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

TL;DR

This paper analyzes the phase transition behavior of q-composite random key graphs, which model secure sensor networks, focusing on properties like connectivity and robustness crucial for networked control applications.

Contribution

It provides a detailed probabilistic analysis of key graph properties and their sharp transitions, offering design guidelines for secure sensor networks in control applications.

Findings

Sharp transition in property probabilities as K_n increases

Probabilities of k-connectivity, robustness, Hamilton cycle, and perfect matching analyzed

Guidelines for designing secure sensor networks for control applications

Abstract

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.