On Connectivity Thresholds in the Intersection of Random Key Graphs on Random Geometric Graphs

TL;DR

This paper analyzes the connectivity thresholds of a combined model of random key graphs and random geometric graphs, providing tight bounds for parameters ensuring almost sure connectivity as the network grows.

Contribution

It introduces a model combining random key graphs with geometric constraints and derives tight bounds for connectivity thresholds in this hybrid setting.

Findings

Derived tight bounds on parameters for asymptotic almost sure connectivity.

Established conditions linking key pool size, key ring size, and transmission radius.

Extended understanding of connectivity in secure wireless sensor networks.

Abstract

In a random key graph (RKG) of $n$ nodes each node is randomly assigned a key ring of $K_n$ cryptographic keys from a pool of $P_n$ keys. Two nodes can communicate directly if they have at least one common key in their key rings. We assume that the $n$ nodes are distributed uniformly in $[0,1]^2.$ In addition to the common key requirement, we require two nodes to also be within $r_n$ of each other to be able to have a direct edge. Thus we have a random graph in which the RKG is superposed on the familiar random geometric graph (RGG). For such a random graph, we obtain tight bounds on the relation between $K_n,$ $P_n$ and $r_n$ for the graph to be asymptotically almost surely connected.