Asymptotic Properties of Random Unlabelled Block-Weighted Graphs

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