无限图上空间Maki–Thompson模型的灭绝、存活与波动

中文摘要 AI 辅助

我们研究有界度无限连通图上的空间Maki–Thompson谣言模型：传播者将谣言传递给无知邻居，但在接触非无知邻居后变为抑制者。我们证明，对于线性增长的Cayley图，当λ、α>0且初始构型中仅有有限个非无知顶点时，谣言必然灭绝；并针对从有限个传播者启动的过程，建立了任意有界度图上的显式灭绝准则。对于超线性增长的Cayley图，当抑制率与传播率的比值低于仅依赖最大度的显式阈值时，我们证明单个传播者即可使谣言存活。对于D≥2次多项式增长的Cayley图，我们进一步证明，谣言传播范围以正概率具有正的下密度；在更强条件下，宏观环带中会存在表面量级的同时活跃传播者，且总持续时间均匀远离零。当α>0时，每个有限区域最终将不再有传播者，因此全局存活意味着谣言必须持续向新区域扩散。在亚临界状态或足够小的抑制率条件下，我们证明最终抑制者密度、总传播者占用时间满足中心极限定理，经验存活函数满足泛函中心极限定理。这些结果源于多项式增长Cayley图上该模型平稳稳定泛函的中心极限定理，且泛函可依赖于观测集之外的场。

英文摘要

We study the spatial Maki--Thompson rumor model on infinite, connected graphs of bounded degree. Spreaders transmit the rumor to ignorant neighbors but become stiflers upon contacting non-ignorant neighbors. We prove extinction on Cayley graphs of linear growth, for every \(λ,α>0\) and every initial configuration with finitely many non-ignorant vertices, and establish an explicit extinction criterion on arbitrary bounded-degree graphs for processes started from finitely many spreaders. On Cayley graphs of superlinear growth, we prove survival from a single spreader whenever the ratio of the stifling to the transmission rate lies below an explicit threshold depending only on the maximum degree. On Cayley graphs of polynomial growth of degree \(D\ge2\), we further show that the range has positive lower density with positive probability. Under a stronger condition, macroscopic annuli contain a surface-order number of simultaneously active spreaders for a total duration bounded uniformly away from zero. When \(α>0\), every finite region eventually contains no spreaders, so global survival forces the rumor to move continually into new regions. Under either subcriticality or a sufficiently small stifling rate, we prove central limit theorems for the final stifler density and the total spreader occupation time, together with a functional central limit theorem for the empirical survival function. These results follow from a central limit theorem for stationary stabilizing functionals of i.i.d.fields on polynomial-growth Cayley graphs; the functionals may depend on the field outside the observation set.

发表机构

• Universidade Federal do ABC (UFABC)（巴西联邦ABC大学）
• Universidade Estadual de Campinas (UNICAMP)（坎皮纳斯州立大学）

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