Transitional Behavior of q-Composite Random Key Graphs with Applications to Networked Control

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 Kn different keys uniformly at random from the same pool of Pn 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 Gq(n,Kn, Pn) models a secure sensor network employing the well-known q-composite key predistribution. For Gq(n,Kn, Pn), we analyze the probabilities of Gq(n,Kn, Pn) 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 Gq(n,Kn, Pn) exhibits a sharp transition for each property: as Kn increases, the probability thatGq(n,Kn, Pn) 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, faulttolerant consensus, and resilient data backup.