The Maki-Thompson Model with Spontaneous Stifling on Symmetric Networks

TL;DR

This work extends the classical Maki–Thompson rumor model by incorporating spontaneous stifling on a broad class of symmetric networks called quasi-transitive graphs. It develops a unified framework where vertex states are tracked by type, leading to nonlinear integral equations that describe the mean-field limit and a functional central limit theorem that yields Gaussian fluctuations with explicit covariance. The authors prove both a Functional Law of Large Numbers and a Functional Central Limit Theorem for densities by vertex type, and they extend the analysis to infinite graphs with subexponential growth using boundary-control and first-passage percolation arguments. Simulations on multi-type graphs and on the square lattice illustrate topology- and time-law-driven effects on outbreak speed and variability, and show that the classical MT model is recovered when spontaneous stifling is absent. The results provide a rigorous link between network topology, stifling-time distributions, and the macroscopic behavior of rumor outbreaks, with potential implications for information diffusion and intervention strategies.

Abstract

We investigate rumor spreading in a generalized Maki-Thompson model with spontaneous stifling, evolving on quasi-transitive networks. Individuals are either ignorants, spreaders, or stiflers; spreaders stop by contact with other spreaders or stiflers or after an independent random waiting time sampled from a given distribution, modeling a spontaneous loss of interest. The topology of the underlying population network is incorporated by modeling it as a broad class of symmetric networks, whose vertices are partitioned into finitely many orbit types. This yields a unified framework for homogeneous and heterogeneous networks. For sequences of finite quasi-transitive graphs, and for infinite quasi-transitive graphs with subexponential growth, we establish a Functional Law of Large Numbers and a Functional Central Limit Theorem for the densities of each vertex type for the three states. The mean-field limit is described by a system of nonlinear integral equations, while fluctuations are asymptotically Gaussian and governed by a system of stochastic integral equations with explicit covariance. Our results show how the topology and the law of spontaneous stifling jointly shape the speed and variability of rumor outbreaks. As a special case, our model reduces to the classical Maki-Thompson model when spontaneous stifling is absent.

Figures (10)

• Figure 1: Graphs $\Gamma_7^{2}$, $\Gamma_{13}^{2}$, $\Gamma_8^{2,4}$ and $\Gamma_{13}^{2,4}$.
• Figure 2: Decorated grid: four-type QTS with degrees: $B=4$, $A=5$, $D=5$, $C=6$. Note that $A$ and $D$ share the same degree but have different types, as their neighbor sets have different type compositions
• Figure 3: Planar graphs with quadratic growth: the square lattice, the dual of the trihexagonal tiling, and a 4-uniform tiling.
• Figure 4: Diagram representation of the model. Here, $\mathcal{X}^n_k=\mathcal{X}^n\cap V^n_k,\mathcal{Y}^n_k=\mathcal{Y}^n\cap V^n_k$ and $\mathcal{Z}^n_k=\mathcal{Z}^n\cap V^n_k$. The arrows above indicate transitions between two states with their corresponding rates above and the arrow below indicate the spontaneous transition with a time that is an independent copy of the $F$-distributed random variable $\eta$. The symbol $\leftrightharpoons$ represents contact between the two sets.
• Figure 5: Schematic representation of the region $B_{n}$ and its inner region $B_{n-g(n)}$.

Theorems & Definitions (30)

• Definition 2.1
• Theorem 3.3: Functional Law of Large Numbers
• Theorem 3.5: Functional Central Limit Theorem
• Lemma 3.6
• Proposition 4.1: Invariance of measure under automorphisms
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4
• proof
• Lemma 4.5