Zero-one laws for connectivity in inhomogeneous random key graphs

TL;DR

This paper introduces a new inhomogeneous random key graph model for heterogeneous wireless sensor networks, establishing conditions under which the network is almost surely connected or has no isolated nodes as the network size grows.

Contribution

It provides new zero-one laws for connectivity and absence of isolated nodes in inhomogeneous random key graphs, improving upon previous models.

Findings

Identified critical scalings for P and K_i parameters

Proved high probability of connectivity under certain conditions

Enhanced previous results on inhomogeneous random intersection graphs

Abstract

We introduce a new random key predistribution scheme for securing heterogeneous wireless sensor networks. Each of the n sensors in the network is classified into r classes according to some probability distribution {\mu} = {{\mu}_1 , . . . , {\mu}_r }. Before deployment, a class-i sensor is assigned K_i cryptographic keys that are selected uniformly at random from a common pool of P keys. Once deployed, a pair of sensors can communicate securely if and only if they have a key in common. We model the communication topology of this network by a newly defined inhomogeneous random key graph. We establish scaling conditions on the parameters P and {K_1 , . . . , K_r } so that this graph i) has no isolated nodes; and ii) is connected, both with high probability. The results are given in the form of zero-one laws with the number of sensors n growing unboundedly large; critical scalings are identified and shown to coincide for both graph properties. Our results are shown to complement and improve those given by Godehardt et al. and Zhao et al. for the same model, therein referred to as the general random intersection graph.