For some classes of random graphs (e.g. defined by a given set of independent properties) one can prove that some (dependent) properties hold for almost all of them.
Almost all graphs are asymmetric.
Almost all graphs have diameter 2.
Almost all Erdős–Rényi graphs have a Poisson degree distribution.
I can imagine the case that
one has defined some class $\Gamma$ of random graphs,
one has generated a large number of such graphs by a random process, and
one experimentally found out that almost all of them have some graph property $P$. But
one has not been successful in proving that $P$ holds for almost all graphs in $\Gamma$.
Maybe this is only a fictious scenario and never occurred. Then it would be interesting to know why. Does a proof of holding almost certainly always lie open? In case it did occur, I ask for specific references: Which are examples of graph properties (in some given class $\Gamma$ of graphs) that are only assumed to hold for almost all graphs in $\Gamma$