Peaks over Threshold in Scale-Free Random Graphs
Arnaud Rousselle, Ercan S\"onmez

TL;DR
This paper investigates the extreme edge lengths in scale-free random graphs with weighted vertices, revealing how the tail behavior of weights influences the maximum edge lengths and their peaks over thresholds.
Contribution
It provides a detailed analysis of the impact of weight tail behavior on extreme edge lengths, including explicit scaling regimes and limit theorems for peaks over thresholds.
Findings
Identifies a three-phase behavior based on the weight-tail parameter β.
Derives Fréchet-type limits for maximum edge lengths.
Establishes POT limit theorems conditioned on hubs.
Abstract
We explore extreme value phenomena in spatial scale-free random graphs in a continuum setting based on a homogeneous Poisson point process in . Vertices carry i.i.d. weights and, conditionally on the vertex set and the weights, edges are present independently with probability . Assuming Pareto-type weight tails with index 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 . We both deduce Fr\'echet-type limits for the maximum edge length…
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Taxonomy
TopicsRandom Matrices and Applications · Stochastic processes and statistical mechanics · Point processes and geometric inequalities
