Normal approximation for subgraph counts in random intersection graphs: the case of cliques

Source abstract

Let $X^{(r)}_n$ denote the number of copies of a complete graph $K_r$ in the random intersection graph model on $n$ vertices. We derive upper bounds on the distance between the standardized version of $X^{(r)}_n$ and the standard normal distribution, measured in both the Kolmogorov and Wasserstein metrics. We complement these results with necessary and sufficient conditions for asymptotic normality. As a byproduct, we provide precise estimates for the variance of $X^{(r)}_n$. In the process, we develop new general techniques for analysing so--called clique covers, which play a crucial role in the study of subgraph probabilities in random intersection graphs.