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

TL;DR

This work analyzes the upper tail behavior of the power-weighted edge-length sum in Poisson-based spatial networks for $\alpha>d$, revealing a condensation mechanism where a small condensate drives the rare event. The authors formulate a variational rate via an influence-zone optimization $A(\varphi,\psi)$ and show $\mathbb{P}(H_n>\mu_\alpha+r)$ decays like $\exp(-|A(\varphi,\psi)| r^{d/\alpha})$, with finite-atom condensation under additional conditions. They verify the framework for key graph classes, notably the directed/undirected/bidirected $k$-nearest neighbor graphs and two-dimensional $\beta$-skeletons, including explicit results for the nearest-neighbor case where a single unit ball is optimal. Moreover, the paper provides conditioned convergence results, proving that on the rare event the excess weight is carried by at most $m_0$ Poisson points, with $m_0=1$ in the nearest-neighbor setting. These insights connect geometric graph deviations to concrete optimization problems, offering a principled view of rare-event structure in spatial networks with potential practical implications for security-sensitive network design and analysis.

Abstract

We study the large-volume asymptotics of the sum of power-weighted edge lengths $\sum_{e \in E}|e|^α$ in Poisson-based spatial random networks. In the regime $α> d$, 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 $β$-skeletons.

Figures (5)

• Figure 1: Two configurations that result in a typical sum (left) and an exceptionally large sum (right) of $\alpha$-power weighted edge lengths with $\alpha =15$. In each configuration, the three vertices inside an observation window with the most distant nearest neighbor are highlighted.
• Figure 2: Illustration of an edge in the $\beta$-skeleton and a random simulation of the $\beta$-skeleton with $\beta=1.2$.
• Figure 3: Illustration of the statement of Lemma \ref{['lemma_b_skel_FIN']} including the inserted node relevant for \ref{['FIN']}. The extended line between $f_1$ and $M(f)$ is tangent to the disk segment.
• Figure 4: Labeling of the boxes in 3D where 27 labels are sufficient and 2D where 9 are sufficient
• Figure 5: Sketch of $U_n$, $W_n$, $W_n^-$ and $W_n^{2-}$.

Theorems & Definitions (36)

• Theorem 1: Upper large deviations
• Theorem 2: Condensation conditioned on rare event
• Remark 1
• Remark 2
• Remark 3
• Lemma 1: Stabilization for $\beta$-skeletons
• proof
• Lemma 2: Disjoint regions for $\beta$-skeletons
• Lemma 3: Exclusion of short edges
• Lemma 4: Size bound for recorded set