Generic properties of neutral networks of RNA secondary structures can be described by means of random graph theory. The success of this approach is dependent on details of the underlying secondary structure. Some of these dependencies are analyzed in this paper. In addition we present an algorithm, which, given a network does conform to the random graph model, allows to determine whether it is a connected graph. The algorithm is linear in time in the sequence length, this being possible because of local connectivity, a special property of graphs under the random graph model.