximum proportion of spreaders throughout the process converges almost surely, as the population size approaches~\(\infty\). For both the classical Daley--Kendall and Maki--Thompson models, the asymptotic proportion of the rumor peak is \(1 - \log 2 \approx 0.3069\).