Correlation between clustering and degree in affiliation networks

Abstract

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 $k^{-\delta }$ as $k\to +\infty$. Our results are mathematically rigorous. The parameter $0\le \delta \le 1$ is determined by the tail indices of power law random weights defining the links between vertices and attributes.