A Modified Rumor Model on Infinite Cayley Trees

Resumen

Some of the earliest references to mathematical rumor models can be found in [1, 4]. The Maki-Thompson rumor model on a connected graph can be informally described as follows. The vertices represent individuals who can be classified into three categories: ignorants, spreaders, and stiflers. A spreader transmits the rumor to any of its nearest ignorant neighbors at a rate of one. At the same rate, a spreader becomes a stifler after contact with other nearest-neighbor spreaders or stiflers. In this work, we consider an extension of the Maki-Thompson rumor model on an infinite Cayley tree, assuming that as soon as an individual hears the rumor, they either spread it with probability p ∈ (0, 1] or remain neutral, becoming a stifler, with probability 1 − p. Of course, if we take p = 1 we recover the basic model. For a review of recent results on trees, we refer the reader to [2, 3]. We focus our attention on the infinite Cayley tree of coordination number d + 1, with d ≥ 2, T = T_d. The model is a continuous-time Markov process (η_t)_t≥0 with state space S = {0, 1, 2}^T. That is, at time t the state of the process is a function η_t : T → {0, 1, 2}. We assume that each vertex v ∈ T represents an individual, and we say that such individual is an ignorant if η(v) = 0, a spreader if η(v) = 1, or a stifler if η(v) = 2. Moreover, if the system is in configuration η ∈ S, the state of vertex v changes according to the following transition rates: transition rate 0 → 1, pn₁(v, η), 0 → 2, (1 − p)n₁(v, η), 1 → 2, n₁(v, η) + n₂(v, η), where n_i(v, η) = ∑_u∼v 1{η(u) = i} is the number of nearest-neighbors of vertex v in state i for the configuration η, for i ∈ {1, 2}. Formally, (1) means that if the vertex v is, say, in state 0 at time t then the probability that it will be in state 1 at time t + h, for small h, is pn₁(x, η)h + o(h), where o(h) represents a function such that lim_h→0 o(h)/h = 0. We call the Markov process (η_t)_t≥0 the Maki-Thompson rumor model on T with probability p of spreading, MT(T, p)-model for short. In addition, we refer to the case when η₀(0) = 1 and η₀(v) = 0 for all v ≠ 0 as the standard initial configuration. [...]