  A random intersection graph is constructed by assigning independently to each
vertex a subset of a given set and drawing an edge between two vertices if and
only if their respective subsets intersect. In this paper a model is developed
in which each vertex is given a random weight, and vertices with larger weights
are more likely to be assigned large subsets. The distribution of the degree of
a given vertex is characterized and is shown to depend on the weight of the
vertex. In particular, if the weight distribution is a power law, the degree
distribution will be so as well. Furthermore, an asymptotic expression for the
clustering in the graph is derived. By tuning the parameters of the model, it
is possible to generate a graph with arbitrary clustering, expected degree and
-- in the power law case -- tail exponent.
