On asymptotically exact probability of $k$-connectivity in random key graphs intersecting Erdős-Rényi graphs

Random key graphs have originally been introduced in the context of a random key predistribution scheme for securing wireless sensor networks (WSNs). Since then, they have appeared in applications spanning recommender systems, social networks, clustering and classification analysis, and cryptanalysis of hash functions. Random key graphs, denoted G(n;K,P ), form a class of random intersection graphs and can be described as follows: With Vn = {v1, . . . , vn} denoting the set of vertices, each vertex vi is assigned a set Si of K distinct keys that are selected uniformly at random from a key pool of size P . An undirected edge is then drawn between any pair of distinct vertices vi and vj if Si ∩Sj 6= ∅. In this paper, we consider random key graphs with unreliable (i.e., randomly deleted) edges. Namely, let H(n; p) denote an Erdős–Rényi graph on vertices Vn = {v1, . . . , vn}, where an edge exists between any distinct pair of vertices vi and vj with probability p, independently from all other edges. The intersection of a random key graph and an Erdős–Rényi graph, denoted Gon(n;K,P, p) = G(n;K,P )∩H(n; p), corresponds to a random key graph with unreliable (Bernoulli) links, and can be a useful model in various real-world applications; e.g., with secure WSN application in mind, link unreliability can be attributed to harsh environmental conditions severely impairing transmissions. With parameters K,P , and p scaling with the number of vertices n, we derive asymptotically exact probabilities for three related graph properties in Gon(n;Kn, Pn, pn): i) kvertex-connectivity, ii) k-edge-connectivity, and iii) the minimum vertex degree being at least k, where a graph is k-vertexconnected (resp. k-edge-connected) if it remains connected despite the deletion of any (k − 1) vertices (resp. edges). Our results extend the literature on random key graphs in several directions, in particular providing the first analysis on the asymptotically exact probability of the connectivity of Gon(n;Kn, Pn, pn).