  We consider uniform random cographs (either labeled or unlabeled) of large
size. Our first main result is the convergence towards a Brownian limiting
object in the space of graphons. We then show that the degree of a uniform
random vertex in a uniform cograph is of order $n$, and converges after
normalization to the Lebesgue measure on $[0,1]$. We finally analyze the vertex
connectivity (i.e. the minimal number of vertices whose removal disconnects the
graph) of random connected cographs, and show that this statistics converges in
distribution without renormalization. Unlike for the graphon limit and for the
degree of a random vertex, the limiting distribution is different in the
labeled and unlabeled settings.
  Our proofs rely on the classical encoding of cographs via cotrees. We then
use mainly combinatorial arguments, including the symbolic method and
singularity analysis.
