Asymptotic properties of random unlabelled block-weighted graphs

TL;DR

This paper investigates the asymptotic geometric structure of unlabelled graphs with weighted components, demonstrating convergence to the Brownian tree and describing phase transitions in connectivity.

Contribution

It establishes the scaling limits of unlabelled graphs with Boltzmann weights, including the emergence of a giant component and convergence to infinite random graphs.

Findings

Graphs converge to the Brownian tree under scaling.

A giant component appears in disconnected graph families.

Small fragments converge to finite random graphs.

Abstract

We study the asymptotic shape of random unlabelled graphs subject to certain subcriticality conditions. The graphs are sampled with probability proportional to a product of Boltzmann weights assigned to their $2$-connected components. As their number of vertices tends to infinity, we show that they admit the Brownian tree as Gromov--Hausdorff--Prokhorov scaling limit, and converge in a strengthened Benjamini--Schramm sense toward an infinite random graph. We also consider a family of random graphs that are allowed to be disconnected. Here a giant connected component emerges and the small fragments converge without any rescaling towards a finite random limit graph.