Articles
Journal of Applied Probability (00219002)60(3)pp. 1096-1111
We study a sceptical rumour model on the non-negative integer line. The model starts with two spreaders at sites 0, 1 and sceptical ignorants at all other natural numbers. Then each sceptic transmits the rumour, independently, to the individuals within a random distance on its right after s/he receives the rumour from at least two different sources. We say that the process survives if the size of the set of vertices which heard the rumour in this fashion is infinite. We calculate the probability of survival exactly, and obtain some bounds for the tail distribution of the final range of the rumour among sceptics. We also prove that the rumour dies out among non-sceptics and sceptics, under the same condition. © The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust.
Indian Journal of Pure and Applied Mathematics (00195588)51(4)pp. 1661-1671
We consider a model of the spread of rumour among sceptical individuals. Let X0, X1,… be a {0, 1}-valued Markov chain and ρ0, ρ1, … a sequence of i.i.d. ℕ valued random variables independent of the Markov chain. An individual located at site i ∈ ℕ*:= ℕ ∪ {0} spreads the rumour to the individuals located in the interval [i, i + ρi] provided (i) Xi = 1 and (ii) if s/he has received the rumour from at least two distinct sources j, k < i with Xj = Xk = 1. To start the process we place two individuals at locations −1 and −2, each of spread the rumour to a distance ρ−1 and ρ−2 respectively to the right of itself. Here ρ−1 and ρ−2 are i.i.d. copies of ρ0. This extends the work of Sajadi and Roy [7] who considered the case when X0, X1,… is a sequence of i.i.d. {0,1} valued random variables, i.e. the believers {i: Xi = 1} and the disbelievers {i: Xi = 0} are located in an i.i.d. fashion. Here we study the case when the the believers and the disbelievers are located in a Markovian fashion. © 2020, Indian National Science Academy.
