论文标题
具有边界的一维软几何图的最长边缘
The longest edge of the one-dimensional soft random geometric graph with boundaries
论文作者
论文摘要
研究的对象是一个软随机几何图，其顶点由泊松点过程在线上和边缘之间的边缘给出，其概率具有多项式衰变，它们之间的距离之间具有多项式衰变。已经对与连接结构相关的此类模型的各个方面进行了广泛的研究。在本文中，我们从极值理论的角度研究了随机图，并着重于单个长边的发生。我们研究的模型具有非周期性边界，并通过正常数$α$进行参数化，这是确定边缘存在的概率多项式衰变的功率。作为主要结果，我们在分布中的渐近行为方面提供了最长边缘的幅度的精确描述。因此，我们说明了对功率$α$的关键依赖性，并且我们恢复了[2]中与完全相同的阶段相吻合的相变。
The object of study is a soft random geometric graph with vertices given by a Poisson point process on a line and edges between vertices present with probability that has a polynomial decay in the distance between them. Various aspects of such models related to connectivity structures have been studied extensively. In this paper we study the random graph from the perspective of extreme value theory and focus on the occurrence of single long edges. The model we investigate has non-periodic boundary and is parameterized by a positive constant $α$, which is the power for the polynomial decay of the probabilities determining the presence of an edge. As a main result we provide a precise description of the magnitude of the longest edge in terms of asymptotic behavior in distribution. Thereby we illustrate a crucial dependence on the power $α$ and we recover a phase transition which coincides with exactly the same phases in [2].
