A modified rumor model on infinite Cayley trees

The Maki-Thompson rumor model on a connected graph can be informally described as follows. The vertices represent individuals that 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 a 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. We focus our attention in 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 states 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.