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 results also extend the work by @cite_7 on the homogeneous random key graph intersecting ER graph to the heterogeneous setting. There, a zero-one law for the property that the graph is @math -connected was established for @math . Considering Theorem and setting @math , i.e., when all nodes belong to the same class and thus receive the same number @math of keys, our results recover Theorem 2 of (See [Theorems 2] Jun K-Connectivity ).
@cite_7
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Random key graphs form a class of random intersection graphs that are naturally induced by the random key predistribution scheme of Eschenauer and Gligor for securing wireless sensor network (WSN) communications. Random key graphs have received much attention recently, owing in part to their wide applicability in various domains, including recommender systems, social networks, secure sensor networks, clustering and classification analysis, and cryptanalysis to name a few. In this paper, we study connectivity properties of random key graphs in the presence of unreliable links. Unreliability of graph links is captured by independent Bernoulli random variables, rendering them to be on or off independently from each other. The resulting model is an intersection of a random key graph and an Erdős–Renyi graph, and is expected to be useful in capturing various real-world networks; e.g., with secure WSN applications in mind, link unreliability can be attributed to harsh environmental conditions severely impairing transmissions. We present conditions on how to scale this model’s parameters so that: 1) the minimum node degree in the graph is at least @math and 2) the graph is @math -connected, both with high probability as the number of nodes becomes large. The results are given in the form of zero-one laws with critical thresholds identified and shown to coincide for both graph properties. These findings improve the previous results by Rybarczyk on @math -connectivity of random key graphs (with reliable links), as well as the zero-one laws by Yagan on one-connectivity of random key graphs with unreliable links.