Minimum node degree in inhomogeneous random key graphs with unreliable links

TL;DR

This paper analyzes the minimum node degree in inhomogeneous random key graphs with unreliable links, providing conditions for ensuring network connectivity properties in heterogeneous wireless sensor networks.

Contribution

It introduces zero-one laws for the minimum node degree in the intersection of inhomogeneous random key graphs and Erdős-Rényi graphs, extending previous models to heterogeneous settings.

Findings

Conditions for minimum node degree at least k with high probability.

Numerical results support the theoretical zero-one laws.

Conditions for k-connectivity align with those for minimum degree.

Abstract

We consider wireless sensor networks under a heterogeneous random key predistribution scheme and an on-off channel model. The heterogeneous key predistribution scheme has recently been introduced by Ya\u{g}an - as an extension to the Eschenauer and Gligor scheme - for the cases when the network consists of sensor nodes with varying level of resources and/or connectivity requirements, e.g., regular nodes vs. cluster heads. The network is modeled by the intersection of the inhomogeneous random key graph (induced by the heterogeneous scheme) with an Erd\H{o}s-R\'enyi graph (induced by the on/off channel model). 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 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 $k$. We also present numerical results to support our results in the finite-node regime. The numerical results suggest that the conditions that ensure $k$-connectivity coincide with those ensuring the minimum node degree being no less than $k$.