Many known networks have structure of affiliation networks, where each of $n$ network nodes (actors) selects an attribute set from a given collection of $m$ attributes, and two nodes (actors) establish adjacency relation whenever they share a common attribute. We study the behavior of a random walk on such networks. For this purpose, we use a common model of such networks, a random intersection graph. We establish the cover time of the simple random walk on the binomial random intersection graph ${\mathcal{ G}}(n,m,p)$ at the connectivity threshold and above it. We consider the range of $(n,m,p)$ , where the typical attribute is shared by a (stochastically) bounded number of actors. [ABSTRACT FROM AUTHOR]