On Connectivity and Robustness in Random Intersection Graphs

Abstract

Random intersection graphs have received much attention recently and been used in a wide range of applications ranging from key predistribution in wireless sensor networks to modeling social networks. For these graphs, each node is equipped with a set of objects in a random manner, and two nodes have an undirected edge in between if they have at least one object in common. In this paper, we investigate connectivity and robustness in a general random intersection graph model. Specifically, we establish sharp asymptotic zero-one laws for k-connectivity and k-robustness, as well as the asymptotically exact probability of k-connectivity, for any positive integer k. The k-connectivity property quantifies how resilient is the connectivity of a graph against node or edge failures, while k-robustness measures the effectiveness of local-information-based consensus algorithms (that do not use global graph topology information) in the presence of adversarial nodes. In addition to presenting the results under the general random intersection graph model, we consider two special cases of the general model, a binomial random intersection graph and a uniform random intersection graph, which both have numerous applications as well. For these two specialized graphs, our results on asymptotically exact probabilities of k-connectivity and asymptotic zero-one laws for k-robustness are also novel in the literature.