Understanding the size of phylogenetic network classes and the typical shape of a random network from a fixed class has been one of the major research focuses in phylogenetics over the last couple of years. In this extended abstract, we consider two subclasses of the (recently introduced) class of semi-simplex phylogenetic networks, namely, semi-simplex tree-child networks and semi-simplex galled trees. We clarify their sizes relative to the (known) sizes of general tree-child networks and galled trees, respectively, and prove limit laws for parameters of random networks from these classes. Additional classes of semi-simplex networks will be considered in the journal version of this paper.