Condensation in scale-free geometric graphs with excess edges

TL;DR

This paper analyzes the probability of large deviations in the number of edges in scale-free geometric graphs, revealing a condensation mechanism where a few vertices dominate connections, supported by limit theorems and empirical distribution analysis.

Contribution

It identifies the large deviation mechanism in scale-free geometric graphs as a condensation effect, extending to various models and providing theoretical verification through limit theorems.

Findings

Large deviation probability is driven by a condensation effect.

A finite number of vertices acquire macroscopic degrees.

Edge-length distribution splits into bulk and traveling wave parts.

Abstract

We identify the upper large deviation probability for the number of edges in scale-free geometric random graph models as the space volume goes to infinity. Our result covers the models of scale-free percolation, the Boolean model with heavy-tailed radius distribution, and the age-dependent random connection model. In all these cases the mechanism behind the large deviation is based on a condensation effect. Loosely speaking, the mechanism randomly selects a finite number of vertices and increases their power, so that they connect to a macroscopic number of vertices in the graph, while the other vertices retain a degree close to their expectation and thus make no more than the expected contribution to the large deviation event. We verify this intuition by means of limit theorems for the empirical distributions of degrees and edge-lengths under the conditioning. We observe that at large finite volumes, the edge-length distribution splits into a bulk and travelling wave part of asymptotically positive proportions.