Upper large deviations for power-weighted edge lengths in spatial random networks

TL;DR

This paper investigates the large deviation behavior of power-weighted edge lengths in spatial random networks, revealing a condensation phenomenon when the power exceeds the dimension, with implications for various network types.

Contribution

It characterizes upper large deviations for power-weighted edge sums in spatial networks, introducing conditions for condensation phenomena and explicit rate functions.

Findings

Condensation phenomenon occurs in large deviations when power exceeds dimension.

Rate functions are explicitly characterized via optimization problems.

Framework applies to various network models like k-nearest neighbor and beta-skeletons.

Abstract

We study the large-volume asymptotics of the sum of power-weighted edge lengths $\sum_{e \in E}|e|^\alpha$ in Poisson-based spatial random networks. In the regime $\alpha > d$, we provide a set of sufficient conditions under which the upper large deviations asymptotics are characterized by a condensation phenomenon, meaning that the excess is caused by a negligible portion of Poisson points. Moreover, the rate function can be expressed through a concrete optimization problem. This framework encompasses in particular directed, bidirected and undirected variants of the $k$-nearest neighbor graph, as well as suitable $\beta$-skeletons.