A zero-one law for k-connectivity in Random K-out graphs intersecting Erdős-Rényi Graphs

We investigate k-connectivity in secure wireless sensor networks under the random pairwise key predistribution scheme with unreliable links; a network is said to be kconnected if it remains connected despite the failure of any of its (k − 1) nodes or links. With wireless communication links modeled as independent on-off channels, this amounts to analyzing a random graph model formed by intersecting a random K-out graph and an Erdős-Rényi graph. We present conditions on how to scale the parameters of this intersection model so that the resulting graph is k-connected with probability approaching to one (resp. zero) as the number of nodes gets large. The resulting zeroone law is shown to improve and sharpen the previous result on the 1-connectivity of the same model. We also provide numerical results to support our analysis and show that even in the finite node regime, our results can provide useful guidelines for designing sensor networks that are secure and reliable.