$k$-connectivity of inhomogeneous random key graphs with unreliable links

TL;DR

This paper analyzes the conditions under which inhomogeneous random key graphs with unreliable links in wireless sensor networks achieve secure and reliable $k$-connectivity, providing zero-one laws and numerical validation.

Contribution

It introduces a unified model combining inhomogeneous random key graphs with unreliable links and derives zero-one laws for $k$-connectivity in this setting.

Findings

Zero-one laws for $k$-connectivity in the intersection graph

Conditions for minimum node degree of at least $k$

Numerical results supporting theoretical conditions

Abstract

We consider secure and reliable connectivity in wireless sensor networks that utilize a heterogeneous random key predistribution scheme. We model the unreliability of wireless links by an on-off channel model that induces an Erd\H{o}s-R\'enyi graph, while the heterogeneous scheme induces an inhomogeneous random key graph. The overall network can thus be modeled by the intersection of both graphs. We present conditions (in the form of zero-one laws) on how to scale the parameters of the intersection model so that with high probability i) all of its nodes are connected to at least $k$ other nodes; i.e., the minimum node degree of the graph is no less than $k$ and ii) the graph is $k$-connected, i.e., the graph remains connected even if any $k-1$ nodes leave the network. We also present numerical results to support these conditions in the finite-node regime. Our results are shown to complement and generalize several previous work in the literature.