# Towards $k$-connectivity in Heterogeneous Sensor Networks under Pairwise Key Predistribution

## TL;DR

This paper analyzes the conditions under which heterogeneous sensor networks using pairwise key predistribution achieve high probability of minimum degree at least k, advancing understanding of their connectivity and robustness.

## Contribution

It establishes a zero-one law for the minimum node degree in inhomogeneous random K-out graphs, aiding the design of secure, reliable sensor networks.

## Key findings

- Critical conditions for network parameters ensuring minimum degree k with high probability.
- Zero-one law for the minimum node degree in inhomogeneous random K-out graphs.
- Numerical results demonstrating practical parameter selection for finite networks.

## Abstract

We study the secure and reliable connectivity of wireless sensor networks under the heterogeneous pairwise key predistribution scheme. This scheme was recently introduced as an extension of the random pairwise key predistribution scheme of Chan et al. to accommodate networks where the constituent sensors have different capabilities or requirements for security and connectivity. For simplicity, we consider a heterogeneous network where each of the $n$ sensors is classified as type-1 (respectively, type-2) with probability $\mu$ (respectively, $1-\mu)$ where $0<\mu<1$. Each type-1 (respectively, type-2) node selects 1 (respectively, $K_n$) other nodes uniformly at random to be paired with; according to the pairwise scheme each pair is then assigned a unique pairwise key so that they can securely communicate with each other. We establish critical conditions on $n, \mu$, and $K_n$ such that the resulting network has minimum node degree of at least $k$ with high probability in the limit of large network size. Our result constitutes a zero-one law for the minimum node degree of the recently introduced inhomogeneous random K-out graph model. This constitutes a crucial step towards establishing a similar zero-one law for the $k$-connectivity of the graph; i.e., for the property that the network remains connected despite the failure of any $k-1$ nodes or links. We present numerical results that indicate the usefulness of our results in selecting the parameters of the scheme in practical settings with finite number of sensors.