Peaks over Threshold in Scale-Free Random Graphs

We explore extreme value phenomena in spatial scale-free random graphs in a continuum setting based on a homogeneous Poisson point process in $\mathbb{R}^d$. Vertices carry i.i.d. weights $(W_x)$ and, conditionally on the vertex set and the weights, edges are present independently with probability $p_{xy}=1-\exp\{-\lambda W_xW_y/|x-y|^\alpha\}$. Assuming Pareto-type weight tails with index $\beta>0$ and working in parameter ranges where degrees are almost surely finite, we study extremes and peaks over thresholds (POT) of edge lengths in a growing observation window. Our focus is the precise impact of the presence of (large) weights on edge lengths, captured through explicit scaling regimes and conditional POT limit theorems. Our main results identify a three-phase behavior governed by the weight-tail parameter $\beta$. We both deduce Fr\'echet-type limits for the maximum edge length itself and we reveal POT structures under a hub conditioning by proving a POT limit theorem. In the finite-mean regime $\beta>1$, the leading scaling agrees with the unweighted model up to a constant. By contrast, for $\beta\le 1$ the weights have a macroscopic effect on extreme edge lengths: for $\beta< 1$ the scaling changes, and the borderline case $\beta=1$ exhibits additional logarithmic corrections. The proofs combine Stein-type Poisson approximation via a Palm-coupling approach with a refined treatment of the dependence created by the conditioning event.