Connectivity in inhomogeneous random key graphs

Osman Yagan

Abstract: We consider 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 a probability distribution μ = {μ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,...μ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> } Before deployment, a class-i sensor is assigned K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> cryptographic keys that are selected uniformly at random from a pool of P keys. Once deployed, a pair of sensors can communicate securely if and only if they have a key in common. The communication topology of this network is modeled by an inhomogeneous random key graph. We establish scaling conditions on the parameters P and {K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,...,K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> } so that this graph is connected with high probability. The result is given in the form of a zero-one law with the number of sensors n growing unboundedly large. Our result is shown to complement and improve those given by Godehardt et al. and Zhao et al. for the same model, therein referred to as the general random intersection graph.