Correlation between clustering and degree in affiliation networks

We are interested in the probability that two randomly selected neighbors of a random vertex of degree (at least) k are adjacent. We evaluate this probability for a power law random intersection graph, where each vertex is prescribed a collection of attributes and two vertices are adjacent whenever they share a common attribute. We show that the probability obeys the scaling k−δ as k → +∞. Our results are mathematically rigorous. The parameter 0 ≤ δ ≤ 1 is determined by the tail indices of power law random weights defining the links between vertices and attributes.