Critical thresholds in stochastic rumors on trees

TL;DR

The paper addresses phase transitions in stochastic rumor spreading on trees under two extensions of the Maki–Thompson model: (i) spreading on the infinite Cayley tree with a fixed spread probability $p$, and (ii) spreading on a three-type inhomogeneous tree with hubs connected by length $h$ paths. The authors reduce the dynamics to an associated branching process and derive explicit thresholds, notably $p_c(d)=\{ \frac{d e^{d+1}}{(d+1)^d}\Gamma(d,d+1) \}^{-1}$ with $p_c(d)\sim\sqrt{2/(\pi d)}$, and an alpha-threshold $\alpha_c(d,k,h)$ with a closed-form expression and asymptotics $h \lesssim \log d/\log k$ (or $h \lesssim \Theta(\log d/\log\log d)$ when $k=\Theta(\log d)$). The results quantify how hub distance and network heterogeneity influence rumor dissemination, offering rigorous phase-transition characterizations and guiding insights for more realistic network topologies. The methods rely on carefully constructed branching processes and generating-function analyses, enabling precise localization of critical parameters and potential extensions to other tree-like networks. Overall, the work provides a rigorous, parameter-specific understanding of when rumors persist versus die out on hierarchical networks.

Abstract

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 $d\geq 2$, 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 $k=o(d)$. 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 $α(d+1)$ hubs, with $α\in (0,1]$, via paths of length $h$. We obtain a phase transition result in $α$ in terms of $d,k,$ and $h$, and we show that in the case of $k=Θ(\log d)$ phase transition occurs iff $h \lesssim Θ( \log d / (\log \log d))$.

Figures (2)

• Figure 1: Possible realization of the MT-model on a tree $\mathbb{T}$. The vertices of the tree represent individuals, each belonging to one of three categories: ignorants (black vertices), spreaders (red vertices), or stiflers (blue vertices). (a) A spreader passes the rumor to any of its nearest ignorant neighbors at rate one. (b) After receiving the rumor, the contacted ignorant becomes a spreader and begins spreading the information. (c)-(d) At the same rate, a spreader turns into a stifler after coming into contact with neighboring spreaders or stiflers.
• Figure 2: Illustration of the first steps in the generation of $\mathbb{T}_{d,k,\alpha,h}$. For the sake of simplicity we consider $d=5, k=4$, and $h=4$. (a) The tree starts with a hub of degree $d+1$, which is chosen as the root and denoted by $\mathbf{0}$. (b) Each neighbor of the root is, with probability $\alpha$, a regular vertex connected by a path to another hub, or, with probability $1-\alpha$, a leaf. In this example, only the vertices $u, v,$ and $w$ are assumed to connect to another hubs. (c) We then reveal the connections of those vertices that link to other hubs. Here we show the path associated with $w$, which connects it to another hub. This new hub, in turn, may also have neighbors connected to other hubs, and the process continues generating the inhomogeneous tree $\mathbb{T}_{d,k,\alpha,h}$.

Theorems & Definitions (25)

• Lemma 2.1
• proof
• Definition 2.1
• Theorem 2.1
• Corollary 2.1
• Theorem 2.2
• Definition 2.2
• Theorem 2.3
• Corollary 2.2
• Corollary 2.3