k-Connectivity in Random Key Graphs with Unreliable Links

Random key graphs form a class of random intersection graphs and 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 interest 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 the edges are captured by independent Bernoulli random variables, rendering edges of the graph to be on or off independently from each other. The resulting model is an intersection of a random key graph and an Erdos-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 i) the minimum node degree in the graph is at least k, and ii) the graph is k-connected, both with high probability as the number of nodes becomes large. The results are given in the form of zeroone laws with critical thresholds identified and shown to coincide for both graph properties. These findings improve the previous results by Rybarczyk on the k-connectivity of random key graphs (with reliable links), as well as the zero-one laws by Yagan on the 1-connectivity of random key graphs with unreliable links.Comment: Published in IEEE Transactions on Information Theor

Resilient Wireless Sensor Networks Using Topology Control: A Review

Wireless sensor networks (WSNs) may be deployed in failure-prone environments, and WSNs nodes easily fail due to unreliable wireless connections, malicious attacks and resource-constrained features. Nevertheless, if WSNs can tolerate at most losing k − 1 nodes while the rest of nodes remain connected, the network is called k − connected. k is one of the most important indicators for WSNs’ self-healing capability. Following a WSN design flow, this paper surveys resilience issues from the topology control and multi-path routing point of view. This paper provides a discussion on transmission and failure models, which have an important impact on research results. Afterwards, this paper reviews theoretical results and representative topology control approaches to guarantee WSNs to be k − connected at three different network deployment stages: pre-deployment, post-deployment and re-deployment. Multi-path routing protocols are discussed, and many NP-complete or NP-hard problems regarding topology control are identified. The challenging open issues are discussed at the end. This paper can serve as a guideline to design resilient WSNs