Rumor Processes in Random Environment on $\mathbb{N}$ and on Galton–Watson Trees

Abstract

Abstract. The aim of this paper is to study rumor processes in random environment. In a rumor process a signal starts from the stations of a fixed vertex (the root) and travels on a graph from vertex to vertex. We consider two rumor processes. In the firework process each station, when reached by the signal, transmits it up to a random distance. In the reverse firework process, on the other hand, stations do not send any signal but they "listen" for it up to a random distance. The first random environment that we consider is the deterministic 1-dimensional tree N with a random number of stations on each vertex; in this case the root is the origin of N. We give conditions for the survival/extinction on almost every realization of the sequence of stations. Later on, we study the processes on Galton-Watson trees with random number of stations on each vertex. We show that if the probability of survival is positive, then there is survival on almost every realization of the infinite tree such that there is at least one station at the root. We characterize the survival of the process in some cases and we give sufficient conditions for survival/extinction.

The Rumor Percolation Model and Its Variations

“…Theorem 6.8 (Bertacchi and Zucca [3]). In the heterogeneous case, [ϕ N (P(R < n + 1)) − ϕ N (P(R < n))]t n .…”

“…We refer to this tree as Td. Bertacchi and Zucca [3] rely in their analisys on associating the processes with random numbers of stations (fireworks or reverse fireworks), with processes with one station per vertex as in Junior et al [9] . Indeed, they consider processes with one station on each vertex x and radius of influenceR x = 1 {Nx≥1} max{R x,j : j = 1, .…”

“…ϕ N (P(R < j)) Theorem 6.12 (Bertacchi and Zucca [3]). Consider a Homogeneous Reverse Fireworks Process.…”

“…Zucca [3]. They consider a type of random environment in the sense that the number of informants in each vertex of are random.…”

“…Random environmentsIn this section we review the Fireworks and the Reverse Fireworks processes, with a random number of stations at each vertex. Bertacchi and Zucca[3] consider an extra source of randomness: the number of individuals sitting on each vertex. They consider two families ofrandom variables {N x } x∈G and {R x,i } i∈N x∈G such that {N x , R x,i } are independent and {R x,i } i∈N are identically distributed for all x ∈ G that is R x,i ∼ R x .In their paper N x represents the random number of individuals at vertex x (in particular N O is the number of individuals at the origin) while {R x,i } Nx i=1 are their radius of influence.…”