We consider wireless sensor networks secured by the heterogeneous random key predistribution scheme under an on off channel model. The heterogeneous random key predistribution scheme considers the case when the network includes sensor nodes with varying levels of resources, features, or connectivity requirements; e.g., regular nodes vs. cluster heads, but does not incorporate the fact that wireless channel are unreliable. To capture the unreliability of the wireless medium, we use an on off channel model; wherein, each wireless channel is either on (with probability @math ) or off (with probability @math ) independently. We present conditions (in the form of zero-one laws) on how to scale the parameters of the network model so that with high probability the network is @math -connected, i.e., the network remains connected even if any @math nodes fail or leave the network. We also present numerical results to support these conditions in the finite-node regime.
Our paper completes the analysis that we started in @cite_24 concerning the minimum node degree of the intersection model @math . There, we presented conditions on how to scale the parameters of the intersection model @math so that the minimum node degree is no less than @math , with high probability when the number of nodes @math gets large. It is clear that a graph can not be @math -connected if its minimum node degree is less than @math . Thus, we readily obtain the zero-law of Theorem by virtue of the zero-law of the minimum node degree being no less than @math [Theorem 3.1] EletrebyISIT . Our paper completes the analysis of @cite_24 by means of establishing the one-law of @math -connectivity; thus, obtaining a fuller understanding of the properties of the intersection model @math . In particular, it was conjectured in [Conjecture 3.2] EletrebyISIT , that under some additional conditions, the zero-one laws for @math -connectivity would resemble those for the minimum node degree being no less than @math . In our paper, we prove that this conjecture is correct and provide the extra conditions needed for it to hold.
@cite_24
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We consider wireless sensor networks under a heterogeneous random key predistribution scheme and an on-off channel model. The heterogeneous key predistribution scheme has recently been introduced by Yagan - as an extension to the Eschenauer and Gligor scheme - for the cases when the network consists of sensor nodes with varying level of resources and or connectivity requirements, e.g., regular nodes vs. cluster heads. The network is modeled by the intersection of the inhomogeneous random key graph (induced by the heterogeneous scheme) with an Erdős-Renyi graph (induced by the on off channel model). We present conditions (in the form of zero-one laws) on how to scale the parameters of the intersection model so that with high probability all of its nodes are connected to at least k other nodes; i.e., the minimum node degree of the graph is no less than k. We also present numerical results to support our results in the finite-node regime. The numerical results suggest that the conditions that ensure k-connectivity coincide with those ensuring the minimum node degree being no less than k.