  We give new examples and describe the complete lists of all measures on the
set of countable homogeneous universal graphs and $K_s$-free homogeneous
universal graphs (for $s\geq 3$) that are invariant with respect to the group
of all permutations of the vertices. Such measures can be regarded as random
graphs (respectively, random $K_s$-free graphs). The well-known example of
Erd\"os--R\'enyi (ER) of the random graph corresponds to the Bernoulli measure
on the set of adjacency matrices. For the case of the universal $K_s$-free
graphs there were no previously known examples of the invariant measures on the
space of such graphs.
  The main idea of our construction is based on the new notions of {\it
measurable universal}, and {\it topologically universal} graphs, which are
interesting themselves. The realization of the construction can be regarded as
two-step randomization for universal measurable graph : {\it "randomization in
vertices"} and {\it "randomization in edges"}. For $K_s$-free, $s\geq 3$ there
is only randomization in vertices of the measurable graphs. The completeness of
our lists is proved using the important theorem by D. Aldous about
$S_{\infty}$-invariant matrices, which we reformulate in appropriate way.
