k-Connectivity of Random Key Graphs

Random key graphs represent topologies of secure wireless sensor networks that apply the seminal Eschenauer– Gligor random key predistribution scheme to secure communication between sensors. These graphs have received much attention and also been used in diverse application areas beyond secure sensor networks; e.g., cryptanalysis, social networks, and recommender systems. Formally, a random key graph with n nodes is constructed by assigning each node Xn keys selected uniformly at random from a pool of Yn keys and then putting an undirected edge between any two nodes sharing at least one key. Considerable progress has been made in the literature to analyze connectivity and k-connectivity of random key graphs; e.g., Yağan and Makowski [ISIT ’09, Trans. IT’12] on connectivity under Xn = Ω( √ lnn ), Rybarczyk [Discrete Mathematics ’11] on connectivity under Xn ≥ 2, and our recent work [CDC ’14] on k-connectivity under Xn = Ω( √ lnn ), where k-connectivity of a graph ensures connectivity even after the removal of k nodes or k edges. Yet, it still remains an open question for k-connectivity in random key graphs under Xn ≥ 2 and Xn = o( √ lnn ) (the case of Xn = 1 is trivial). In this paper, we answer the above problem by providing an exact analysis of k-connectivity in random key graphs under Xn ≥ 2.