Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Peaks over Threshold in Scale-Free Random Graphs

Arnaud Rousselle, Ercan Sönmez

Abstract

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 $p_{xy}=1-\exp\{-λW_xW_y/|x-y|^α\}$. Assuming Pareto-type weight tails with index $β>0$ 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 $β$. We both deduce Fréchet-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 $β>1$, the leading scaling agrees with the unweighted model up to a constant. By contrast, for $β\le 1$ the weights have a macroscopic effect on extreme edge lengths: for $β< 1$ the scaling changes, and the borderline case $β=1$ 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.

Peaks over Threshold in Scale-Free Random Graphs

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 Fréchet-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 , the leading scaling agrees with the unweighted model up to a constant. By contrast, for the weights have a macroscopic effect on extreme edge lengths: for the scaling changes, and the borderline case 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.
Paper Structure (14 sections, 18 theorems, 274 equations)

This paper contains 14 sections, 18 theorems, 274 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Intermediate and further results
  5. Proof of Theorem \ref{['thm:uncond-max']}
  6. A construction of the scale-free random connection model
  7. The largest unconditional edge-length
  8. The expected number of exceedances
  9. Proof of Theorem \ref{['thm:POT-hub']}
  10. Proof of Lemma \ref{['lem:POT-rooted-unrooted']}
  11. Proof of Lemma \ref{['lem:rooted-exceed-asymptotics']}
  12. Proof of Lemma \ref{['le:jointedgesweights']}
  13. Conclusuion
  14. Acknowledgements.

Key Result

Theorem 1.1

Consider the continuum model in which $V$ is a homogeneous Poisson point process on $\mathbb{R}^d$ with unit intensity, and let the connection probabilities be given by eq:conn-prob-intro. For $n\in\mathbb{N}$ set $B_n=[-n,n]^d$ and define as well as the exceedance count $F(n,r)$ from eq:Fnr-intro. Assume that the weight distribution satisfies $\mathbb{P}(W>w)=w^{-\beta}$ for $w\ge 1$, with tail

Theorems & Definitions (32)

  • Theorem 1.1: Unconditional exceedances and maximum edge length
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Peaks over threshold under a hub conditioning
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • Theorem 3.1
  • proof
  • ...and 22 more