k-Connectivity of Random Key Graphs

TL;DR

This paper provides an exact analysis of the conditions for k-connectivity in random key graphs, which model secure wireless sensor networks, especially in the challenging regime where each node has at least two keys and the number of keys per node grows slowly.

Contribution

It addresses the open problem of k-connectivity in random key graphs for the case where each node has at least two keys and the number of keys grows slower than the square root of logarithm of the number of nodes.

Findings

Derived exact conditions for k-connectivity in the specified regime.

Extended understanding of network robustness in secure sensor networks.

Provided theoretical foundation for future network design and analysis.

Abstract

Random key graphs represent topologies of secure wireless sensor networks that apply the seminal Eschenauer-Gligor random key predistribution scheme to secure communication between sensors. These graphs have received much attention and also been used in diverse application areas beyond secure sensor networks; e.g., cryptanalysis, social networks, and recommender systems. Formally, a random key graph with $n$ nodes is constructed by assigning each node $X_n$ keys selected uniformly at random from a pool of $Y_n$ keys and then putting an undirected edge between any two nodes sharing at least one key. Considerable progress has been made in the literature to analyze connectivity and $k$-connectivity of random key graphs, where $k$-connectivity of a graph ensures connectivity even after the removal of $k$ nodes or $k$ edges. Yet, it still remains an open question for $k$-connectivity in random key graphs under $X_n \geq 2$ and $X_n = o(\sqrt{\ln n})$ (the case of $X_n=1$ is trivial). In this paper, we answer the above problem by providing an exact analysis of $k$-connectivity in random key graphs under $X_n \geq 2$.