  Random key graphs have received considerable attention and been used in
various applications including secure sensor networks, social networks, the
study of epidemics, cryptanalysis, and recommender systems. In this paper, we
investigate a $q$-composite random key graph, whose construction on $n$ nodes
is as follows: each node independently selects a set of $K_n$ different keys
uniformly at random from the same pool of $P_n$ distinct keys, and two nodes
establish an undirected edge in between if and only if they share at least $q$
key(s). Such graph denoted by $G_q(n,K_n,P_n)$ models a secure sensor network
employing the well-known $q$-composite key predistribution. For
$G_q(n,K_n,P_n)$, we analyze the probabilities of $G_q(n,K_n,P_n)$ having
$k$-connectivity, $k$-robustness, a Hamilton cycle and a perfect matching,
respectively. Our studies of these four properties are motivated by a detailed
discussion of their applications to networked control. Our results reveal that
$G_q(n,K_n,P_n)$ exhibits a sharp transition for each property: as $K_n$
increases, the probability that $G_q(n,K_n,P_n)$ has the property sharply
increases from $0$ to $1$. These results provide fundamental guidelines to
design secure sensor networks for different control-related applications:
distributed in-network parameter estimation, fault-tolerant consensus, and
resilient data backup.
