k-connectivity of Random Graphs and Random Geometric Graphs in Node Fault Model

TL;DR

This paper analyzes the k-connectivity and network breakdown probability in random and geometric graphs under node failures, revealing phase transitions relevant to wireless sensor network reliability.

Contribution

It provides universal asymptotic bounds on network breakdown probability for various random graph models, including Erdős-Rényi, random intersection, and geometric graphs.

Findings

Universal bounds for network breakdown probability

Existence of phase transitions in network connectivity

Applicability to multiple random graph models

Abstract

k-connectivity of random graphs is a fundamental property indicating reliability of multi-hop wireless sensor networks (WSN). WSNs comprising of sensor nodes with limited power resources are modeled by random graphs with unreliable nodes, which is known as the node fault model. In this paper, we investigate k-connectivity of random graphs in the node fault model by evaluating the network breakdown probability, i.e., the disconnectivity probability of random graphs after stochastic node removals. Using the notion of a strongly typical set, we obtain universal asymptotic upper and lower bounds of the network breakdown probability. The bounds are applicable both to random graphs and to random geometric graphs. We then consider three representative random graph ensembles: the Erdos-Renyi random graph as the simplest case, the random intersection graph for WSNs with random key predistribution schemes, and the random geometric graph as a model of WSNs generated by random sensor node deployment. The bounds unveil the existence of the phase transition of the network breakdown probability for those ensembles.