  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.
