Connectivity of Wireless Sensor Networks Secured by Heterogeneous Key Predistribution Under an On/Off Channel Model

We investigate the connectivity of a wireless sensor network secured by the heterogeneous key predistribution scheme under an independent on/off channel model. The heterogeneous scheme induces an inhomogeneous random key graph, denoted by $\mathbb{K}(n;\pmb{\mu},\pmb{K},P)$ and the on/off channel model induces an Erd\H{o}s-R\'enyi graph, denoted by $\mathbb{H}(n,\alpha)$. Hence, the overall random graph modeling the WSN is obtained by the intersection of $\mathbb{K}(n;\pmb{\mu},\pmb{K},P)$ and $\mathbb{H}(n,\alpha)$. We present conditions on how to scale the parameters of the intersecting graph with respect to the network size $n$ such that the graph i) has no isolated nodes and ii) is connected, both with high probability as the number of nodes gets large. Our results are supported by a simulation study demonstrating that i) despite their asymptotic nature, our results can in fact be useful in designing finite-node wireless sensor networks so that they achieve secure connectivity with high probability; and ii) despite the simplicity of the on/off communication model, the probability of connectivity in the resulting wireless sensor network approximates very well the case where the disk model is used.