Critical thresholds in stochastic rumors on trees

The vertices of a tree represent individuals in one of three states: ignorant, spreader, or stifler. A spreader transmits the rumor to any of its nearest ignorant neighbors at rate one. At the same rate, a spreader becomes a stifler after contacting nearest-neighbor spreaders or stiflers. The rumor survives if, at all times, there exists at least one spreader. We consider two extensions and prove phase transition results for rumor survival. First, we consider the infinite Cayley tree of coordination number $d+1$, with $d\ge 2$, and assume that as soon as an ignorant hears the rumor, the individual becomes spreader with probability $p$, or stifler with probability $1-p$. Using coupling with branching processes we prove that for any $d$ there is a phase transition in $p$ and localize the critical parameter. By refining this approach, we extend the study to an inhomogeneous tree with hubs of degree $d+1$ and other vertices of degree at most $k=o\left(d\right)$. The purpose of this extension is to illustrate the impact of the distance between hubs on the dissemination of rumors in a network. To this end, we assume that each hub is, on average, connected to $\alpha (d+1)$ hubs, with $\alpha \in (0,1]$, via paths of length $h$. We obtain a phase transition result in $\alpha$ in terms of $d,k,$ and $h$, and we show that in the case of $k=\Theta (logd)$ phase transition occurs iff $h\lesssim \Theta (logd/(loglogd))$.