<inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-Connectivity in Random Key Graphs With Unreliable Links

Abstract

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-Rényi 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 k and 2) the graph is k-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 k-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.