Subgraph densities and scaling limits of random graphs with a prescribed modular decomposition

We consider large uniform labeled random graphs in different classes with prescribed decorations in their modular decomposition. Our main result is the estimation of the number of copies of every graph as an induced subgraph. As a consequence, we obtain the convergence of a uniform random graph in such classes to a Brownian limit object in the space of graphons. Our proofs rely on combinatorial arguments, computing generating series using the symbolic method and deriving asymptotics using singularity analysis.