The maximum proportion of spreaders in stochastic 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).