  We introduce two models of inclusion hierarchies: Random Graph Hierarchy
(RGH) and Limited Random Graph Hierarchy (LRGH). In both models a set of nodes
at a given hierarchy level is connected randomly, as in the Erd\H{o}s-R\'{e}nyi
random graph, with a fixed average degree equal to a system parameter $c$.
Clusters of the resulting network are treated as nodes at the next hierarchy
level and they are connected again at this level and so on, until the process
cannot continue. In the RGH model we use all clusters, including those of size
$1$, when building the next hierarchy level, while in the LRGH model clusters
of size $1$ stop participating in further steps. We find that in both models
the number of nodes at a given hierarchy level $h$ decreases approximately
exponentially with $h$. The height of the hierarchy $H$, i.e. the number of all
hierarchy levels, increases logarithmically with the system size $N$, i.e. with
the number of nodes at the first level. The height $H$ decreases monotonically
with the connectivity parameter $c$ in the RGH model and it reaches a maximum
for a certain $c_{max}$ in the LRGH model. The distribution of separate cluster
sizes in the LRGH model is a power law with an exponent about $-1.25$. The
above results follow from approximate analytical calculations and have been
confirmed by numerical simulations.
