Asymptotic properties of random unlabelled block-weighted graphs
Combinatorics
2017-12-06 v1 Probability
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 -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.
Keywords
Cite
@article{arxiv.1712.01301,
title = {Asymptotic properties of random unlabelled block-weighted graphs},
author = {Benedikt Stufler},
journal= {arXiv preprint arXiv:1712.01301},
year = {2017}
}