Absence of isolated nodes in inhomogeneous random key graphs

We introduce a new random key predistribution scheme for securing heterogeneous wireless sensor networks. Each of the n sensors in the network is classified into r classes according to some probability distribution μ = {μ₁, ..., μ_r}. Before deployment, a class i sensor is assigned K_i cryptographic keys that are selected uniformly at random from a common pool of P keys, for each i = 1, ..., r. Once deployed, a pair of sensors can establish a secure communication channel if and only if they have a key in common. We model the topology of this network by an inhomogeneous random key graph. We establish scaling conditions on the parameters P and {K₁, ..., K_r} so that the this graph has no isolated nodes with high probability. The result is given in the form of a zero-one law with the number of sensors n growing unboundedly large. An analogous result is also conjectured for the property of graph connectivity.