In this paper, we investigate the secure connectivity of wireless sensor networks utilizing the heterogeneous random key predistribution scheme, where each sensor node is classified as class- @math with probability @math for @math with @math and @math . A class- @math sensor is given @math cryptographic keys selected uniformly at random from a key pool of size @math . After deployment, two nodes can communicate securely if they share at least one cryptographic key. We consider the wireless connectivity of the network using a heterogeneous on-off channel model, where the channel between a class- @math node and a class- @math node is on (respectively, off) with probability @math (respectively, @math ) for @math . Collectively, two sensor nodes are adjacent if they i) share a cryptographic key and ii) have a wireless channel in between that is on. We model the overall network using a composite random graph obtained by the intersection of inhomogeneous random key graphs (IRKG) @math with inhomogeneous Erd o s-R 'enyi graphs (IERG) @math . The former graph is naturally induced by the heterogeneous random key predistribution scheme, while the latter is induced by the heterogeneous on-off channel model. More specifically, two nodes are adjacent in the composite graph if they are i) adjacent in the IRKG i.e., share a cryptographic key and ii) adjacent in the IERG, i.e., have an available wireless channel. We investigate the connectivity of the composite random graph and present conditions (in the form of zero-one laws) on how to scale its parameters so that it i) has no secure node which is isolated and ii) is securely connected, both with high probability when the number of nodes gets large. We also present numerical results to support these zero-one laws in the finite-node regime.
We introduce a new random key predistribution scheme for securing heterogeneous wireless sensor networks. Each of the @math sensors in the network is classified into @math classes according to some probability distribution @math . Before deployment, a class- @math sensor is assigned @math cryptographic keys that are selected uniformly at random from a common pool of @math keys. Once deployed, a pair of sensors can communicate securely if and only if they have a key in common. We model the communication topology of this network by a newly defined inhomogeneous random key graph. We establish scaling conditions on the parameters @math and @math so that this graph: 1) has no isolated nodes and 2) is connected, both with high probability. The results are given in the form of zero-one laws with the number of sensors @math growing unboundedly large; critical scalings are identified and shown to coincide for both graph properties. Our results are shown to complement and improve those given by and for the same model, therein referred to as the general random intersection graph.