Rumor Processes on $$\mathbb {N}$$ N and Discrete Renewal Processes

Abstract

We study two rumor processes on N, the dynamics of which are related to an SI epidemic model with long range transmission. Both models start with one spreader at site 0 and ignorants at all the other sites of N, but differ by the transmission mechanism. In one model, the spreaders transmit the information within a random distance on their right, and in the other the ignorants take the information from a spreader within a random distance on their left.We obtain the probability of survival, information on the distribution of the range of the rumor and limit theorems for the proportion of spreaders. The key step of our proofs is to show that, in each model, the position of the spreaders on N can be related to a suitably chosen discrete renewal process.Date: December 18, 2013.

Frog models on trees through renewal theory

“…Lemma 1 in [6] states that the probability that the firework process on N reaches site n is equal to the probability that an undelayed renewal sequence (see Section 4 above) Y with hazard rate…”

The Rumor Percolation Model and Its Variations

“…The use of a non-standard version of Borel-Cantelli lemma helped in the task of finding conditions for the processes to die out. Gallo et al [5] based the proofs of their results on a clever relationship between the rumor processes and a specific discrete time renewal process. With this technique they were able to obtain more precise results for homogeneous versions of the processes.…”

“…(iii) Reverse fireworks processes on Z. Individuals throw their radius of influence to every direction as in (1.1) (See Gallo et all [5]).…”

“…(iv) If E(N) < ∞ and E(R) < ∞ then P(V ) = 0. (v) If lim inf n→∞ nP(R ≥ n)ϕ ′ N (P(R < n)) > 1 then P(V ) > 0.A consequence of Theorem 1 from Gallo et al[5] is the following result It is possible to have E(N) = ∞, E(R) = ∞ and P(V ) = 0. Take P(N ≥ n) ∼ 1 n when n → ∞ and P(R ≥ n) = 1 n ln n ln(lnn) 6.1.2.…”

“…Gallo et al[5]). If µ < ∞ thenZ(n) n a.s. −→ µ −1 ,and thus, with probability one, µ −1 is the final proportion of spreaders.…”