论文标题
在一个维
Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension
论文作者
论文摘要
我们考虑在实际线路上考虑不均匀的空间随机图。每个顶点都带有I.I.D.绘制重量和边缘，以使其短边和边缘到具有较大重量的顶点，概率更高。这允许研究具有远距离效应和重尾分布的模型。我们引入了一个新系数$δ_\ text {eff} $，该{eff} $量化了重尾学位对远程连接的影响。我们表明$Δ_\ text {eff} <2 $足以在模型中存在超临界渗透阶段，并且$Δ__\ text {eff}> 2 $总是暗示没有渗透率。特别是，我们的结果补充了Gracar等人。 （Adv。Appl。prob。，2021），在软布尔模型中给出了足够的条件，以及对于存在和不存在亚临界渗透阶段的年龄依赖性随机连接模型。我们的结果进一步提供了巨型组件在大图图中存在或不存在的标准。
We consider inhomogeneous spatial random graphs on the real line. Each vertex carries an i.i.d. weight and edges are drawn such that short edges and edges to vertices with large weights occur with higher probability. This allows the study of models with long-range effects and heavy-tailed degree distributions. We introduce a new coefficient $δ_\text{eff}$ which quantifies the influence of heavy-tailed degrees on long-range connections. We show that $δ_\text{eff}<2$ is sufficient for the existence of a supercritical percolation phase in the model and that $δ_\text{eff}>2$ always implies the absence of percolation. In particular, our results complement those in Gracar et al. (Adv. Appl. Prob., 2021), where sufficient conditions were given for the soft Boolean model and the age-dependent random connection model for both the existence and the absence of a subcritical percolation phase. Our results further provide a criterion for the existence or non-existence of a giant component in large finite graphs.
