Correlation between clustering and degree in affiliation networks

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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.