Rumour Propagation Among Sceptics: The Markovian Case

Abstract

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.