k-Connectivity in Random Key Graphs with Unreliable Links

Random key graphs form a class of random intersection graphs and are naturally induced by the random key predistribution scheme of Eschenauer and Gligor for securing wireless sensor network (WSN) communications. Random key graphs have received much interest recently, owing in part to their wide applicability in various domains including recommender systems, social networks, secure sensor networks, clustering and classification analysis, and cryptanalysis to name a few. In this paper, we study connectivity properties of random key graphs in the presence of unreliable links. Unreliability of the edges are captured by independent Bernoulli random variables, rendering edges of the graph to be on or off independently from each other. The resulting model is an intersection of a random key graph and an Erdos-Renyi graph, and is expected to be useful in capturing various real-world networks; e.g., with secure WSN applications in mind, link unreliability can be attributed to harsh environmental conditions severely impairing transmissions. We present conditions on how to scale this model's parameters so that i) the minimum node degree in the graph is at least k, and ii) the graph is k-connected, both with high probability as the number of nodes becomes large. The results are given in the form of zeroone laws with critical thresholds identified and shown to coincide for both graph properties. These findings improve the previous results by Rybarczyk on the k-connectivity of random key graphs (with reliable links), as well as the zero-one laws by Yagan on the 1-connectivity of random key graphs with unreliable links.Comment: Published in IEEE Transactions on Information Theor

Performance of the Eschenauer-Gligor key distribution scheme under an ON/OFF channel

We investigate the secure connectivity of wireless sensor networks under the random key distribution scheme of Eschenauer and Gligor. Unlike recent work which was carried out under the assumption of full visibility, here we assume a (simplified) communication model where unreliable wireless links are represented as on/off channels. We present conditions on how to scale the model parameters so that the network i) has no secure node which is isolated and ii) is securely connected, both with high probability when the number of sensor nodes becomes large. The results are given in the form of full zero-one laws, and constitute the first complete analysis of the EG scheme under non-full visibility. Through simulations these zero-one laws are shown to be valid also under a more realistic communication model, i.e., the disk model. The relations to the Gupta and Kumar's conjecture on the connectivity of geometric random graphs with randomly deleted edges are also discussed.Comment: Submitted to IEEE Transactions on Information Theory in November, 201