The longest edge of the one-dimensional soft random geometric graph with boundaries

Abstract

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 $\alpha$, 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 $\alpha$ and we recover a phase transition which coincides with exactly the same phases in [2].