In this work, we overview some results concerning communication combinatorial properties in random intersection graphs and uniform random intersection graphs. These properties relate crucially to algorithmic design for important problems (like secure communication and frequency assignment) in distributed networks characterized by dense, local interactions and resource limitations, such as sensor networks. In particular, we present and discuss results concerning the existence of large independent sets of vertices whp in random instances of each of these models. As the main contribution of our paper, we introduce a new, general model, which we denote G(V, χ, f). In this model, V is a set of vertices and χ is a set of m vectors in ℝm. Furthermore, f is a probability distribution over the powerset 2χ of subsets of χ. Every vertex selects a random subset of vectors according to the probability f and two vertices are connected according to a general intersection rule depending on their assigned set of vectors. Apparently, this new general model seems to be able to simulate other known random graph models, by carefully describing its intersection rule.