Stable graphs: distributions and line-breaking construction

TL;DR

This paper studies the geometric structure of $oldsymbol{ ext{α}}$-stable graphs, revealing their kernel distributions, their construction from stable trees, and introducing a line-breaking method akin to Aldous' for the Brownian CRT.

Contribution

It provides a detailed analysis of the distributional properties and construction methods of $oldsymbol{ ext{α}}$-stable graphs, including a novel line-breaking construction.

Findings

Distribution of the kernel and finite-dimensional marginals

Representation of the graph as stable trees glued onto the kernel

A new line-breaking construction for $ ext{α}$-stable graphs

Abstract

For $\alpha \in (1,2]$, the $\alpha$-stable graph arises as the universal scaling limit of critical random graphs with i.i.d. degrees having a given $\alpha$-dependent power-law tail behavior. It consists of a sequence of compact measured metric spaces (the limiting connected components), each of which is tree-like, in the sense that it consists of an $\mathbb R$-tree with finitely many vertex-identifications (which create cycles). Indeed, given their masses and numbers of vertex-identifications, these components are independent and may be constructed from a spanning $\mathbb R$-tree, which is a biased version of the $\alpha$-stable tree, with a certain number of leaves glued along their paths to the root. In this paper we investigate the geometric properties of such a component with given mass and number of vertex-identifications. We (1) obtain the distribution of its kernel and more generally of its discrete finite-dimensional marginals; we will observe that these distributions are related to the distributions of some configuration models (2) determine the distribution of the $\alpha$-stable graph as a collection of $\alpha$-stable trees glued onto its kernel and (3) present a line-breaking construction, in the same spirit as Aldous' line-breaking construction of the Brownian continuum random tree.