Random generations of the countable random graph

We consider random processes more general than those considered by Erdös and Rényi for generating the countable random graph. It is proved that, in the category sense, almost all random processes we consider generate the countable random graph with probability 1. Under a weak boundedness assumption we give a criterion for the random processes which generate the countable random graph almost surely. We also consider further questions asked by Jackson regarding the outcome graphs when the process fails to produce the countable random graph.